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functional analysis

L² & Hilbert spaces, distributions, orthonormal bases & completeness, hermitian & unitary operators, spectral theory, boundary value problems, Bessel & Legendre functions.

hilbert spaces · orthonormal bases · distributions · operators & projections · spectra · spectral theorem · unbounded operators · boundary value problems · special functions

L² & Hilbert spaces

The space. \(L^2(\Omega)=\big\{f\colon\Omega\to\C\ \big|\ \int_\Omega|f|^2\dd x\lt\infty\big\}\) with inner product \(\langle f|g\rangle=\int_\Omega\overline{f(x)}\,g(x)\dd x\) and norm \(\|f\|=\sqrt{\langle f|f\rangle}\). Contains every continuous function on a finite-volume domain, step functions, and unbounded functions: \(|x|^a\in L^2(-1,1)\) iff \(a\gt-\tfrac12\).

Convergence & Cauchy sequences. \(|f_n\rangle\to|f\rangle\) means \(\|f_n-f\|\to0\) — the area between the graphs shrinks, not pointwise agreement. Rectangles of height \(n^a\), width \(n^{-b}\): \(\|f_n\|^2=n^{2a-b}\to0\) iff \(2a\lt b\), even though \(f_n(x)\to0\) pointwise for every \(x\gt0\) regardless. Cauchy: \(\|v_n-v_m\|\to0\) as \(n,m\to\infty\); convergent \(\Rightarrow\) Cauchy (triangle inequality). Limits respect the structure: \(v_n\to v\Rightarrow\langle w|v_n\rangle\to\langle w|v\rangle\) and \(\|v_n\|\to\|v\|\) (Cauchy–Schwarz).

Hilbert space. An inner product space in which every Cauchy sequence converges in the space — no holes; the \(\R\)-vs-\(\Q\) distinction lifted to vectors (infinite sums like decimal expansions demand the completed space). Every finite-dimensional inner product space is Hilbert; the notion only bites in infinite dimensions. Riesz–Fischer: \(L^2(\Omega)\) is a Hilbert space for any \(\Omega\subseteq\R^n\).

Nice functions are not enough. \(C^1\cap L^2\) is not complete: ramps with a parabolic cap of width \(1/n\) converge in \(L^2(-1,1)\) to \(|x|\), which is not \(C^1\). Continuous, smooth, and test functions form dense — but not closed — subspaces; completeness forces admitting every \(L^2\) limit.

Orthonormal bases & completeness

Plane waves are orthonormal. For integers \(k_1,k_2\): \(\langle e^{ik_1x}|e^{ik_2x}\rangle=\int_0^{2\pi}e^{i(k_2-k_1)x}\dd x=2\pi\,\delta_{k_1k_2}\), so \(\{(2\pi)^{-1/2}|e^{ikx}\rangle\}_{k\in\Z}\) is orthonormal in \(L^2(0,2\pi)\). In \(d\) dimensions the integral factorizes coordinate-by-coordinate (Fubini): \(\langle e^{i\vv k\cdot\vv x}|e^{i\vv n\cdot\vv x}\rangle=(2\pi)^d\,\delta_{\vv k\vv n}\) on \([0,2\pi]^d\).

Orthonormal basis. \(\{|e_i\rangle\}\) with \(\langle e_i|e_j\rangle=\delta_{ij}\) such that every \(|h\rangle\) is an \(L^2\)-convergent sum \(\sum_ia_i|e_i\rangle\); necessarily \(a_j=\langle e_j|h\rangle\). Every Hilbert space has one (Gram–Schmidt, run forever). Completeness restated as an operator identity:

Parseval. \(\|h\|^2=\sum_i|\langle e_i|h\rangle|^2\) and \(\langle h_1|h_2\rangle=\sum_i\langle h_1|e_i\rangle\langle e_i|h_2\rangle\) (absolutely convergent) — norms and inner products are computed coordinate-wise in any orthonormal basis.

Fourier bases. Orthonormal bases: \(\{(2\pi)^{-1/2}e^{ikx}\}_{k\in\Z}\) of \(L^2(0,2\pi)\) and its \(d\)-dimensional version; shifted/scaled \(\{(b-a)^{-1/2}e^{2\pi ik(x-a)/(b-a)}\}_{k\in\Z}\) of \(L^2(a,b)\); sines \(\{\sqrt2\sin(k\pi x)\}_{k\ge1}\) of \(L^2(0,1)\); cosines \(\{1\}\cup\{\sqrt2\cos(k\pi x)\}_{k\ge1}\) of \(L^2(0,1)\) — cosine completeness via the even extension to \((-1,1)\), which turns exponentials into cosines.

Example (step function). \(H=-1\) on \((0,\pi)\), \(+1\) on \((\pi,2\pi)\): \(H=\sum_{k\ \mathrm{odd},\,k\ge1}\big(-\tfrac4{\pi k}\big)|\sin(kx)\rangle\) — a discontinuous function as an \(L^2\)-limit of smooth ones. Parseval with \(\|H\|^2=2\pi\) evaluates the series \(\sum_{k\ \mathrm{odd}}k^{-2}=\pi^2/8\).

Linear functionals & distributions

Continuous linear functional. \(\varphi\colon H\to\C\) linear with \(|\varphi(v)|\le C\|v\|\); equivalent to sequential continuity (\(v_n\to v\Rightarrow\varphi(v_n)\to\varphi(v)\)), and lets \(\varphi\) pass through infinite sums term by term. Automatic in finite dimensions; in infinite dimensions it weeds out pathological functionals.

Bras & Riesz representation. Each \(|w\rangle\) defines the bra \(\langle w|\colon|v\rangle\mapsto\langle w|v\rangle\) (kets = column vectors, bras = conjugated row vectors; all composition is matrix multiplication). Riesz representation theorem: on a Hilbert space these are all of them, \(H^*=\{\langle w|:|w\rangle\in H\}\) — every bra comes from a ket.

Test functions. \(\mathcal F\) = smooth functions vanishing outside a bounded set (wave functions). A subspace of \(L^2(\R^n)\), dense but not closed: every \(L^2\) function is an \(L^2\)-limit of test functions.

Distributions. Linear functionals on \(\mathcal F\) continuous in the weaker sense \(|\langle\varphi|f\rangle|\le C\sum_{k\le N}\max|\nabla^kf|\) (constants depending on the support radius). Strictly more than bras: the Dirac delta \(\langle\delta|f\rangle:=f(0)\), written \(\int\delta(x)f(x)\dd x=f(0)\) although \(\delta\) is no function — a bra with no ket, meaningful only under an integral; rapidly growing multipliers like \(\langle e^{|x|^2}|\) (finite against compactly supported \(f\)); the principal value \(\langle\mathrm{P.V.}\tfrac1x|f\rangle=\lim_{\varepsilon\to0^+}\int_\varepsilon^1\tfrac{f(x)-f(-x)}x\dd x\), convergent by Taylor cancellation, not absolutely. Every \(L^2\) function is a distribution.

Distributional derivative. \(\langle\varphi'|f\rangle:=-\langle\varphi|f'\rangle\) (integration by parts; boundary terms vanish on \(\mathcal F\)) — every distribution, hence every \(L^2\) function, is differentiable as a distribution. The Heaviside step \(H=\mathbf 1_{x\gt0}\) has \(\langle H'|=\langle\delta|\).

Operators, adjoints & projections

Continuous linear operator. \(T\colon H_1\to H_2\) linear with \(\|Tv\|\le C\|v\|\); same equivalences as functionals; passes through infinite sums. Integral operators \((Kf)(x)=\int K(x,y)f(y)\dd y\) are continuous whenever the kernel is square-integrable: Cauchy–Schwarz twice gives \(\|Kf\|\le\|K\|_{L^2}\|f\|\).

Ket-bras & the sandwich formula. The outer product \(|f\rangle\langle e|\colon|v\rangle\mapsto|f\rangle\langle e|v\rangle\) ("column times row" = matrix). With orthonormal bases \(\{|e_j\rangle\}\) of \(H_1\), \(\{|f_i\rangle\}\) of \(H_2\), every continuous operator is an infinite matrix: \(T=\mathrm{Id}\,T\,\mathrm{Id}=\sum_{i,j}a_{ij}|f_i\rangle\langle e_j|\) with \(a_{ij}=\langle f_i|T|e_j\rangle\). Diagonal sums \(\sum_i\lambda_i|f_i\rangle\langle e_i|\) are continuous iff \(\{\lambda_i\}\) is bounded. E.g. the position operator \(\hat x\) on \(L^2(0,1)\) in the plane-wave basis: \(a_{kk}=\tfrac12\), \(a_{jk}=\tfrac1{2\pi i(k-j)}\) off the diagonal.

Adjoint. \(T^\dagger\colon H_2\to H_1\) defined by \(\langle v|T|w\rangle=\overline{\langle w|T^\dagger|v\rangle}\); the matrix of \(T^\dagger\) is the conjugate transpose, \(\big(\sum a_{ij}|f_i\rangle\langle e_j|\big)^\dagger=\sum\overline{a_{ij}}\,|e_j\rangle\langle f_i|\). Usually fastest from the definition: \(\langle f|\hat x|g\rangle=\int\overline f\,x\,g=\overline{\langle g|\hat x|f\rangle}\Rightarrow\hat x^\dagger=\hat x\), no matrix needed. Hermitian (self-adjoint): \(T^\dagger=T\). Unitary: \(U^\dagger=U^{-1}\) (preserves inner products; e.g. multiplication by \(e^{i\theta(x)}\), \(\theta\) real). Normal: \(T^\dagger T=TT^\dagger\).

Compact operators. "Almost finite-dimensional." A diagonal operator \(\sum_i\lambda_i|f_i\rangle\langle e_i|\) is compact iff \(\lambda_i\to0\); finite-rank operators are compact; Hilbert–Schmidt operators (\(\sum_{ij}|a_{ij}|^2\lt\infty\), equivalently integral operators with \(L^2\) kernel) are compact. Not compact if the diagonal fails to vanish — e.g. \(\hat x\). Heuristic: compact = gains regularity (smoothing, e.g. integration) on a bounded domain.

Kernels invert boundary value problems. \(K(x,y)=1-\max(x,y)\) on \(L^2(0,1)\): \(g=Kf\) solves \(g''=-f\) with \(g(1)=g'(0)=0\) — integral kernels are the inverses (Green's functions) of differential operators plus boundary conditions.

Orthogonal projections. \(P^2=P=P^\dagger\). Onto the closed span of an orthonormal set: \(P=\sum_i|e_i\rangle\langle e_i|\); \(\mathrm{Id}-P\) projects onto \(V^\perp=\{w:\langle v|w\rangle=0\ \forall v\in V\}\). A subspace admits a projection iff it is closed (contains its limits) iff it is itself a Hilbert space; dense-but-not-closed subspaces like \(C(0,1)\) or \(\mathcal F\) admit none. Examples: onto symmetric matrices, \(P|A\rangle=\tfrac12(A+A^t)\), complement the antisymmetric matrices; onto \(\{f\in L^2(0,\infty):\langle e^{-x}|f\rangle=0\}\), \(P=\mathrm{Id}-2|e^{-x}\rangle\langle e^{-x}|\) (normalize the constraint vector, subtract its ket-bra).

Spectra: three ways invertibility fails

Spectrum. \(\sigma(T)=\{\lambda\in\C: T-\lambda\,\mathrm{Id}\ \text{is not invertible}\}\). Three failure modes: (i) point spectrum — \(\lambda\) is an eigenvalue, \(T|v\rangle=\lambda|v\rangle\) with \(|v\rangle\ne0\) (injectivity fails); (ii) continuous spectrum — no eigenvector, but \(\operatorname{range}(T-\lambda)\) is a dense proper subspace (invertible on almost the whole space, never all of it); (iii) residual spectrum — the range is not even dense. Normal operators have no residual spectrum (a residual \(\lambda\) would give an eigenvector of \(T^\dagger\), hence of \(T\)). For normal \(T\), \(\lambda\) lies in the continuous spectrum iff it has approximate eigenvectors: unit \(|v_n\rangle\) with \((T-\lambda)|v_n\rangle\to0\).

General facts. For a continuous normal operator, \(\sigma(T)\) is non-empty, bounded, and closed. Hermitian \(\Rightarrow\sigma\subset\R\) (\(\lambda\langle v|v\rangle=\langle v|T|v\rangle=\overline\lambda\langle v|v\rangle\)); unitary \(\Rightarrow\sigma\subset\{|z|=1\}\) (norm preservation forces \(|\lambda|=1\), also for approximate eigenvalues).

Spectral theorem, compact normal case. A compact normal operator has an orthonormal basis of eigenvectors: \(T=\sum_i\lambda_i|v_i\rangle\langle v_i|\), each nonzero eigenvalue of finite multiplicity, \(\lambda_i\to0\) — infinite-dimensional diagonalization that behaves exactly like matrices.

Where eigenvalues fail. Multiplication by \(e^{ix}\) on \(L^2(0,1)\) is unitary with no eigenvalues: \((e^{ix}-\lambda)f=0\) forces \(f\) to be an impulse at a single point — a Dirac delta, not a function. Yet \(\sigma=\{e^{ix}:x\in[0,1]\}\), all continuous spectrum: the space is too small to hold the eigenvectors but the spectrum still sees them. Likewise the position operator on \(L^2(\R)\) has purely continuous spectrum \(\R\) with generalized eigenfunctions \(|\delta_\lambda\rangle\). At the other extreme, \(\mathrm{Id}\) has the single eigenvalue \(1\) with infinite multiplicity.

Spectral theorem & functions of operators

Projection-valued decomposition. For a unitary \(U\) (Hermitian \(H\)) there is a unique family of orthogonal projections \(P_\theta\) (\(P_\lambda\)), increasing, commuting with the operator, running from the zero projection to \(\mathrm{Id}\), with

Continuous eigenbasis (informal). The projections are the rigorous objects; for computation write \(\dd P_\lambda=|v_\lambda\rangle\langle v_\lambda|\dd\lambda\), a continuous basis of generalized (approximate) eigenvectors — possibly distributions — with continuous completeness \(\int\dd P_\lambda=\mathrm{Id}\) replacing \(\sum_i|e_i\rangle\langle e_i|=\mathrm{Id}\), and monotonicity of \(P_\lambda\) playing the role of orthogonality. Individual objects \(|v_\lambda\rangle\langle v_\lambda|\) are not honest projections; only integrals against them are meaningful.

Functions of operators. Apply \(f\) to the spectrum: \(f(T)=\sum_if(\lambda_i)|v_i\rangle\langle v_i|\) (discrete) or \(f(T)=\int f(\lambda)\,\dd P_\lambda\) (continuous) — powers, hence power series, act on eigenvalues only, never on normalization constants. Matrices: diagonalize \(A=SDS^{-1}\), then \(f(A)=S\,f(D)\,S^{-1}\), e.g. \(e^{tA}\) by exponentiating eigenvalues. For Hermitian \(H\), \(e^{itH}=\int e^{it\lambda}\dd P_\lambda\) is unitary since \(|e^{it\lambda}|=1\) — the mathematics behind quantum time evolution \(e^{-itH/\hbar}\).

Unbounded operators & the Fourier transform

Definition. Differentiation makes no sense on all of \(L^2\); an unbounded operator is defined only on the dense test space, \(T\colon\mathcal F\to\mathcal F\), linear plus a closability condition standing in for continuity. Examples: \(\partial_x\) (test functions are smooth); multiplication by growing functions — \(e^{x^2}\) maps \(\mathcal F\to\mathcal F\) but \((1+x^2)^{-1}\in L^2(\R)\) while \(e^{x^2}(1+x^2)^{-1}\notin L^2\), so no continuous extension exists.

Adjoints by integration by parts. On \(\mathcal F\) boundary terms vanish, so \((\partial_x)^\dagger=-\partial_x\) (anti-hermitian); hence \(\hat p:=-i\hbar\,\partial_x\) is Hermitian (the two sign flips cancel), \(\hat x^\dagger=\hat x\), \((\hat p^2/2M)^\dagger=\hat p^2/2M\). Hermitian unbounded operators still have real spectrum and obey the spectral theorem above.

Commutators & uncertainty. \([A,B]:=AB-BA\); the product rule gives \([\hat x,\hat p]=i\hbar\,\mathrm{Id}\) (a conjugate pair). For any Hermitian conjugate pair and unit \(\psi\): \(0\le\|(A+izB)\psi\|^2=\avg{A^2}-\hbar z+z^2\avg{B^2}\) for all real \(z\), so the discriminant forces \(\avg{A^2}\avg{B^2}\ge\hbar^2/4\); applied to \(A-\avg A\), \(B-\avg B\) (still conjugate): \(\sigma_A^2\sigma_B^2\ge\hbar^2/4\), where \(\avg A=\langle\psi|A|\psi\rangle\) and \(\sigma_A^2=\avg{A^2}-\avg A^2\) — the Heisenberg uncertainty relation as pure operator theory.

Fourier transform. \(\hat f(\xi)=\int e^{-2\pi i\xi x}f(x)\dd x\), inverse \(\check f(x)=\int e^{2\pi ix\xi}f(\xi)\dd\xi\); inversion rests on the distributional identity \(\int e^{2\pi i\xi(x-y)}\dd\xi=\delta(x-y)\). The transform is unitary on \(L^2(\R)\). Rescaled version \(\hat\psi(p)=(2\pi\hbar)^{-1/2}\int e^{-ipx/\hbar}\psi(x)\dd x\): same structure, momentum in place of frequency.

Position & momentum diagonalized. Neither has true eigenfunctions in \(L^2(\R)\): \((x-\lambda)\psi=0\) forces \(\psi\propto\delta_\lambda\); \(-i\hbar\psi'=\lambda\psi\) gives \(e^{i\lambda x/\hbar}\), never normalizable (\(\|e^{i\lambda x/\hbar}\|^2=\int1\dd x=\infty\)), with non-real \(\lambda\) rejected outright (exponential growth). Both have purely continuous spectrum \(\R\) and distributional eigenbases

normalizations fixed by demanding \(\int\dd P=\mathrm{Id}\) (sandwich between test functions). Spectral projections: for \(\hat x\), \(P_y\) multiplies by \(\mathbf 1_{x\le y}\) — cuts off in position; for \(\hat p\), \(P_y\) zeroes the Fourier transform above \(p=y\) — cuts off in momentum. \(\hat p^2/2M\) has spectrum \([0,\infty)\), each \(\lambda\gt0\) doubly degenerate (\(e^{\pm i\sqrt{2M\lambda}\,x/\hbar}\)).

Boundary value problems

Model problem. \(-\psi''=\lambda\psi\) on \((0,a)\), \(\psi(0)=\psi(a)=0\): the constant/exponential branches die on the boundary conditions, leaving \(\psi_n=\sqrt{2/a}\,\sin(n\pi x/a)\), \(\lambda_n=n^2\pi^2/a^2\), \(n=1,2,\dots\) Because \(\{\psi_n\}\) is an orthonormal basis of \(L^2(0,a)\), the list is complete: any further eigenfunction expands in it and integration by parts forces it to be a multiple of one \(\psi_N\). Scaled by \(\hbar^2/2m\) this is the particle-in-a-box Hamiltonian, \(E_n=\hbar^2\pi^2n^2/(2ma^2)\).

Dirichlet Laplacian. On bounded \(\Omega\subset\R^d\): \(-\Delta\psi=\lambda\psi\) in \(\Omega\), \(\psi=0\) on \(\partial\Omega\). Uniqueness for \(-\Delta\psi=f\): the difference \(w\) of two solutions satisfies \(0=-\int_\Omega w\,\Delta w=\int_\Omega|\nabla w|^2\) (divergence theorem, boundary term zero), so \(w\equiv0\). The solution operator \((-\Delta)^{-1}\) is Hermitian and compact (gains a derivative, \(\int|\nabla\psi|^2\le\|\psi\|\|f\|\), on a bounded domain), so the compact spectral theorem applies: \(-\Delta=\sum_iE_i|\psi_i\rangle\langle\psi_i|\) with eigenfunctions an orthonormal basis of \(L^2(\Omega)\), each \(E_i=\int|\nabla\psi_i|^2\gt0\) of finite multiplicity, \(E_i\to\infty\), and no continuous spectrum.

Separation of variables. Guess a product solution; the PDE splits into ODEs equated through a shared separation constant \(C_{\mathrm{sep}}\); split the boundary conditions too. Solve the easy equation first, casing on the sign of \(C_{\mathrm{sep}}\): Dirichlet or periodic conditions kill the linear (\(C_{\mathrm{sep}}=0\)) and real-exponential (\(C_{\mathrm{sep}}\lt0\)) branches and quantize the oscillatory one.

Rectangle. \([0,a]\times[0,b]\): \(\psi_{nm}=\tfrac2{\sqrt{ab}}\sin\tfrac{n\pi x}a\sin\tfrac{m\pi y}b\), \(\lambda_{nm}=\pi^2\big(\tfrac{n^2}{a^2}+\tfrac{m^2}{b^2}\big)\), \(n,m\ge1\) — an orthonormal basis, so the spectrum is completely found.

Disc (radius \(R\)). Polar Laplacian \(\Delta=\partial_r^2+\tfrac1r\partial_r+\tfrac1{r^2}\partial_\theta^2\). \(2\pi\)-periodicity forces the angular part \(e^{im\theta}\), \(C_{\mathrm{sep}}=m^2\), \(m\in\Z\); the radial equation \(r^2\psi''+r\psi'+(\lambda r^2-m^2)\psi=0\) becomes Bessel's equation under \(r\mapsto\sqrt\lambda\,r\). Regularity at \(0\) discards \(Y_m\) (\(\sim r^{-m}\)); the boundary condition pins \(\sqrt\lambda\,R\) to a zero \(j_{m,n}\) of \(J_m\): \(\lambda_{nm}=j_{m,n}^2/R^2\), \(\psi_{nm}=J_m(j_{m,n}r/R)\,e^{im\theta}\), with \(\pm m\) degenerate for \(m\ne0\).

Ball (radius \(R\)). Spherical coordinates (volume element \(\rho^2\sin\theta\,\dd\rho\,\dd\theta\,\dd\varphi\)), \(\Delta=\partial_\rho^2+\tfrac2\rho\partial_\rho+\tfrac1{\rho^2}\big[\partial_\theta^2+\cot\theta\,\partial_\theta+\tfrac1{\sin^2\theta}\partial_\varphi^2\big]\). Separate twice: azimuthal part \(e^{im\varphi}\); the polar equation under \(x=\cos\theta\) becomes the generalized Legendre equation, regular at the poles only for \(C_{\mathrm{sep}}=\ell(\ell+1)\), \(\ell\ge|m|\) — the angular parts combine into spherical harmonics \(Y_\ell^m\); the radial equation \(\rho^2\psi''+2\rho\psi'+(\lambda\rho^2-\ell(\ell+1))\psi=0\) is the spherical Bessel equation. Result: \(\psi_{n\ell m}=j_\ell(x_{n,\ell}\,\rho/R)\,Y_\ell^m(\theta,\varphi)\), \(\lambda_{n\ell}=x_{n,\ell}^2/R^2\) with \(x_{n,\ell}\) the \(n\)-th zero of \(j_\ell\); each \((n,\ell)\) eigenvalue is \((2\ell+1)\)-fold degenerate (\(m=-\ell,\dots,\ell\)).

Special functions

Frobenius method. For ODEs with polynomial coefficients seek \(y=x^k\sum_{n\ge0}a_nx^n\), \(a_0\ne0\): the lowest power fixes \(k\) (indicial equation), the remaining coefficients obey a two-step recurrence. A terminating recurrence gives polynomial solutions; a non-terminating one fixes the radius of convergence (ratio test).

Bessel functions. \(x^2y''+xy'+(x^2-m^2)y=0\); general solution \(AJ_m+BY_m\) with \(J_m\sim(x/2)^m/m!\) regular at \(0\) and \(Y_m\sim x^{-m}\) singular. Series \(J_m(x)=\sum_{\ell\ge0}\frac{(-1)^\ell}{\ell!\,(\ell+m)!}\big(\tfrac x2\big)^{2\ell+m}\); real order by replacing \((\ell+m)!\) with \(\Gamma(\ell+m+1)\). Oscillatory with slowly decaying amplitude: \(J_m(x)\approx\sqrt{\tfrac2{\pi x}}\cos\big(x-\tfrac{m\pi}2-\tfrac\pi4\big)\) for \(x\gg1\). Recurrence \(J_{m-1}+J_{m+1}=\tfrac{2m}xJ_m\) (any real order); \(J_{1/2}=\sqrt{2/\pi x}\,\sin x\), \(J_{-1/2}=\sqrt{2/\pi x}\,\cos x\). The zeros \(j_{m,n}\) set the disc eigenvalues.

Spherical Bessel functions. \(x^2y''+2xy'+(x^2-\ell(\ell+1))y=0\); the substitution \(y=x^{-1/2}F\) turns it into Bessel's equation of order \(\ell+\tfrac12\), so \(j_\ell(x)=\sqrt{\tfrac\pi{2x}}\,J_{\ell+1/2}(x)\) (singular partner \(y_\ell\)). Series \(j_\ell(x)=\sum_{n\ge0}\frac{(-1)^n\,x^{2n+\ell}}{(2n)!!\,(2n+2\ell+1)!!}\), so \(j_\ell\sim x^\ell/(2\ell+1)!!\) at \(0\). Closed forms: \(j_0=\tfrac{\sin x}x\), \(j_1=\tfrac{\sin x}{x^2}-\tfrac{\cos x}x\); same recurrence \(j_{\ell-1}+j_{\ell+1}=\tfrac{2\ell}xj_\ell\). Amplitude decays like \(x^{-1}\) (vs \(x^{-1/2}\) for \(J_m\)).

Legendre. Generalized Legendre equation \((1-x^2)y''-2xy'+\big[\lambda-\tfrac{m^2}{1-x^2}\big]y=0\) on \((-1,1)\); solutions regular at \(x=\pm1\) exist only for \(\lambda=\ell(\ell+1)\), integer \(\ell\ge|m|\): peel off the singular behaviour with \(y=(1-x^2)^{m/2}F(x)\), then the recurrence \(a_{n+2}=\frac{(m+n)(m+n+1)-\lambda}{(n+1)(n+2)}a_n\) must terminate (a non-terminating series has radius of convergence \(1\) and blows up at \(\pm1\)), leaving \(F\) a polynomial of degree \(\ell-m\). The solutions are the associated Legendre functions, via the Rodrigues formula

Legendre polynomials \(P_\ell=P_\ell^0\): \(P_0=1\), \(P_1=x\), \(P_2=\tfrac12(3x^2-1)\), \(P_3=\tfrac12(5x^3-3x)\); also \(P_1^1=-\sqrt{1-x^2}\), \(P_2^2=3(1-x^2)\). For odd \(m\), \(P_\ell^m\) is not a polynomial (carries a \(\sqrt{1-x^2}\) factor); \(P_\ell^m=0\) for \(m\gt\ell\); conventions differ by sign/normalization, with \(P_\ell^{-m}\propto P_\ell^m\).

Spherical harmonics. \(Y_\ell^m(\theta,\varphi)=\sqrt{\tfrac{2\ell+1}{4\pi}\tfrac{(\ell-|m|)!}{(\ell+|m|)!}}\,P_\ell^m(\cos\theta)\,e^{im\varphi}\) — the angular eigenfunctions on the sphere, an orthonormal basis of \(L^2(S^2)\) with surface measure \(\sin\theta\,\dd\theta\,\dd\varphi\). First few: \(Y_0^0=\tfrac1{\sqrt{4\pi}}\), \(Y_1^0=\sqrt{\tfrac3{4\pi}}\cos\theta\), \(Y_1^{\pm1}=\mp\sqrt{\tfrac3{8\pi}}\sin\theta\,e^{\pm i\varphi}\), \(Y_2^0=\sqrt{\tfrac5{16\pi}}(3\cos^2\theta-1)\).

Gamma function. \(\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\dd t\) for \(x\gt0\); \(\Gamma(x+1)=x\Gamma(x)\) (integration by parts), so \(\Gamma(n)=(n-1)!\) — the factorial in Bessel series of real order. Double factorial: \(n!!=n(n-2)(n-4)\cdots\) down to \(2\) or \(1\).