Mathematical Methods Handbook

The definitions of a convergent sequence, a Cauchy sequence and a complete space, together with the three facts about real sequences used everywhere later: the Cauchy criterion, monotone convergence, and the existence of limsup and liminf.

Partial sums, the vanishing-terms and Cauchy criteria, and the comparison, ratio, root, integral and alternating-series tests, each with the hypotheses it needs and the cases in which it decides nothing.

A series whose absolute values converge is convergent and its sum is unchanged by any rearrangement; a convergent series of real terms that is not absolutely convergent can be rearranged to any sum whatever, and the Cauchy product of two absolutely convergent series converges to the product of their sums.

Uniform convergence is convergence at a rate independent of the point; it is detected by the Cauchy criterion or the Weierstrass M-test, and it is the hypothesis under which a limit of continuous functions is continuous.

Uniform convergence on a finite interval lets a limit or a sum be taken through an integral, while term-by-term differentiation needs the differentiated series to converge uniformly; both fail without those hypotheses.

A power series has a radius of convergence given by Hadamard's formula, inside which it converges absolutely, uniformly on closed subdiscs, and may be differentiated and integrated term by term; at a boundary point where it converges, its sum is the limit of its values along the radius.

The Taylor polynomial of order N approximates a function with an error given exactly by the Lagrange or the integral remainder; the Taylor series represents the function only where that remainder tends to zero, and for some smooth functions it never does.

Defines partial derivatives and the total derivative of a map between Euclidean spaces, identifies the total derivative with the Jacobian matrix, and gives the chain rule as a matrix product together with the $C^2$ condition under which mixed partials commute.

Gives the second-order Taylor expansion of a $C^2$ function of $n$ variables in Peano and mean-value form, and classifies a critical point as a strict local minimum, a strict local maximum or a saddle according as the Hessian there is positive definite, negative definite or indefinite, with no conclusion when it is singular and semidefinite.

A map of class $C^k$ between open subsets of $\mathbb{R}^n$ whose Jacobian determinant is non-zero at a point is a $C^k$ diffeomorphism of some neighbourhood of that point onto an open set, and an equation $\Phi(x,y) = c$ whose partial Jacobian in $y$ is invertible at a solution determines $y$ locally as a $C^k$ function $g$ of $x$, with $Dg(x) = -(\partial\Phi/\partial y)^{-1}(\partial\Phi/\partial x)$ evaluated at $(x, g(x))$.

If a $C^1$ function has a local extremum, among the points of the common zero set of $m < n$ constraints, at a point where the $m$ constraint gradients are linearly independent, then its gradient there is a unique linear combination of them; this gives the standard recipe for constrained extrema together with the points at which the recipe can miss them.

States when a multiple integral may be computed as an iterated integral and gives the change-of-variables formula for a $C^1$ diffeomorphism, with the absolute Jacobian determinant as the local volume factor, specialised to plane polar, cylindrical and spherical coordinates.

If $f$ and all its first partial derivatives in the parameters $x \in U \subseteq \mathbb{R}^n$ are continuous on $U \times [a,b]$, then $F(x) = \int_a^b f(x,t)\,dt$ is of class $C^1$ with $\partial F/\partial x_i = \int_a^b \partial f/\partial x_i\,dt$; with differentiable limits of integration the derivative acquires the two boundary terms of the Leibniz rule.

Collects the one-dimensional Gaussian integral and its moments, the effect of completing the square in the exponent, and the $n$-dimensional formula $\int \exp(-\tfrac12 x^{\mathsf T}Ax + b^{\mathsf T}x)\,d^n x = (2\pi)^{n/2}(\det A)^{-1/2}\exp(\tfrac12 b^{\mathsf T}A^{-1}b)$ for symmetric positive definite $A$, with the recipe for reducing an exponent to that form.

Fixes the set and map notation used throughout the handbook, including which of the image and inverse-image identities hold without hypotheses, and gives the comparison of cardinalities together with Cantor's theorem and the standard list of countable and uncountable sets.

States Zorn's lemma on partially ordered sets, proves that it implies the well-ordering principle, the axiom of choice and the Hausdorff maximal principle, and gives the standard recipe for extracting a maximal object from it.

Defines a topology and the derived notions of closed set, neighbourhood, interior, closure and boundary, gives the metric topology and the subspace topology, and states what a basis is and what the Hausdorff condition buys.

Defines continuity by inverse images of open sets, lists the equivalent formulations including the epsilon-delta condition on metric spaces, and defines homeomorphisms and what it means for a property to be topological.

Gives compactness by finite subcovers with its permanence properties, the three equivalent forms it takes in a metric space, the Heine-Borel characterisation of the compact subsets of R^n as exactly the closed bounded ones, Bolzano-Weierstrass, and the extreme value theorem for a continuous real function on a nonempty compact space.

Defines connected and path-connected spaces, their components, homotopy of paths and the fundamental group, and says precisely what "simply connected" means, how the definition used in plane complex analysis differs from the homotopy one, and which standard regions satisfy it.

The axioms of a vector space over the real or complex scalars, and the facts that make coordinates possible: a finitely spanned space has a basis, all its bases have the same size, and coordinates in a fixed ordered basis are unique.

Linear maps, their kernel and image, the matrix of a map in a chosen pair of ordered bases, and the rules by which composition becomes matrix multiplication and a change of basis becomes a similarity transformation.

The determinant as the unique alternating multilinear function of the columns taking the value one at the identity, with its behaviour under transposition, elementary operations, cofactor expansion and products, the non-vanishing criterion for invertibility, and its reading as a signed volume.

Gaussian elimination with partial pivoting as a recipe for the rank and the complete solution set of Ax = b, together with the rank-nullity law and the Fredholm alternative that say in advance how many solutions and how many consistency conditions there are.

Real and complex inner products in the convention used throughout this handbook, the norm they induce, and the Cauchy-Schwarz and triangle inequalities with their exact cases of equality.

The Gram-Schmidt recipe that turns a linearly independent list into an orthonormal one spanning the same nested subspaces, the expansion of a vector in an orthonormal basis, and the QR factorization the recipe produces.

Orthogonal projection onto a finite-dimensional subspace, its characterisation as the nearest point of that subspace, and the normal equations that this turns into for the least-squares problem of minimising the norm of Ax minus b.

Defines eigenvalues, eigenvectors, eigenspaces and the characteristic polynomial, distinguishes algebraic from geometric multiplicity, and gives the diagonalizability criterion together with trace and determinant as the sum and product of the eigenvalues.

A Hermitian matrix has real eigenvalues, orthogonal eigenvectors for distinct eigenvalues, and an orthonormal eigenbasis, so some unitary matrix of eigenvector columns diagonalizes it; a real symmetric matrix can be diagonalized by a real orthogonal matrix.

Defines unitary and orthogonal matrices, shows they are exactly the matrices that preserve the inner product and have orthonormal columns, and records that their eigenvalues have modulus one and that the real ones reduce to rotations and reflections.

A complex matrix has an orthonormal eigenbasis exactly when it is normal, and two normal matrices can be given a common orthonormal eigenbasis exactly when they commute.

Every square matrix whose eigenvalues lie in the ground field is similar to a block diagonal matrix of Jordan blocks, unique up to the order of the blocks; the block sizes are read off the nullities of the powers of $A - \lambda I$, and the form turns powers and power series of a defective matrix into polynomial-times-exponential formulas.

How to reduce a quadratic or Hermitian form to its principal axes, decide its definiteness from the eigenvalue signs or from Sylvester's criterion on the leading principal minors, and bound the eigenvalues by the Rayleigh quotient.

States the spectral theory of the pencil Ax = lambda Bx for Hermitian A and positive definite B, and applies it to the small oscillations of a conservative system, whose normal frequencies and normal modes are the eigenvalues and B-orthonormal eigenvectors of the pencil formed by the potential and kinetic energy matrices.

Every complex m-by-n matrix factors as A = USV* with U, V unitary and S non-negative diagonal; the singular values are the square roots of the min(m,n) largest eigenvalues of A*A, truncating the factorisation gives the best approximation of each rank, and inverting the non-zero part gives the pseudoinverse that solves least squares with minimum norm.

Defines a function of a matrix by its power series, states when the series converges, collects the properties of exp(A) including the exact condition — that A and B commute — for exp(sA+tB) = exp(sA)exp(tB) to hold for all s and t near 0, which is more than the single identity exp(A+B) = exp(A)exp(B) requires, and gives the recipes for computing exp(tA) by diagonalization and for defective matrices.

Defines induced matrix norms and the spectral norm, defines the condition number and expresses it through the singular values, and states how far the solution of a linear system can move when the data are perturbed.

Gives the scalar, vector and triple products of vectors in $\mathbb{R}^3$ both geometrically and in components, introduces the summation convention with the Kronecker delta and the Levi-Civita symbol, and proves the $\varepsilon$-$\delta$ identity that reduces every repeated cross product to dot products.

Defines the operator $\nabla$ and the gradient of a scalar field, identifies $\nabla\phi\cdot\hat{\mathbf{a}}$ as the rate of change of $\phi$ per unit distance in the direction $\hat{\mathbf{a}}$, and shows that at a point where it does not vanish $\nabla\phi$ points along the steepest increase of $\phi$ and normal to the level surface through that point.

Defines $\nabla\cdot\mathbf{F}$, $\nabla\times\mathbf{F}$ and $\nabla^2\phi$ in Cartesian coordinates and proves that for a $C^1$ field the divergence at a point is the limit of outward flux per unit volume over shrinking cubes, while the component of the curl along a unit vector is the limit of circulation per unit area over shrinking squares normal to it.

Collects the six product rules for $\nabla$, $\nabla\cdot$ and $\nabla\times$ together with $\nabla\times\nabla\phi = \mathbf{0}$, $\nabla\cdot(\nabla\times\mathbf{a}) = 0$ and $\nabla\times(\nabla\times\mathbf{a}) = \nabla(\nabla\cdot\mathbf{a}) - \nabla^2\mathbf{a}$, and gives a mechanical index-notation recipe that proves each of them in a few lines.

Defines the line integral of a vector field along a piecewise $C^1$ curve and proves that path independence, the existence of a single-valued potential and exactness of $\mathbf{F}\cdot d\mathbf{r}$ are the same condition, that each forces $\nabla\times\mathbf{F} = \mathbf{0}$, and that the converse holds on a star-shaped (more generally simply connected) domain.

Defines a regular parametrized surface patch, its scalar surface element $dS = \lvert \mathbf{r}_u \times \mathbf{r}_v\rvert\,du\,dv$ and its vector element $d\mathbf{S} = \hat{\mathbf{n}}\,dS$, proves that both are independent of the parametrization, fixes the orientation conventions for closed and open surfaces, and gives the graph and projection formulas used to compute a flux integral.

States Gauss's theorem — the outward flux of a $C^1$ field through the boundary of a bounded region equals the integral of its divergence over the region — proves it for regions bounded by $C^1$ graphs in each coordinate direction, derives the gradient and curl companion identities, and uses it to turn conservation of mass into the continuity equation.

Proves Green's theorem in the plane for a region bounded by graphs in both coordinate directions, then obtains Stokes' theorem — the circulation of a $C^1$ field round the perimeter of an oriented surface equals the flux of its curl through the surface — by pulling the circulation back to the parameter domain, and fixes the orientation convention that makes the two sides agree in sign.

Derives Green's first and second identities from the divergence theorem and uses the first to prove that the Dirichlet problem for Poisson's equation has at most one solution, that the Neumann problem has one up to an additive constant and only when its data satisfy the compatibility condition $\oint_S h\,dS = \int_V f\,dV$, and that the Robin problem with positive coefficient has at most one.

States and proves the two theorems of potential theory as Arfken gives them — a field is determined by its divergence, its curl and its boundary normal component on a simply connected region, and a field whose source and circulation densities vanish at infinity splits into an irrotational part $-\nabla\varphi$ and a solenoidal part $\nabla\times\mathbf{A}$ with the two Poisson integrals as potentials — and describes the gauge freedom in $\varphi$ and $\mathbf{A}$.

Introduces the delta as the limit of a normalized family of peaked functions, proves that the Gaussian family sifts the value of any bounded function continuous at the point, collects the working rules (evenness, rescaling, $\delta(g(x))$, the derivative, $\theta' = \delta$), defines the three-dimensional delta, and proves $\nabla^2(1/r) = -4\pi\delta^{(3)}(\mathbf{r})$ from Green's second identity.

Builds the orthonormal tangent frame and the scale factors of a curvilinear coordinate system, gives the line, area and volume elements they produce, and derives the four general formulas for grad, div, curl and the Laplacian, with a fully worked example in parabolic cylindrical coordinates.

Tabulates grad, div, curl and the Laplacian, together with the line, area and volume elements, in cylindrical and spherical polar coordinates, and derives the Laplacian twice — from the general curvilinear formula and directly by the chain rule — with worked examples in pipe flow and electrostatics.

Polar form and de Moivre's theorem, the exponential as a power series with its addition theorem and period 2 pi i, the trigonometric and hyperbolic functions built from it, and the logarithm, argument and complex powers as its multivalued inverses.

A complex function is differentiable at a point only if its real and imaginary parts satisfy the Cauchy-Riemann equations there, and it is analytic on an open set if those equations hold with continuous first partial derivatives; the two parts are then harmonic conjugates.

How a multivalued function such as a logarithm or a fractional power is made single-valued and analytic: locate the branch points, where continuation around a loop changes the value, and cut the plane so that no such loop is left.

The integral of a continuous complex function along a piecewise smooth path, its independence of the parametrization, the ML bound on its size, and the fact that it depends only on the endpoints exactly when the integrand is the derivative of an analytic function.

The integral of an analytic function around a closed contour in a simply connected region is zero; consequently a contour may be deformed freely within a region of analyticity, and in a multiply connected region the integral reduces to one loop around each hole.

An analytic function and all of its derivatives at an interior point are given by integrals of the function over the surrounding contour; consequently an analytic function is infinitely differentiable, and conversely a continuous function all of whose closed contour integrals vanish is analytic.

Cauchy's estimates bound the derivatives of an analytic function by the maximum of its modulus on a circle; a bounded entire function is therefore constant, every non-constant polynomial has a root, and the modulus of a non-constant analytic function attains its maximum only on the boundary.

An analytic function is the sum of its Taylor series on every open disc about a point that lies inside its domain of analyticity, so the radius of convergence is at least the distance to the boundary; consequently the zeros of a non-constant analytic function are isolated.

A function analytic in an annulus is the sum of a unique two-sided power series there, whose coefficients are the Cauchy integrals over any circle in the annulus; in practice the coefficients are found by algebra on known expansions rather than by integration.

An isolated singularity is removable, a pole of finite order, or essential, according to whether the principal part of the Laurent series is empty, finite or infinite; equivalently according to whether the function stays bounded, tends to infinity, or comes arbitrarily close to every complex value.

Two analytic functions that agree on a set with an accumulation point agree wherever both are defined, so an analytic function has at most one extension to any larger region; extensions are built by chains of overlapping expansions or by functional relations, as for the Gamma function.

The integral of an analytic function over a closed contour equals $2\pi i$ times the sum of its residues at the enclosed singularities, each weighted by the winding number; at a pole of order $n$ the residue is a single derivative of $(z-z_0)^n f(z)$.

An integral of a rational function of sine and cosine over one full period becomes a contour integral on the unit circle under $z = e^{i\theta}$, and is then $2\pi i$ times the sum of the residues, inside the circle, of the transformed integrand $R(\cdot,\cdot)/(iz)$.

A real integral over the whole line is computed by closing the contour with a large semicircle and summing the residues in that half-plane; the semicircle is negligible when $zf(z)\to0$, and, for an integrand carrying a factor $e^{iax}$, already when $f(z)\to0$, which is Jordan's lemma.

For an integrand containing a fractional power or a logarithm, a keyhole contour hugging the cut converts the integral into a residue sum, because the two edges of the cut carry values of the multivalued factor differing by a known constant; a dogbone contour does the same for a cut joining two branch points.

A simple pole on the path of integration is handled by deleting a symmetric interval around it, which defines the Cauchy principal value; indenting the contour by a small semicircle contributes exactly half a residue, so the principal value equals $2\pi i$ times the residues in the half-plane plus $i\pi$ times those on the axis.

Multiplying a function by $\pi\cot\pi z$, whose residue at every integer is $1$, turns an infinite sum over the integers into a contour integral that vanishes; the sum is then minus the residues at the poles of the function, and $\pi\csc\pi z$ does the same for alternating sums.

The integral of $f'/f$ around a contour counts the zeros minus the poles of $f$ inside it, each with multiplicity; if one function dominates another on the contour, the two have the same number of zeros inside, which locates the roots of polynomials without solving them.

The index of a closed curve about a point is the integral of $dz/(z-a)$ divided by $2\pi i$; it is an integer, constant on each complementary region and zero far away, unchanged by continuous deformation, and it is the weight with which each point enters the general form of Cauchy's theorem.

An analytic map with non-vanishing derivative turns and rescales a small figure without distorting it, Mobius maps carry circles and lines to circles and lines, and a harmonic function composed with such a map stays harmonic — which is how an awkward two-dimensional Laplace problem is transported to a geometry one can solve by inspection.

A response that cannot precede its cause has a susceptibility analytic in the upper half-plane; Cauchy's formula on a contour that skirts the real axis then expresses its real part as a principal-value integral of its imaginary part and conversely — the Kramers-Kronig relations, which tie refraction to absorption without any model of the medium.

A function meromorphic in the whole plane can be written as a sum of its singular parts plus an entire remainder, and an entire function as a product over its zeros with exponential convergence factors; carrying both out for the sine gives the expansion of $\pi\cot\pi z$ and the product $\sin\pi z = \pi z\prod(1-z^2/n^2)$.

The domain on which a multivalued function becomes an honest single-valued analytic function, built by stacking cut copies of the plane and gluing them across the cuts at the branch points, with the surfaces of $z^{1/n}$ and $\log z$ worked out and parametrised by a single variable.

Gives the six standard closed-form methods for a first-order first-degree equation — separation, exactness, integrating factors, the linear formula, the substitutions for homogeneous and for Bernoulli equations — each with the hypotheses it needs, the reason it works, and worked examples with checkable numbers.

Proves that an initial-value problem $y' = f(x,y)$, $y(x_0)=y_0$ has exactly one solution on an explicit interval around $x_0$ when $f$ is continuous and Lipschitz in $y$, by showing that Picard's successive approximations converge uniformly, and shows by example what fails when the Lipschitz condition is dropped.

Gronwall's inequality turns the self-referential integral bound satisfied by the gap between two solutions into an explicit exponential, giving the separation estimate $|Y(t)-Z(t)| \le |Y(t_0)-Z(t_0)|e^{K(t-t_0)}$, continuity of the flow in the initial point and in parameters, an error bound for a perturbed vector field, and the fact that a solution on a bounded maximal interval leaves every compact subset of its domain.

For a linear equation of order n whose leading coefficient does not vanish, the solutions of the homogeneous equation form a vector space of dimension exactly n, any n solutions are independent precisely when their Wronskian is nonzero at one point, and Abel's formula gives that Wronskian up to a constant without solving the equation.

One identity, $L(e^{\lambda x}q) = e^{\lambda x}\sum_j p^{(j)}(\lambda)q^{(j)}/j!$, delivers the whole method — a fundamental set $x^{r}e^{\lambda_j x}$ read off the roots of the characteristic polynomial, the real form for a complex pair, and a polynomial-times-exponential particular solution whose degree rises by the multiplicity of the forcing exponent, which is resonance.

Given a fundamental set of the homogeneous equation, replacing its constants by functions and imposing $n-1$ convenient constraints turns the inhomogeneous equation into a linear system whose determinant is the Wronskian; it yields a particular solution as an integral, for any continuous forcing and any order.

One known solution $\varphi_1$ of a linear homogeneous equation, nonvanishing on an interval, lowers the order by one through the substitution $y = u\varphi_1$; for second-order equations this gives the second solution in closed form, $\varphi_2 = \varphi_1\int \varphi_1^{-2}e^{-\int a_1}$, which is Abel's formula integrated once.

An equation in which each term carries $x^{k}$ with the $k$-th derivative is scale invariant, so powers $|x|^{r}$ solve it; the exponents are the roots of an indicial polynomial, a root of multiplicity $m$ contributes $|x|^{r}$ times $1, \log|x|, \dots, (\log|x|)^{m-1}$, and the substitution $x = e^{t}$ turns the equation into one with constant coefficients.

The equation $\ddot x + 2\gamma\dot x + \omega_0^2 x = f(t)$ worked out in full: the three free regimes fixed by the sign of $\gamma - \omega_0$, the decaying transient plus periodic steady state under harmonic driving, the amplitude and phase as functions of frequency, and the quality factor $Q = \omega_0/2\gamma$ read three ways.

At a point where the normalised coefficients are analytic, every solution is analytic: substituting a power series turns the equation into a recurrence for the coefficients, the first $n$ of which are the initial data, and the series converges at least as far as the coefficient series do.

At a regular singular point the substitution $y = x^{\sigma}\sum_n a_nx^n$ works: the lowest power gives a quadratic indicial equation for $\sigma$, the rest a recurrence, and the larger exponent always yields a solution — while equal exponents, or exponents differing by an integer, can block the second one and force a logarithm.

A point is ordinary, regular singular or irregular singular according to how fast $p$ and $q$ blow up there; the point at infinity is classified by substituting $x = 1/t$ and looking at $t = 0$; an equation all of whose singular points in the extended plane are regular is Fuchsian, and with three such points it is the Riemann equation, whose six exponents sum to one.

Gives the solution of the first-order linear system x' = A(t)x in terms of a fundamental matrix, shows that for constant A the fundamental matrix is exp(tA) and the unique solution of the initial-value problem is exp(tA)x_0, and turns this into a recipe that reads the solution off the eigenvalues and eigenvectors of A, including the two real solutions carried by each complex conjugate pair.

Gives the solution of a forced linear system as the free evolution of the initial state plus a convolution of the forcing with the propagator, namely the Duhamel integral x(t) = exp(tA)x_0 plus the integral of exp((t-s)A)g(s) ds for constant A, and its fundamental-matrix form for time-dependent A, together with a recipe, the resonance that appears when the forcing matches an eigenvalue, and the reason the same formula reappears for the heat and wave equations.

For a linear system whose coefficient matrix has period T, the fundamental matrix normalised at the origin factors as a T-periodic matrix times the exponential of a constant one, so the system is a periodic change of variable away from a constant-coefficient system; the eigenvalues of the monodromy matrix — the Floquet multipliers — do not depend on any choice made and decide decay, boundedness and the existence of periodic solutions, which for Hill's equation reduces to the single test |tr M| < 2.

Turns a constant-coefficient initial-value problem into one algebraic equation for the transform of the unknown, the subsidiary equation p(s)Y(s) = Q(s) + F(s), whose numerator carries the initial data, and shows how partial fractions, the shifting theorem and the convolution rule recover the solution, including for forcing that is switched on at a later time or delivered as an impulse.

The formal adjoint of a second-order differential operator, Lagrange's identity and Green's formula with the boundary term written out, the criterion for formal self-adjointness, and the integrating factor that puts any second-order equation into self-adjoint (Sturm-Liouville) form.

For a regular Sturm-Liouville problem the eigenvalues are real and simple and form an increasing sequence tending to infinity, and eigenfunctions belonging to different eigenvalues are orthogonal with respect to the weight in the equation.

The eigenfunctions of a regular Sturm-Liouville problem form a complete orthogonal family, so every square-integrable function has a mean-square convergent expansion in them with coefficients given by weighted inner products, and the expansion turns a forced boundary-value problem into a term-by-term division by $\lambda_n-\sigma$.

The zeros of two independent solutions of a second-order linear equation interlace, a solution of the equation with the larger coefficient must vanish between consecutive zeros of a solution of the equation with the smaller one, and these two facts bound the spacing of zeros, decide whether solutions oscillate, and count the zeros of Sturm-Liouville eigenfunctions.

Defines $\Gamma$ by Euler's integral on the right half-plane, collects the functional equation $\Gamma(z+1) = z\Gamma(z)$ with the factorial and half-integer values, Gauss's limit formula, and the reflection and duplication formulas, and shows how to reduce an integral or a half-integral argument to a tabulated value.

The Beta integral $B(p,q) = \int_0^1 t^{p-1}(1-t)^{q-1}dt$ equals $\Gamma(p)\Gamma(q)/\Gamma(p+q)$, proved by writing the product of two Gamma integrals in polar coordinates; the trigonometric and $(1+u)^{-p-q}$ forms that follow give closed values for powers of sine and cosine and for a large family of algebraic integrals.

Defines $\psi = \Gamma'/\Gamma$ and its derivatives, derives the series $\psi(z) = -\gamma - 1/z + \sum_{k\ge1}(1/k - 1/(z+k))$ and $\psi^{(m)}(z) = (-1)^{m+1}m!\sum_{k\ge0}(z+k)^{-m-1}$ from the Weierstrass product, collects the values at integers and half-integers, and turns a convergent series of rational terms into a finite combination of digamma values.

Defines $\zeta(s)=\sum_{n\ge1}n^{-s}$ for real $s>1$ and the Bernoulli numbers $B_n$ by the generating function $t/(e^t-1)$, and proves Euler's identity $\zeta(2k)=(-1)^{k+1}(2\pi)^{2k}B_{2k}/\bigl(2\,(2k)!\bigr)$, which turns a rational recursion into the closed values $\pi^2/6$, $\pi^4/90$, $\pi^6/945$ and gives every $\zeta$ at an even integer.

Defines the incomplete Gamma functions $\gamma(a,x)$ and $\Gamma(a,x)$ and the error functions $\operatorname{erf}$ and $\operatorname{erfc}$ as the case $a=\tfrac12$, then gives both of the expansions a computation needs — a convergent power series for small argument and an asymptotic series with a rigorous remainder bound for large argument.

For any positive weight with finite moments the monomials can be orthogonalised into a unique family of polynomials of degrees 0, 1, 2, ...; the family obeys a three-term recurrence, each member has exactly its own degree many simple zeros inside the interval, and the classical families all come from one Rodrigues formula.

The polynomial solutions of Legendre's equation, with the four descriptions that are used interchangeably in practice - the differential equation, Rodrigues' formula, the generating function and the three-term recurrence - together with their orthogonality on [-1, 1] and the normalisation 2/(2l+1).

The regular solutions of the associated Legendre equation, built from the Legendre polynomials by m-fold differentiation and a factor (1-x^2)^{m/2}, with their two orthogonality relations, their behaviour under m -> -m, and an explicit account of the Condon-Shortley sign convention that divides the textbooks.

The orthonormal basis of L^2 of the sphere made from the associated Legendre functions and e^{i m phi}: its definition and conventions, the Laplace series that expands any square-integrable function on the sphere, and the addition theorem that turns a Legendre polynomial of the angle between two directions into a sum of 2l+1 products.

The polynomials generated by $\mathrm{e}^{-t^{2}+2tx}$: their Rodrigues formula, three-term recurrence, Hermite equation and orthogonality with weight $\mathrm{e}^{-x^{2}}$ on the whole line, the Hermite functions they build, and the two incompatible conventions (physicists' $H_n$, probabilists' $He_n$).

The polynomials orthogonal on the half-line with weight $x^{k}\mathrm{e}^{-x}$: their Rodrigues formula and generating function, the associated Laguerre equation they solve, the two conventions in circulation, and the hydrogen radial functions they build.

Disguised cosines: $T_n(\cos\theta)=\cos n\theta$ gives polynomials with a three-term recurrence, orthogonality for the weight $(1-x^{2})^{-1/2}$, and the minimax property that makes $2^{1-n}T_n$ the monic polynomial of degree $n$ that stays closest to zero on $[-1,1]$.

The second solution Q_n of Legendre's equation for integer degree: its series definition and normalisation, the closed form P_n(x) artanh(x) minus a polynomial, the logarithmic blow-up at x = ±1 with the harmonic number H_n as its constant, and the Wronskian (1-x^2)^{-1} with P_n.

Bessel's equation has the Frobenius solution $J_\nu(x) = \sum_m (-1)^m (x/2)^{\nu+2m}/[m!\,\Gamma(\nu+m+1)]$, convergent for all $x$ and regular at the origin when $\nu \ge 0$; for integer order the whole family is packed into the generating function $e^{(x/2)(t-1/t)} = \sum_n J_n(x)t^n$, and four recurrence relations connect neighbouring orders and derivatives.

The Neumann function $Y_\nu = (\cos\nu\pi\,J_\nu - J_{-\nu})/\sin\nu\pi$, defined at integer order by its limit, is a second solution of Bessel's equation, unbounded at the origin — like $x^{-\nu}$ for $\nu > 0$, like $\ln x$ at $\nu = 0$; the Wronskian $J_\nu Y_\nu' - J_\nu' Y_\nu = 2/\pi x$ is independent of $\nu$ and never vanishes, and the combinations $H^{(1,2)}_\nu = J_\nu \pm iY_\nu$ are the cylindrical analogues of $e^{\pm ix}$, carrying energy outward and inward.

For fixed order $\nu > -1$ the functions $J_\nu(\alpha_{\nu j}r/a)$, with $\alpha_{\nu j}$ the positive zeros of $J_\nu$, are orthogonal on $[0,a]$ with weight $r$ and have squared norm $\tfrac12a^{2}J_{\nu+1}(\alpha_{\nu j})^{2}$, so any reasonable function on the disc expands in a Fourier-Bessel series with explicitly computable coefficients.

Rotating the argument of Bessel's equation through a right angle turns the oscillating solutions into a growing one and a decaying one: $I_\nu(x) = i^{-\nu}J_\nu(ix)$ is finite at the origin and grows like $e^{x}/\sqrt{2\pi x}$, $K_\nu(x) = \tfrac{\pi}{2}i^{\nu+1}H^{(1)}_\nu(ix)$ is singular at the origin and decays like $\sqrt{\pi/2x}\,e^{-x}$, and their Wronskian is exactly $-1/x$.

The radial part of the Helmholtz equation in spherical coordinates is Bessel's equation of half-integer order; normalised as $j_n(x) = \sqrt{\pi/2x}\,J_{n+1/2}(x)$ the solutions are elementary — $j_0 = \sin x/x$ and all the rest obtained from it by the Rayleigh formula $f_n = (-1)^nx^n(x^{-1}d/dx)^nf_0$ — and they expand a plane wave as $e^{ikr\cos\theta} = \sum_l(2l+1)i^lj_l(kr)P_l(\cos\theta)$.

At small argument every Bessel function is a power or a logarithm fixed by its leading series term; at large argument all of them are $x^{-1/2}$ times an exponential, with the whole family generated by two divergent-but-asymptotic series $P_\nu$ and $Q_\nu$ in $1/x$ — so that $J_\nu(x) \sim \sqrt{2/\pi x}\cos(x - \nu\pi/2 - \pi/4)$ and $K_\nu(x)\sim\sqrt{\pi/2x}\,e^{-x}$.

The loop integral $J_\nu(z) = (2\pi i)^{-1}\int_C e^{(z/2)(t-1/t)}t^{-\nu-1}dt$ over a contour wrapped round the cut on the negative axis reproduces $J_\nu$ at every order when $\operatorname{Re}z>0$, collapses to Bessel's integral $\tfrac1\pi\int_0^\pi\cos(n\theta - z\sin\theta)\,d\theta$ when $\nu$ is an integer, and splits along two halves of its contour into the Hankel functions.

The Gauss series ${}_2F_1(a,b;c;x)$ and the three-singular-point equation it solves, with Euler's integral representation, Gauss's closed form at $x=1$, and the reduction of elementary and classical-polynomial functions to it.

Kummer's equation, its entire solution $M(a,c,x)$ and the second solution $U(a,c,x)$, obtained by merging two singular points of the hypergeometric equation, together with the reduction of Hermite, Laguerre, Bessel, error and incomplete-Gamma functions to them.

The incomplete and complete elliptic integrals of the first and second kinds, their binomial series and hypergeometric forms, and the exact period of a pendulum of arbitrary amplitude.

Airy's equation $y''=zy$, its two standard solutions $\mathrm{Ai}$ and $\mathrm{Bi}$ as series, contour integrals and Bessel functions of order $1/3$, their opposite behaviour on the two sides of the origin, and the role of $\mathrm{Ai}$ at a WKB turning point.

Defines the order symbols and Poincare's notion of an asymptotic expansion, shows that such an expansion determines its coefficients but not its function, says which operations on it are legitimate, and gives the optimal-truncation rule that makes a divergent series into a numerical method.

Turns a sum over integers into an integral plus a short, explicitly bounded series of boundary derivative terms built from the Bernoulli numbers, so that a slowly convergent series can be evaluated to eight figures from a handful of terms.

Shows that the large-parameter behaviour of a Laplace-type integral is fixed entirely by the behaviour of its integrand at the endpoint, so that expanding the integrand near the origin and replacing each power by a Gamma function produces, term for term, a valid asymptotic expansion.

Gives the leading large-parameter behaviour of an integral of the form $\int g\,e^{xf}$ whose exponent has a single non-degenerate interior maximum, by showing that the integral is controlled by a shrinking neighbourhood of that maximum on which the exponent is a Gaussian.

Gives the large-parameter behaviour of an oscillatory integral $\int g\,e^{i\lambda f}$: each interior point where the phase is stationary and its second derivative is non-zero contributes a term of size $\lambda^{-1/2}$ carrying a fixed $\pm\pi/4$ twist, each endpoint contributes a term of size $\lambda^{-1}$, and everywhere else the oscillation cancels to all orders.

Evaluates $\int_C g(z)e^{\lambda f(z)}dz$ for large $\lambda$ by deforming the contour onto a path through a saddle point of $f$ on which the imaginary part of $f$ is frozen, so that the oscillation disappears and the integral becomes a real Gaussian; gives the leading-order formula $g(z_0)e^{\lambda f(z_0)}e^{i\theta}\sqrt{2\pi/(\lambda|f''(z_0)|)}$ and the rule that fixes the direction $\theta$.

Derives $\Gamma(x+1)\sim\sqrt{2\pi x}\,x^{x}e^{-x}$ from the Euler integral by Laplace's method, carries the expansion of the exponent far enough to produce the first correction $1+1/(12x)$, records the full Stirling series in Bernoulli numbers together with its range of validity and its divergence, and tests the result on a binomial coefficient.

Expands the unknown of a problem carrying a small parameter $\epsilon$ in powers of $\epsilon$ and solves the resulting hierarchy order by order: the unit gives the non-degeneracy condition that makes every coefficient computable, the recipe, and the secular terms that destroy the approximation after a time of order $1/\epsilon$.

Treats problems in which setting the small parameter to zero lowers the order of the equation, so that a boundary condition can no longer be imposed: the unit proves that the solution of a linear model problem splits into a smooth outer part plus a correction confined to a layer of width $\epsilon$, gives the recipe of inner and outer expansions and matching, and shows how dominant balance fixes the width of the layer.

Builds two approximate solutions of the equation $y'' + q(x)y = 0$ when $q$ varies slowly, by letting the phase accumulate as $\int\sqrt{q}$ and the amplitude follow as $q^{-1/4}$; gives the exact residual that measures the error, the precise validity criterion, the connection formulas that carry a solution through a turning point, and the quantization condition they produce.

Defines the Fourier coefficients and Fourier series of an integrable function on an interval of any length, in both the complex-exponential and the real sine-cosine form, shows that the trigonometric system is orthogonal and that the coefficients of a uniformly convergent trigonometric series are forced to be these integrals, and sets up the even and odd extensions that turn a function given on half an interval into a cosine or sine half-range series.

Writes the symmetric partial sum as an average of the function against the Dirichlet kernel and proves that it converges at any point where the one-sided limits and the one-sided derivatives of the function exist, the limit being the average of the two one-sided values, so that a piecewise-smooth function is recovered everywhere except that each jump is split down the middle.

Computes exactly what a truncated Fourier series does next to a jump: the partial sum of the square wave reaches 2Si(pi)/pi = 1.17898 times the half-jump at a point just under half a wavelength of the highest retained harmonic away from the discontinuity, so it overshoots each one-sided value by 8.95 percent of the jump, and the spike narrows without shrinking as more terms are taken.

Averaging the Fourier partial sums replaces the sign-changing Dirichlet kernel by the non-negative Fejér kernel, and the averages converge to the function at every point of continuity, uniformly when the function is continuous; this makes the trigonometric polynomials dense in the continuous periodic functions.

The partial sums of a Fourier series are the best mean-square approximations by trigonometric polynomials, they always converge to the function in mean square, and the energy of the function equals the sum of the squared moduli of its coefficients, from which the Riemann-Lebesgue lemma follows at once.

Differentiating a periodic function multiplies its Fourier coefficients by $in$, so each extra continuous derivative buys one more power of decay; this makes term-by-term integration of a Fourier series always legitimate, while term-by-term differentiation is legitimate only when the differentiated series still converges uniformly.

Defines the Fourier transform of an integrable function on the line in the handbook's convention, records the six conventions used by the cited books and the dictionary between them, proves the operational rules for translation, modulation, dilation, reflection, conjugation, differentiation and multiplication by the variable, and tabulates the standard transform pairs.

Proves that a rapidly decreasing function is recovered from its Fourier transform by the inverse integral, using the Gaussian as an approximate identity and the multiplication formula, shows that the transform is a bijection of the Schwartz class whose square is reflection times a constant, and records the weaker versions that hold when the transform is merely integrable or the function merely piecewise smooth.

Defines the convolution and the cross-correlation of two functions on the line, proves that the transform turns convolution into pointwise multiplication and correlation into the product of one transform with the conjugate of the other, and shows how the two results reduce a linear translation-invariant system, a deconvolution and a power spectrum to algebra in the transform variable.

Proves that the Fourier transform preserves the $L^2$ inner product up to the factor $2\pi$ fixed by the convention, by transforming the cross-correlation and evaluating the inversion formula at the origin, then extends the transform from the Schwartz class to a unitary map of $L^2$ onto itself and uses the resulting Parseval relation to evaluate integrals in closed form.

A normalised function and its Fourier transform cannot both be sharply concentrated: the product of their two variances is at least $1/16\pi^2$, with equality exactly for Gaussians.

Wrapping a rapidly decreasing function around the circle and sampling its Fourier transform at the integers produce the same periodic function, so a slowly convergent lattice sum can be traded for a rapidly convergent one.

How to compute Fourier transforms on $\mathbb{R}^n$: factorise when the function is a product, reduce to a single radial integral when it is spherically symmetric, and read off the transforms of the Yukawa and Coulomb potentials.

The one-sided transform $\bar f(s)=\int_0^\infty f(t)e^{-st}dt$, the half-plane on which it converges, its standard pairs and its derivative, shift and scaling rules, together with the Bromwich contour integral that inverts it.

Shows that every rational function splits uniquely into a polynomial plus one singular part per pole, gives the cover-up rule and its derivative extension for the coefficients, the real form that keeps complex-conjugate pairs together as quadratics, and turns the decomposition into a recipe for inverting Laplace and $z$-transforms term by term.

Defines the Hankel transform of order $\nu$ and the Mellin transform in the normalisation of the cited books, shows that each is the Fourier transform seen through a change of variable — Hankel is the plane Fourier transform restricted to radial functions, Mellin is the line Fourier transform after $t = e^{x}$ — and reads the inversion formula for each off that identification.

Defines the discrete Fourier transform of $N$ samples and its competing normalisations, proves the discrete orthogonality relation and from it the inversion formula, the discrete Parseval identity and the circular convolution theorem, derives the aliasing formula that says exactly which Fourier coefficients a finite sample confuses with which, and sets out the even-odd splitting that turns the $O(N^2)$ sum into the $O(N\log N)$ radix-2 FFT.

Proves that a function whose Fourier transform vanishes outside $[-a,a]$ is recovered exactly from its values at the points $k/2a$ by a sum of shifted sinc functions, identifies $2a$ as the Nyquist rate below which distinct frequencies become indistinguishable, and shows what the same series returns when the function is not band-limited.

Defines test functions and distributions as continuous linear functionals on them, proves that every locally integrable function is a distribution while the delta is not, defines the distributional derivative and derives the jump formula for it, shows that a distribution with vanishing derivative is a constant, and ends with a worked derivative and the notion of a weak solution.

Introduces the Schwartz space and the tempered distributions on it, defines the Fourier transform by duality and shows it is a bijection of that space obeying the usual operational rules, gives four routes for computing a transform, and derives the pairs for the delta, constants and powers, the Heaviside step and the principal value of 1/x, ending with a distributional solution of an ordinary differential equation.

Collects the working identities for the Dirac delta — scaling, evenness, multiplication by a smooth function, composition with a function, the derivative rules, the transformation of the multidimensional delta under a change of coordinates, and the standard representations as limits — each with a proof in the distributional sense and a statement of the hypothesis it needs.

Defines the Heaviside step, the sign, ramp and rectangle functions and the identities relating them, proves the chain of distributional derivatives that ends at the delta and the reciprocal statement that the step is the integral of the delta, and gives a recipe with a worked trapezoidal pulse for transforming a switched-on signal by differentiating it into deltas.

Defines elliptic, parabolic and hyperbolic second-order equations by the sign of the discriminant (and, in many variables, by the eigenvalue signs of the coefficient matrix), identifies the type with the number of real characteristic directions, reduces the two-variable constant-coefficient equation to Laplace, wave or heat canonical form, and states what well-posedness in Hadamard's sense asks for.

Turns a first-order quasilinear equation with data on a curve into a system of ordinary differential equations along characteristics, gives the recipe and the non-characteristic condition under which it determines a unique local solution, proves it, and works out the transport equation and three further examples.

Shows how a scalar conservation law in one space dimension loses its classical solution at a computable breaking time, gives the weak formulation that lets the solution be continued past it, derives and proves the Rankine-Hugoniot speed of the resulting jump and the entropy condition that picks out the physical one, and carries a Burgers ramp through from the data to a checked shock position.

Defines the Dirichlet, Neumann, Robin, mixed, Cauchy and periodic conditions, tabulates which of them give a unique and stable solution for elliptic, parabolic and hyperbolic equations on open and on closed boundaries, and explains each verdict — with a proof of uniqueness for the elliptic case and the compatibility condition the Neumann problem needs.

Shows how a dilation symmetry of a partial differential equation forces solutions of the form t^{-alpha} v(x/t^beta), reduces the equation to an ordinary differential equation for the profile v, and carries this out for the diffusion equation, for the porous medium equation (the Barenblatt solution) and for the Airy equation.

Reduces a partial differential equation to an ordinary differential equation for a profile moving at constant speed, shows when the speed is free and when it is selected by the equation, derives the Korteweg-de Vries soliton and an explicit bistable reaction-diffusion front, and states the existence theorem for such fronts.

Gives the general solution $F(x-ct)+G(x+ct)$ of $u_{tt}=c^2u_{xx}$ and d'Alembert's formula for the initial-value problem, proves that the formula solves it and is the only solution, reads off the domains of dependence and influence, and extends both to a half-line or an interval by odd and even reflection.

Defines the energy of a solution of the wave equation, proves it is conserved on the line, in all of space and on a bounded domain with fixed or free boundary, and uses that conservation to prove that the initial/boundary-value problem has at most one solution.

Shows that the spherical means of a solution of the wave equation satisfy the Euler-Poisson-Darboux equation, turns that into a half-line problem in one dimension to derive Kirchhoff's formula for the Cauchy problem in three dimensions, reads off Huygens' principle, and descends to the two-dimensional Poisson formula, where Huygens' principle fails.

Separates the variables in the wave equation for a string with fixed ends, shows that the only separated solutions are the standing waves $\sin(n\pi x/L)$ oscillating at frequency $nc/2L$, gives the recipe for fitting arbitrary initial data by a Fourier sine series, and says when the resulting series is a genuine solution.

Convolving bounded continuous initial data with the Gaussian kernel (4 pi k t)^{-1/2} exp(-x^2/4kt) solves the heat equation on the line, reproduces the data as t tends to 0, makes the solution positive everywhere at once, and bounds its m-th derivative by t^{-m/2}.

A solution of the heat equation in a bounded space-time cylinder takes its largest and smallest values on the initial slice or the lateral boundary, which gives uniqueness and a quantitative stability bound for the initial-boundary-value problem without solving it.

How to solve the heat equation on an interval by separating variables: the eigenvalue problem set by the boundary condition, the exponentially decaying modes it produces, the rate at which the solution settles to its equilibrium, and two ways to handle boundary data that are not zero.

Adding transport and reaction to the diffusion equation: the Peclet number that says which of advection and diffusion wins, the change to a moving frame that solves the linear problem exactly, and the travelling fronts that a bistable reaction term supports.

Shows that a harmonic function equals its average over every sphere and every ball lying in its domain, that the converse holds for $C^2$ functions, that consequently it attains its maximum and minimum only on the boundary unless it is constant, and that the Dirichlet problem for Poisson's equation therefore has at most one solution, which depends continuously on its data.

Finds all radial harmonic functions, normalizes them into the fundamental solution $\Phi$ with $-\nabla^2\Phi = \delta_0$ — the logarithm in the plane and $1/(4\pi r)$ in space — and proves that for $f \in C_c^2(\mathbb{R}^n)$ the Newtonian potential $u = \Phi * f$ is $C^2$ and solves Poisson's equation $-\nabla^2 u = f$ in all of $\mathbb{R}^n$.

Solves the Dirichlet problem for Laplace's equation in a disk by separating variables and summing the resulting Fourier series into the Poisson kernel, proves from the kernel's three properties that the integral has the prescribed continuous boundary values, records the corresponding formula for a ball in three dimensions, and deduces that every harmonic function is infinitely differentiable inside its domain.

Gives the recipe for reducing Laplace's and Helmholtz's equations to ordinary differential equations by seeking product solutions, proves the lemma that makes the separation constants constant, carries the spherical case through in full, and catalogues which special functions - trigonometric, hyperbolic, Bessel, Legendre and spherical harmonic - each of the Cartesian, cylindrical and spherical systems produces.

Expands the potential of a localized source outside a sphere containing it, first in Legendre polynomials from the generating function and then in spherical harmonics by the addition theorem, defines the monopole, dipole and quadrupole moments that the terms carry, and shows that the lowest nonvanishing moment is independent of the choice of origin while all higher ones are not.

Reduces the wave and diffusion equations to the Helmholtz equation by separating a time-harmonic factor, collects its elementary solutions - plane waves, cylindrical waves and the spherical waves built from spherical Bessel and Hankel functions - and states and proves the Sommerfeld radiation condition that picks out the outgoing solution and forces a field that is source-free on all of space to vanish.

States the time-dependent and time-independent Schrodinger equations, shows that separating the time turns the first into an eigenvalue problem whose solutions are the stationary states, works out the free particle, the infinite well and the harmonic oscillator with their normalizations and energy levels, and derives the probability current and the continuity equation that makes the normalization persist.

Introduces the Klein-Gordon equation as the wave equation with a mass term, derives its dispersion relation and reads the relativistic energy-momentum relation off it, verifies its one-dimensional source function across the light cone and records the three-dimensional one, shows that the massless limit gives back the wave equation and the non-relativistic limit the Schrodinger equation, and computes the conserved density that fails to be positive.

Defines the weak derivative of a locally integrable function, the Sobolev spaces $H^1$ and $H^1_0$ built from it, and the weak formulation of the Poisson problem, and shows that a smooth weak solution is a classical one.

Proves that a bounded coercive bilinear form on a Hilbert space represents every bounded linear functional exactly once, proves the Poincare inequality that makes the Dirichlet form coercive, and combines the two into existence, uniqueness and an a priori bound for weak solutions of second-order elliptic Dirichlet problems.

Proves the basic energy estimates for the forced heat and wave equations on a bounded domain, deduces uniqueness and continuous dependence on the data from them, and shows by the backward heat equation and Hadamard's Cauchy problem for Laplace's equation what the estimates exclude.

Reads a linear evolution equation as an ordinary differential equation in a Banach space, proves that the solution operators of such an equation form a contraction semigroup whose generator is densely defined and closed and whose resolvent is the Laplace transform of the semigroup, states the Hille-Yosida characterisation of the operators that generate one, and identifies the heat semigroup and its generator explicitly.

Defines the integral of a function over every hyperplane of $\mathbb{R}^n$, proves the projection-slice theorem that identifies its one-dimensional Fourier transform with a radial slice of the $n$-dimensional one, inverts it by filtering and back-projecting, and applies the inversion to computed tomography and to the plane-wave solution of the wave equation.

Defines the Green's function of a second-order two-point boundary-value problem by the homogeneous equation off the source point, continuity, the jump $1/a_2(\xi)$ in the first derivative and the homogeneous boundary conditions; derives the jump from $LG = \delta$, constructs $G$ explicitly from two homogeneous solutions and their Wronskian, proves that $u(x) = \int_a^b G(x,\xi)f(\xi)\,d\xi$ is the unique solution, and proves the reciprocity relation $G^*(x,\xi) = G(\xi,x)$ and hence the symmetry of $G$ for a self-adjoint problem.

States and proves the dichotomy governing a boundary-value problem whose homogeneous problem has nontrivial solutions — either a unique solution for every forcing, or solvability exactly when the forcing is orthogonal to every solution of the homogeneous adjoint problem — gives the model matrix case with a complete proof, and builds the modified Green's function, the pseudoinverse that returns the minimum-norm solution when the problem is solvable and the minimum-norm least-squares solution when it is not.

Derives the bilinear series $G(x,\xi;\lambda) = \sum_n u_n(x)\overline{u_n(\xi)}/(\lambda_n-\lambda)$ for the resolvent of a regular Sturm–Liouville problem and the special case $G = \sum u_n(x)\overline{u_n(\xi)}/\lambda_n$ at $\lambda=0$, shows that $G$ is meromorphic in $\lambda$ with simple poles at the eigenvalues whose residues are $-u_n(x)\overline{u_n(\xi)}$, and uses that to read the whole spectrum off a closed-form Green's function.

Builds the retarded Green's function $G(t,\tau) = H(t-\tau)\,u_\tau(t)$ of a linear initial-value problem, proves that $u(t) = \int_a^t G(t,\tau)f(\tau)\,d\tau$ solves it with zero initial data, specialises to the impulse response of a constant-coefficient system so that the solution becomes a convolution with transfer function $\hat G(\omega)$, works the damped oscillator in full, and shows that vanishing before the impulse makes $\hat G$ analytic in the upper half-plane.

Defines the Dirichlet and Neumann Green's functions of a bounded region for $\nabla^2$, derives the representation formula that expresses the solution of $\nabla^2 u = \rho$ from the data alone, and proves that the Dirichlet Green's function is symmetric in its two arguments.

Builds the corrector of a Dirichlet Green's function out of point sources placed outside the region, and carries the construction through for a half-space (one image of strength $-1$) and for a sphere of radius $a$ (one image at the inverse point $a^2\mathbf{r}_0/r_0^2$ of strength $-a/r_0$), recovering the Poisson kernels of both.

Computes the free-space outgoing Green's function of $\nabla^2+k^2$ by Fourier transforming and evaluating the remaining integral by residues with the poles displaced off the real axis, giving $-\tfrac{i}{2k}e^{ik|x|}$ in one dimension, $-\tfrac{i}{4}H_0^{(1)}(k\varrho)$ in two and $-e^{ikr}/(4\pi r)$ in three, and reads off the modified Helmholtz case by $k\mapsto i\kappa$.

Shows that the causal Green's function of the heat operator is the heat kernel and that of the three-dimensional wave operator is $\delta(t-r/c)/(4\pi c^2 r)$, a delta function on the light cone, derives the retarded potential from it, and identifies Duhamel's principle as convolution in time with the causal Green's function.

Turns the stationary scattering problem into the Lippmann-Schwinger integral equation by inverting $\nabla^2+k^2$ with the outgoing Green's function, reads the scattering amplitude off the far field, identifies its Neumann series as the Born series, and evaluates the first Born amplitude $-\tfrac{1}{4\pi}\mathcal{F}U(\mathbf q)$ at momentum transfer $q=2k\sin(\theta/2)$ on a Gaussian potential and on a one-dimensional delta potential where the whole series can be summed.

The vocabulary of linear integral equations — first and second kind, Fredholm and Volterra, the kernel and the classes of kernel that make the integral operator bounded or compact — together with the recipe that turns a second-order linear ODE with its data into an equivalent integral equation, Volterra for initial conditions and Fredholm for boundary conditions.

For a second-kind integral equation with a bounded kernel and a small enough parameter, substituting the equation into itself converges: the solution is the Neumann series in the iterated kernels, summed by the resolvent kernel, with an explicit error bound from the contraction mapping theorem and three usable sufficient conditions for convergence.

A kernel that is a finite sum of products of a function of x with a function of t turns a Fredholm equation of the second kind into an n-by-n linear system, so the characteristic values are the roots of a polynomial and the Fredholm alternative is exactly the solvability trichotomy for that system.

For a Hermitian Hilbert-Schmidt kernel the characteristic values are real, the eigenfunctions are orthogonal and span the range of the operator, the kernel itself expands bilinearly in them, and the inhomogeneous second-kind equation is solved term by term with a resolvent whose poles are exactly the characteristic values.

A second-kind Volterra equation with a continuous kernel has exactly one continuous solution for every value of the parameter, so it has no characteristic values at all; the proof is that some power of the iteration map is a contraction, with a factorial in the bound. Abel's weakly singular equation of the first kind is then inverted in closed form.

Defines functionals on spaces of curves, the admissible variations and the first variation, proves that the first variation vanishes at an extremum, and proves the fundamental lemma that turns this into a pointwise equation.

A continuously differentiable curve with fixed endpoints that makes an integral functional stationary satisfies the Euler-Lagrange equation, and it is automatically twice differentiable near every point where F_{y'y'} != 0 along it.

When the integrand does not involve the unknown function or the independent variable, the second-order Euler-Lagrange equation reduces to a first-order equation, which solves the brachistochrone and the minimal surface of revolution in closed form.

A functional of several unknown functions gives one Euler-Lagrange equation per function, and an integrand containing derivatives up to order m gives the Euler-Poisson equation of order 2m.

A function of several variables that makes a multiple-integral functional stationary under fixed boundary values satisfies a second-order partial differential equation in divergence form, which gives the minimal-surface equation and the field equations of a Lagrangian density.

When an endpoint value is not prescribed, a stationary curve must satisfy an extra boundary condition, F_{y'} = 0 at a free end and the transversality condition when the endpoint slides along a given curve, so the problem still has as many conditions as constants.

To make one integral stationary while another integral is held fixed, add a constant Lagrange multiplier times the constraint integrand and solve the Euler-Lagrange equation of the sum; this gives the circular arc of Dido's problem and the catenary of a hanging chain.

To make an integral stationary over curves forced to lie on a surface g = 0, add to the integrand g times a multiplier that is a function of the independent variable; for arc length this shows that geodesics are the curves whose acceleration is normal to the surface, such as great circles on a sphere and helices on a cylinder.

Newton's equations for particles moving in a potential are exactly the Euler–Lagrange equations of the action, the time integral of kinetic minus potential energy, and they keep this form in any generalized coordinates, including moving ones.

If the action is unchanged by a one-parameter family of transformations of time and coordinates, an explicit combination of momenta and energy is conserved along every motion; time shifts, translations and rotations give conservation of energy, momentum and angular momentum.

When velocities can be solved for in terms of momenta, the Legendre transformation turns the n second-order Euler–Lagrange equations into Hamilton's 2n first-order canonical equations, and first integrals are then recognized by a vanishing Poisson bracket.

The action, viewed as a function of the endpoint, solves the first-order Hamilton–Jacobi equation; conversely, a complete integral of that equation produces all solutions of the canonical equations by differentiation and elimination alone (Jacobi's theorem).

At a weak minimum of a functional the second variation is nonnegative in every direction, which forces F_{y'y'} >= 0 at every point of the curve (Legendre's condition); the unit also writes the second variation in the form used for sufficient conditions.

Along an extremal with F_{y'y'} > 0, the second variation is positive definite exactly when the Jacobi equation has no conjugate point, which gives Jacobi's necessary condition and a set of sufficient conditions for a weak minimum.

The eigenvalues of a regular Sturm-Liouville problem are the successive minima of the Rayleigh quotient, and each one is also a min-max over subspaces, which gives upper bounds and comparison theorems without solving the equation.

Restricting the Rayleigh quotient to the span of N trial functions turns a Sturm-Liouville eigenvalue problem into a small matrix problem K c = Lambda M c whose eigenvalues are guaranteed upper bounds for the first N eigenvalues and decrease to them as the trial space grows.

Gives the two objects every integration theory rests on — a sigma-algebra of sets we agree to measure and a countably additive measure on it — together with the monotonicity, subadditivity and continuity rules that let you pass to the limit inside a measure, and the vocabulary of null sets and "almost everywhere".

Builds Lebesgue measure from the length of an interval by covering sets from outside and keeping the sets that split every test set additively, and records what the result does and does not do — complete, regular, translation invariant, and necessarily undefined on some subsets of the line.

Defines a measurable function as one whose preimages of Borel sets are measurable, shows that the class survives sums, products, suprema and pointwise limits, and builds every non-negative measurable function as an increasing pointwise limit of simple functions — the staircase on which the Lebesgue integral is erected.

Builds the integral against a measure in three stages — finite sums for simple functions, a supremum for non-negative measurable functions, a difference of two such for the rest — records the properties that make it a linear order-preserving functional on $L^1$, and shows that on a compact interval it reproduces every proper Riemann integral while reaching functions the Riemann theory cannot.

For non-negative measurable integrands, an increasing pointwise limit may always be taken outside the integral, a non-negative series may always be integrated term by term, and for an arbitrary sequence the integral of the lower limit never exceeds the lower limit of the integrals.

If a sequence of measurable functions converges almost everywhere and is held under one fixed integrable envelope, the limit is integrable and the integrals converge — indeed in the $L^1$ norm — and the same envelope condition licenses taking a limit, or a derivative in a parameter, inside an integral.

Gives the conditions under which a double integral may be computed as an iterated integral in either order — non-negativity for Tonelli, absolute integrability for Fubini, sigma-finiteness for both — and shows by counterexample what goes wrong when each is dropped.

Defines almost-everywhere, almost-uniform, in-measure and $L^p$ convergence, settles exactly which of them implies which, proves Egorov's theorem and the almost-everywhere subsequence theorem, and gives the four standard counterexamples that block every remaining implication.

Establishes the two inequalities that make $\|\cdot\|_p$ a norm, proves that $L^p$ is complete for $1 \le p \le \infty$ (Riesz-Fischer), and identifies the simple and continuous functions as dense subspaces, so that every $L^p$ function can be approached by something concrete.

Gives the exact condition under which one measure is an integral of a density against another, constructs that density as the Radon-Nikodym derivative $d\nu/d\mu$, and separates off the singular part of a measure that has no density at all.

Gives the distance axioms that let convergence be defined on any set, the Cauchy criterion that tests for convergence without naming the limit, and the proofs that the continuous functions on a closed interval are complete in the sup metric but not in the integral metric.

A map that shrinks all distances by a fixed factor less than one, on a complete metric space, has exactly one fixed point, reached by iterating from anywhere; the proof supplies two computable error bounds, one usable before the iteration starts and one after each step.

Gives the norm axioms and the metric they induce, the series test for completeness, the standard Banach spaces and the standard incomplete ones, and proves that on a finite-dimensional space every norm is equivalent to every other, so that in finite dimensions the choice of norm never matters and in infinite dimensions it always may.

A Hilbert space is an inner product space that is complete; this unit says which of the standard inner product spaces are complete and which are not, proves the completeness of $\ell^2$, and uses the parallelogram law as the test that decides whether a given Banach space is a Hilbert space at all.

Shows that a non-empty closed convex subset of a Hilbert space contains exactly one point nearest any given vector, that for a closed subspace the error of that nearest point is orthogonal to the subspace, and hence that $H = M \oplus M^\perp$, which is what makes "drop a perpendicular" a legitimate move in infinitely many dimensions.

Gives Bessel's inequality for any orthonormal family, the exact criterion for a series $\sum a_k e_k$ to converge, and the equivalence of maximality, Parseval's identity and expandability that defines an orthonormal basis, with the consequence that a separable Hilbert space is $\ell^2$ in disguise.

Shows that every bounded linear functional on a Hilbert space is $u \mapsto (u,z)$ for one and only one vector $z$, of the same norm, so that the dual space is a conjugate copy of the space itself — the statement that makes Dirac's bra a vector and underwrites weak formulations, adjoints and Green's functions.

Defines weak convergence as convergence of every bounded linear functional along the sequence, collects what survives from the norm theory (uniqueness of the limit, boundedness, lower semicontinuity of the norm) and what does not (the norm itself is not continuous), and gives the working test in $\ell^2$ and $\ell^p$ — bounded plus coordinatewise.

Defines the operator norm, proves that a linear map between normed spaces is continuous exactly when it is bounded, collects the algebra of the norm (submultiplicativity, completeness of $\mathcal{B}(X,Y)$, extension from a dense domain), and computes the norm of the Volterra integral operator exactly.

Builds the Hilbert adjoint $T^*$ from the Riesz representation theorem, proves the algebraic rules including $\|T^*T\| = \|T\|^2$, and characterises the three classes it defines — self-adjoint, unitary and orthogonal projection operators — together with the elementary consequences a physicist uses, namely real expectation values, norm preservation, and the correspondence between projections and closed subspaces.

Splits the complex plane into the resolvent set and the point, continuous and residual spectrum of a bounded operator, shows the spectrum is a non-empty compact subset of the disc of radius $\|T\|$, and proves that a bounded self-adjoint operator has real spectrum and no residual spectrum.

Defines the operators that carry bounded sets to sets with compact closure, shows that they form a closed two-sided ideal containing every bounded finite-rank operator, identifies them on a separable Hilbert space as exactly the norm limits of finite-rank operators, and gives the working criterion that an integral operator with a square-integrable kernel is compact.

A compact self-adjoint operator has real eigenvalues of finite multiplicity that can only accumulate at zero, the largest in modulus equals the operator norm and is reached by maximising the Rayleigh quotient, and the eigenvectors form an orthonormal basis in which the operator is diagonal — the infinite-dimensional replacement for diagonalising a Hermitian matrix, and the reason a self-adjoint boundary-value problem has a complete set of normal modes.

Two of the three pillars of linear functional analysis, both consequences of Baire's category theorem: a family of bounded operators on a Banach space that is bounded at every point is bounded in norm, and a bounded operator from one Banach space onto another maps open sets to open sets — from which the bounded inverse and closed graph theorems follow, and with them the fact that a symmetric operator defined on a whole Hilbert space cannot be unbounded.

Extends a linear functional from a subspace to the whole space without increasing the bound that controls it, and draws the consequences that make the dual space useful: a normed space always carries enough bounded functionals to reproduce its norm, to separate its points, to detect the distance to a closed subspace, and to embed the space isometrically in its bidual.

Sets up the bookkeeping that unbounded operators force on us: an operator is a linear map on a dense domain, its graph decides whether it is closed or closable, its adjoint exists only because the domain is dense, and the adjoint determines the closure; the two running examples are multiplication by $x$ and $-i\,d/dx$.

Separates the identity $(T\varphi,\psi)=(\varphi,T\psi)$ from the equality of operators $T=T^{*}$, proves the criterion that decides between them — a closed symmetric operator is self-adjoint exactly when $T^{*}\varphi = \pm i\varphi$ has only the zero solution — introduces the deficiency indices that count the failures, and settles the momentum operator on an interval, where a circle of self-adjoint extensions exists, and on a half-line, where none does.

States and proves the spectral theorem for a possibly unbounded self-adjoint operator in its three equivalent forms - unitary equivalence with a multiplication operator, a Borel functional calculus, and a projection-valued measure - and shows on the momentum operator how the three are read off from one another.

Identifies the strongly continuous one-parameter unitary groups on a Hilbert space with the self-adjoint operators: every self-adjoint $A$ exponentiates to a group $U(t)=e^{itA}$ whose strong derivative at the origin recovers $A$ on exactly its own domain, and conversely every such group is of that form for one and only one self-adjoint operator, which is the mathematical content of the statement that a quantum Hamiltonian generates the time evolution.

Assembles the Hilbert-space model of a quantum system - unit vectors up to a phase as states, densely defined self-adjoint operators as observables, the projection-valued measure of an observable as the probability distribution of measuring it, and a strongly continuous unitary group as the dynamics - and derives from it the expectation and variance of a measurement, the commutator inequality, and the Heisenberg uncertainty relation, with the statistics of two concrete states computed in full.

Sets up the triple of sample space, sigma-field and probability measure on which every later probabilistic statement is made, derives the elementary consequences of the axioms including continuity along monotone sequences, and defines conditioning and independence, from which the partition theorem and Bayes' theorem follow in two lines each.

Defines a random variable as a measurable function on a probability space and its distribution function as the object that carries all the probabilistic information about it, characterises which functions are distribution functions, separates the discrete and continuous cases, and catalogues the standard distributions with the mechanism each one models.

Defines the expectation of a random variable and the three derived numbers that summarise a distribution and the link between two of them, proves the rules that make expectation linear and variance quadratic, and establishes the Cauchy-Schwarz bound that confines the correlation coefficient to the interval from minus one to one.

Gives the working recipe for turning the joint density of several random variables into the density of functions of them — invert the map, multiply by the modulus of the Jacobian, integrate out what is not wanted — and derives from it the convolution formula for the density of a sum of independent variables.

Defines the conditional distribution and the conditional expectation $\mathbb{E}(Y \mid X)$ as a random variable, proves the tower, pull-through and conditional-variance rules, shows that $\mathbb{E}(Y \mid X)$ is the function of $X$ closest to $Y$ in mean square, and works a mixture example whose checks are $\log 2$ and $\tfrac12$.

Defines the $N(\mu, V)$ density on $\mathbb{R}^n$, proves its normalisation, mean and covariance matrix by orthogonal diagonalisation, shows that linear images, marginals and conditionals are all normal with explicit parameters, and works a three-dimensional example whose conditional law is $N(9/5, 13/5)$.

Encodes a distribution in its probability, moment and cumulant generating functions, turns convolution into multiplication and random sums into composition, and applies the composition rule to branching processes, extracting the extinction probability $1/2$ for a worked offspring law.

Defines $\varphi_X(t)=\mathbb{E}(e^{itX})$, proves that it is bounded, uniformly continuous and non-negative definite and that it turns independent sums into products and affine maps into phases, states the inversion and continuity theorems and the moment expansion, and inverts $\sin^2 t/t^2$ to the triangular density as a worked check.

Bounds the probability that a random variable is far from its mean using only one or two of its moments, or its whole moment generating function, and shows on one worked tail which of the four bounds is worth using.

Defines the four modes in which a sequence of random variables can converge, proves the implications between them, shows by explicit counterexamples that no others hold, and gives the two Borel-Cantelli lemmas, which are the standard tool for deciding almost-sure convergence.

Proves that the average of independent random variables converges in probability to the average of their means under a simple variance condition, states the strong law in the sharp form that almost sure convergence of the average of an i.i.d. sequence holds exactly when the mean is finite, and distinguishes what each version promises.

Proves that the standardised sum of independent identically distributed variables with finite non-zero variance converges in distribution to the standard normal, by expanding the characteristic function to second order and applying the continuity theorem, and gives the Berry-Esseen rate, the continuity correction and the Cauchy example where the theorem fails.

Gives the exponential rate at which the probability of a sum straying a fixed fraction away from its mean decays, identifies that rate as the Fenchel-Legendre transform of the cumulant generating function, and computes it on a worked tail against the exact answer.

Sets up a chain on a countable state space through its transition matrix, derives the Chapman-Kolmogorov equations that make the n-step probabilities matrix powers, classifies states by recurrence, mean recurrence time and period, and characterises the stationary distribution as the left eigenvector of the transition matrix with eigenvalue one.

Proves by coupling that the n-step probabilities of an irreducible aperiodic chain converge to the reciprocal of the mean recurrence time, shows that a distribution in detailed balance with the transition matrix is automatically stationary and makes the chain reversible, and gives a geometric bound on the remaining total-variation distance in terms of the second-largest eigenvalue modulus.

Solves the gambler's ruin problem for a simple random walk between two absorbing barriers, and proves Polya's trichotomy: the symmetric nearest-neighbour walk returns to its starting point with probability one on the line and on the plane, but not in space, where the return probability is about 0.34.

Gives the infinitesimal and interarrival definitions of the Poisson process and shows they describe the same process, derives the Poisson law of the counts and the exponential law of the gaps, proves the conditional property that fixes the number of points and leaves their positions an independent sample, and proves that superposing independent Poisson streams and independently thinning one both return Poisson processes with the obvious intensities.

Defines a continuous-time Markov chain through its transition semigroup and its generator, shows that the chain is an exponential holding time in each state followed by a jump governed by a discrete chain, derives the Kolmogorov forward and backward equations and the solution by a matrix exponential, and gives the stationary condition and the birth-death criterion.

Defines strong and weak stationarity and the autocovariance function, proves that an autocorrelation function is non-negative definite and therefore, when continuous at the origin, is the characteristic function of a distribution on the frequency axis, and identifies that distribution's density as the spectral density of the process.

Defines the Wiener process, shows that it is the scaling limit of a random walk in the sense that the rescaled walk's finite-dimensional distributions converge to it, sketches the proof that a version with continuous paths exists, and establishes the path properties - covariance $\sigma^2\min\{s,t\}$, quadratic variation $\sigma^2 t$, and total variation infinite on every interval.

Defines filtrations, martingales, sub- and supermartingales and stopping times, proves that a stopped martingale is a martingale and then the optional stopping theorem in two usable forms, shows by the doubling strategy exactly which hypothesis fails when the conclusion does, and uses the theorem to compute the mean waiting time for a pattern in coin tosses.

Builds the Ito integral against a Wiener process by extending the elementary sum over predictable step functions along the Ito isometry, shows that the resulting integral is a martingale with second moment equal to the mean integrated square of the integrand, and states and sketches Ito's formula, the chain rule with its extra second-derivative term.

Gives a recipe for solving a linear stochastic differential equation with additive noise by an integrating factor, proves that its solution is Gaussian with an explicit mean, variance and autocovariance, applies this to the Ornstein-Uhlenbeck process and the Langevin equation, and explains how the Ito and Stratonovich readings of the same equation differ and how to convert between them.

States the forward (Fokker-Planck) and backward Kolmogorov equations satisfied by the transition density of a one-dimensional diffusion, derives each from Ito's formula - the forward equation by testing against compactly supported functions and integrating by parts, the backward equation from the martingale property of the conditional expectation - and gives the zero-flux recipe for the stationary density, which produces the Boltzmann distribution.

Defines a statistic and its sampling distribution, fixes the two rival conventions for the sample variance, and proves that for a normal sample the sample mean and the sample variance are independent with laws $N(\mu,\sigma^2/n)$ and $\sigma^2\chi^2(n-1)/(n-1)$, from which the Student $t$ and Fisher $F$ densities follow; worked on an eight-point sample giving $t=1.063$ and $\chi^2=5.731$.

Defines bias, consistency, efficiency and the standard error of an estimator, introduces the score and the Fisher information of a sample, and proves the Cramer-Rao (Fisher) inequality bounding the variance of any estimator from below; worked on twelve Poisson counts, where the sample mean attains the bound $\hat\lambda = 6.833 \pm 0.755$ while the sample variance reaches only $6.3\%$ of it.

Defines the likelihood and the maximum-likelihood estimator, proves that maximum likelihood commutes with reparametrisation and that any efficient estimator must be a function of it, states the large-sample normal form of the likelihood with its curvature error bar, and locates a lighthouse from fifteen Cauchy-distributed flashes at $1.163 \pm 0.184$ km where the sample mean gives $-0.64$ km.

Sets up the linear fitting problem with a known noise covariance, solves it in closed form for the parameters and their full covariance matrix, proves the Gauss-Markov theorem that this is the best linear unbiased estimator whatever the noise distribution, shows that for Gaussian noise the minimised chi-squared has $n-M$ degrees of freedom, and fits Planck's constant from six photoelectric points as $h = (6.615 \pm 0.093)\times10^{-34}\,\mathrm{J\,s}$ with $\chi^2 = 2.20$ on $4$ degrees of freedom.

Gives the first-order rule for the variance of a function of uncertain inputs, in the matrix form $V_Y = S V_X S^{\mathsf T}$ that keeps the correlations, says exactly when the rule is exact and when its neglected second-order terms matter, and works two examples ending in $g = 9.811 \pm 0.044\,\mathrm{m\,s^{-2}}$ and a predicted-value error of $0.134$ that the naive uncorrelated formula overstates by a factor of $3.2$.

Builds interval estimates by the pivotal method, gives the exact normal-sample intervals for a mean (known and unknown variance) and for a variance, says precisely what the confidence level asserts and what it does not, and works nine speed-of-sound readings to $\mu \in [342.62, 344.18]\,\mathrm{m\,s^{-1}}$ and $\sigma \in [0.684, 1.940]\,\mathrm{m\,s^{-1}}$ at the 95% level.

Sets up null and alternative hypotheses, rejection regions, significance, power and the $p$-value, proves the Neyman-Pearson lemma, derives Student's $t$-test as a generalised likelihood-ratio test and gives the chi-squared goodness-of-fit test, with eight dark-current readings giving $t = 3.000$ and $p = 0.0199$ and five count bins giving $\chi^{2} = 4.625$ on four degrees of freedom.

Turns Bayes' theorem into a method of parameter estimation - prior, likelihood, evidence, posterior - proves that the posterior updates sequentially and that the beta and normal families are conjugate, defines the credible interval and sets it against the confidence interval, and works a detector efficiency to a posterior mean of $9/14 = 0.643$ with a 95% credible interval $[0.386, 0.861]$.

Builds the sample covariance matrix of a centred data matrix, shows that its eigenvectors are the directions of successively largest sample variance and that they are the right singular vectors of the data matrix itself, reads the eigenvalues as the variance each direction explains, and identifies the truncated expansion as the best approximation of its rank — which is what an EOF analysis of a space-time field computes.

Defines the periodogram of a finite record as the squared modulus of its discrete Fourier transform, shows that it is exactly the transform of the sample autocovariance and that its expectation is the true spectral density smoothed by the Fejer kernel of the data window, computes its distribution for Gaussian white noise to show that it never becomes consistent however long the record, and sets out the two repairs — tapering against leakage and segment averaging against variance.

Shows that a linear time-invariant operation on a signal is a convolution, that complex exponentials pass through it unchanged except for multiplication by its transfer function, and that it multiplies the power spectral density of a stationary process by the squared gain; then turns those facts into a design recipe for moving averages and for low-pass, high-pass and band-pass filters, with the Gibbs ripple and the causality constraint that limit what can be built.

Builds the AR(1) process from uncorrelated noise, derives its autocorrelation $\alpha^{|m|}$ and its power spectrum, obtains the Yule-Walker equations that fit an AR($p$) scheme to data, and uses the fitted AR(1) spectrum as the red-noise null against which a spectral peak is tested, with a worked twelve-point series whose apparent cycle survives a white-noise test and fails a red-noise one.

Sets up the linear Gaussian state-space model, proves by induction on the Gaussian conditioning formula that the filtering distribution stays normal and obeys a forecast step and an update step with gain $K_k = P_k^{f}H_k^{\mathsf T}S_k^{-1}$, identifies the update with a Bayesian posterior and with weighted least squares, and runs the recursion on a four-point scalar record where the forecast variance falls $1.875 \to 1.0380 \to 1.0023 \to 1.00015$ towards the fixed point $1$ and the final estimate $-0.000376 \pm 0.7071$ reproduces the exact nine-dimensional Gaussian conditional.

Derives the best linear unbiased analysis $x_a = x_b + K(y-Hx_b)$ with $K = BH^{\mathsf T}(HBH^{\mathsf T}+R)^{-1}$ by completing the square in the gain, proves that it also minimises the 3D-Var cost function and that the two routes agree exactly when $B$ is invertible, and interpolates a temperature analysis from two stations, obtaining weights $0.740445$ and $-0.135366$, an analysis of $1.9490\,\mathrm{K}$ and an analysis error variance $0.425475\,\mathrm{K^2}$ against a background variance of $1$.

Gives the group axioms and the uniqueness and cancellation laws they force, the one-line test for a subset to be a subgroup, and the order of an element together with the cyclic subgroup it generates.

Shows that the left cosets of a subgroup cut a group into equal-sized disjoint pieces, so that in a finite group the order of a subgroup divides the order of the group, and draws out the consequences for orders of elements and for groups of prime order.

Defines structure-preserving maps of groups, identifies their kernels as exactly the normal subgroups, builds the quotient group on the cosets of a normal subgroup, and proves that every homomorphic image is such a quotient.

Gives the vocabulary of a group acting on a set — orbit, stabilizer, transitivity — proves that the orbit of a point is in bijection with the cosets of its stabilizer, and derives the counting lemma that averages fixed points to count orbits.

Defines conjugation, conjugacy classes, the centralizer and the centre, shows that a class has as many elements as the centralizer has cosets, and assembles these counts into the class equation.

Defines the group of permutations of n symbols, establishes the disjoint-cycle decomposition and its uniqueness, and constructs the sign homomorphism whose kernel is the alternating group of index two.

Classifies the finite symmetry groups of a figure about a fixed point in two and three dimensions, and proves the crystallographic restriction, which allows a lattice only two-, three-, four- and six-fold rotation axes.

Defines rings, ideals, quotient rings, integral domains and fields, records that a quotient by an ideal is a field exactly when the ideal is maximal, and identifies the fields the handbook uses.

Defines the quaternion algebra with its conjugate and norm, shows it is a division algebra, and proves that conjugation by a unit quaternion is a rotation of three-space, giving the two-to-one map onto SO(3) and the identification of the unit quaternions with SU(2).

A representation turns the abstract elements of a group into invertible matrices; for a finite group every complex representation can be made unitary and splits into a direct sum of irreducible pieces, which is Maschke's theorem.

A linear map that intertwines two irreducible representations is either zero or an isomorphism, and over the complex numbers any map commuting with an irreducible representation is a multiple of the identity.

The matrix entries of the inequivalent irreducible representations of a finite group are orthogonal functions on the group, and their traces — the characters — form an orthonormal set that decides, by one sum, how any representation decomposes.

Shows how the square array of irreducible characters of a finite group is pinned down by orthogonality of its rows and of its columns together with a few arithmetic constraints, and how, once built, it reduces any representation to a list of multiplicities.

Builds the tensor product of two representations of a finite group, shows that its character is the pointwise product of the two characters, decomposes it into irreducibles by the Clebsch-Gordan series, and turns the resulting multiplicity of the trivial representation into a rule that forbids matrix elements.

Defines a matrix Lie group as a closed subgroup of the invertible matrices, shows that the set of $X$ with $e^{tX}$ in the group for every real $t$ is a real Lie algebra under the commutator, explains how the exponential map ties the two together near the identity, and fixes the structure constants and the two sign conventions (mathematicians' $e^{X}$, physicists' $e^{i\varphi S}$) that the rest of the chapter uses.

Computes the generators of $SO(3)$ and of $SU(2)$, shows that the two Lie algebras are the same three-dimensional algebra $[J_j,J_k]=i\epsilon_{jkl}J_l$, and proves that $U\mapsto R(U)$ defined by $U(\mathbf{x}\cdot\boldsymbol{\sigma})U^{\dagger}=(R\mathbf{x})\cdot\boldsymbol{\sigma}$ is a surjective homomorphism $SU(2)\to SO(3)$ with kernel $\{\pm I\}$ — the two-to-one covering that makes a spinor change sign under a rotation through $2\pi$.