Special Functions Catalog
A reference catalog of the special functions of mathematical physics, classical analysis, and numerical computation. Primary references: NIST Digital Library of Mathematical Functions (DLMF), Olver, Lozier, Boisvert, and Clark (eds.) (2010). NIST Handbook of Mathematical Functions, Cambridge University Press; Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions, NBS; Gradshteyn-Ryzhik (2014); Erdélyi et al. Higher Transcendental Functions (Bateman manuscript project, 1953); Watson (1922) Theory of Bessel Functions; Whittaker-Watson (1927) A Course of Modern Analysis.
Notation: z, w ∈ C; ν, μ orders; n, m integers; Pochhammer (a)_n = Γ(a+n)/Γ(a) = a(a+1)…(a+n-1) with (a)_0 = 1.
1. Gamma and related functions
1.1 Gamma Γ(z)
Euler integral (Re z > 0): Γ(z) = ∫_0^∞ t^{z-1} e^{-t} dt
Properties:
- Recurrence: Γ(z+1) = z Γ(z); Γ(n+1) = n!
- Reflection (Euler): Γ(z) Γ(1-z) = π / sin(π z)
- Duplication (Legendre): Γ(z) Γ(z + 1/2) = sqrt(π) 2^{1-2z} Γ(2z)
- Multiplication (Gauss): Γ(nz) = (2π)^{(1-n)/2} n^{nz - 1/2} Π_{k=0}^{n-1} Γ(z + k/n)
- Weierstrass product: 1/Γ(z) = z e^{γz} Π_{n=1}^∞ (1 + z/n) e^{-z/n}
- Stirling asymptotic: log Γ(z) ~ (z - 1/2) log z - z + (1/2) log(2π) + Σ B_{2k}/(2k(2k-1) z^{2k-1})
- Lanczos approximation (1964): numerical formula good to machine precision
Reference: Artin, E. (1964). The Gamma Function, Holt, Rinehart and Winston.
1.2 Digamma ψ(z) and polygamma ψ^{(m)}(z)
ψ(z) = Γ’(z)/Γ(z) = -γ + Σ_{n=0}^∞ (1/(n+1) - 1/(n+z))
- ψ(1) = -γ (Euler-Mascheroni constant ≈ 0.5772156649)
- ψ(1/2) = -γ - 2 log 2
- Polygamma: ψ^{(m)}(z) = (d/dz)^{m+1} log Γ(z) = (-1)^{m+1} m! Σ_{n=0}^∞ 1/(n+z)^{m+1}
- ψ^{(1)} is the trigamma function
1.3 Beta function B(p, q)
B(p, q) = ∫_0^1 t^{p-1} (1-t)^{q-1} dt = Γ(p) Γ(q) / Γ(p+q), Re p, Re q > 0.
Incomplete beta: B(x; p, q) = ∫_0^x t^{p-1} (1-t)^{q-1} dt; regularized I_x(p, q) = B(x; p, q)/B(p, q) is the CDF of Beta(p, q).
1.4 Incomplete gamma
γ(s, x) = ∫_0^x t^{s-1} e^{-t} dt (lower) Γ(s, x) = ∫_x^∞ t^{s-1} e^{-t} dt (upper) γ(s, x) + Γ(s, x) = Γ(s); regularized P(s, x) = γ(s, x)/Γ(s) is the CDF of Gamma(s, 1).
1.5 Barnes G-function and multiple gamma
Barnes G(z+1) = G(z) Γ(z); G(1) = 1. Multiple gamma functions Γ_n satisfy Γ_{n+1}(z+1) = Γ_n(z) / Γ_{n+1}(z) (Barnes 1900, Vardi 1988).
2. Error and related functions
2.1 Error function erf(z)
erf(z) = (2/sqrt(π)) ∫_0^z e^{-t²} dt
- erf(-z) = -erf(z); erf(∞) = 1
- Complementary: erfc(z) = 1 - erf(z) = (2/sqrt(π)) ∫_z^∞ e^{-t²} dt
- Imaginary: erfi(z) = -i erf(iz) = (2/sqrt(π)) ∫_0^z e^{t²} dt
- Scaled: erfcx(x) = e^{x²} erfc(x); useful for large x to avoid overflow
- Asymptotic: erfc(x) ~ e^{-x²}/(x sqrt(π)) [1 - 1/(2x²) + 1·3/(2x²)² - …]
Reference: NIST DLMF Chapter 7.
2.2 Fresnel integrals
C(x) = ∫_0^x cos(π t² /2) dt; S(x) = ∫_0^x sin(π t²/2) dt; both → 1/2 as x → ∞ (Cornu spiral).
2.3 Dawson function
F(x) = e^{-x²} ∫_0^x e^{t²} dt = (sqrt(π)/2) e^{-x²} erfi(x).
2.4 Voigt profile
Convolution of Gaussian and Lorentzian; related to Faddeeva function w(z) = e^{-z²} erfc(-iz). Used heavily in spectroscopy.
2.5 Exponential integral Ei(x)
Ei(x) = -∫_{-x}^∞ e^{-t}/t dt (Cauchy principal value); E_1(x) = ∫_x^∞ e^{-t}/t dt = -Ei(-x); E_n(x) = ∫_1^∞ e^{-xt}/t^n dt. Cosine integral Ci, sine integral Si, logarithmic integral li(x) = Ei(log x).
3. Bessel functions
3.1 Bessel of the first kind J_ν
Bessel’s ODE: z² y” + z y’ + (z² - ν²) y = 0.
J_ν(z) = Σ_{k=0}^∞ (-1)^k (z/2)^{ν+2k} / (k! Γ(ν+k+1))
Generating: e^{(z/2)(t - 1/t)} = Σ_{n=-∞}^∞ J_n(z) t^n.
- Jacobi-Anger: e^{i z cos θ} = Σ_n i^n J_n(z) e^{i n θ}
- Recurrence: J_{ν-1}(z) + J_{ν+1}(z) = (2ν/z) J_ν(z); J_{ν-1} - J_{ν+1} = 2 J’_ν
- Orthogonality on [0, a]: ∫0^a J_ν(α_m r) J_ν(α_n r) r dr = δ{mn} (a²/2) [J_{ν+1}(α_m a)]² where α_n are zeros
3.2 Bessel of the second kind Y_ν (Neumann / Weber)
Y_ν(z) = [cos(νπ) J_ν(z) - J_{-ν}(z)] / sin(νπ), with limit for integer ν.
Singular at z = 0: Y_0(z) ~ (2/π) log z; Y_n(z) ~ -(n-1)!(2/z)^n /π for n ≥ 1.
3.3 Hankel functions H_ν^(1), H_ν^(2)
H_ν^(1)(z) = J_ν(z) + i Y_ν(z); H_ν^(2)(z) = J_ν(z) - i Y_ν(z). Outgoing/incoming-wave representations.
3.4 Modified Bessel I_ν, K_ν
z² y” + z y’ - (z² + ν²) y = 0. I_ν(z) = i^{-ν} J_ν(iz); K_ν(z) = (π/2) i^{ν+1} H_ν^(1)(iz).
- I_ν(z) > 0 for real z > 0, growing exponentially; K_ν(z) decaying exponentially
- K_{1/2}(z) = sqrt(π/(2z)) e^{-z} (half-integer reduces to elementary)
- I_{1/2}(z) = sqrt(2/(π z)) sinh(z); I_{-1/2}(z) = sqrt(2/(π z)) cosh(z)
3.5 Spherical Bessel j_n, y_n, h_n^{(1,2)}
j_n(z) = sqrt(π/(2z)) J_{n+1/2}(z); y_n(z) = sqrt(π/(2z)) Y_{n+1/2}(z).
- j_0(z) = sin z / z; j_1(z) = sin z / z² - cos z / z
- y_0(z) = -cos z / z; y_1(z) = -cos z / z² - sin z / z
- Plane-wave expansion: e^{i k · r} = Σ_l (2l+1) i^l j_l(kr) P_l(cos θ)
3.6 Riccati-Bessel functions
ψ_n(z) = z j_n(z); χ_n(z) = -z y_n(z); used in Mie scattering theory (Mie 1908).
3.7 Airy Ai(z), Bi(z)
y” = z y. Ai(z) = (1/π) ∫_0^∞ cos(t³/3 + z t) dt Bi(z) = (1/π) ∫_0^∞ [exp(-t³/3 + z t) + sin(t³/3 + z t)] dt
Asymptotic: Ai(x) ~ (1/(2 sqrt(π) x^{1/4})) exp(-(2/3) x^{3/2}) for x → +∞. Solutions to z = z(x) in WKB; appear in caustics, turning-point analysis (Airy 1838).
3.8 Struve, Anger-Weber, Coulomb wave, Kelvin
- Struve H_ν, modified L_ν: inhomogeneous Bessel equation
- Anger J_ν(z) and Weber E_ν(z): integral representations from Anger (1855), Weber (1879)
- Coulomb wave F_l(η, ρ), G_l(η, ρ): solutions to (d²/dρ² + 1 - 2η/ρ - l(l+1)/ρ²) y = 0
- Kelvin ber, bei, ker, kei (real and imaginary parts of J_0, K_0 at e^{iπ/4} x)
4. Hypergeometric and confluent hypergeometric
4.1 Gauss hypergeometric ₂F₁(a, b; c; z)
₂F₁(a, b; c; z) = Σ_{n=0}^∞ (a)_n (b)_n / [(c)_n n!] z^n, |z| < 1.
Euler integral: ₂F₁(a, b; c; z) = [Γ(c)/(Γ(b) Γ(c-b))] ∫_0^1 t^{b-1} (1-t)^{c-b-1} (1-tz)^{-a} dt.
Hypergeometric ODE: z(1-z) y” + [c - (a+b+1) z] y’ - a b y = 0.
Connection formulas, contiguous relations, Pfaff and Euler transformations. Specializations: log, arctan, Legendre, Jacobi, Chebyshev. Gauss summation (1812): ₂F₁(a, b; c; 1) = Γ(c) Γ(c-a-b) / [Γ(c-a) Γ(c-b)].
4.2 Confluent hypergeometric ₁F₁(a; c; z) = M(a, c, z)
M(a, c, z) = Σ_n (a)_n / [(c)_n n!] z^n; Kummer’s transformation M(a, c, z) = e^z M(c-a, c, -z).
Second solution: Tricomi U(a, c, z). Kummer’s ODE: z y” + (c - z) y’ - a y = 0.
Specializations: error function, Bessel I/K, parabolic cylinder, Laguerre.
4.3 Generalized pFq
_pF_q(a_1, …, a_p; b_1, …, b_q; z) = Σ_n [(a_1)_n … (a_p)_n / ((b_1)_n … (b_q)_n n!)] z^n.
Convergence: ∞ if p ≤ q; |z| < 1 if p = q+1; only z = 0 if p > q+1.
4.4 Meijer G
G^{m,n}_{p,q}(z | a_1,…,a_p ; b_1,…,b_q) — Mellin-Barnes contour integral; contains all_pF_q as special cases; basis of Mathematica’s MeijerG. Reference: Meijer, C.S. (1936-1946) series of papers.
4.5 Fox H-function
Further generalization with arbitrary positive coefficients in the contour integrand; arises in fractional calculus, anomalous diffusion (Mainardi 2010).
4.6 q-hypergeometric (basic hypergeometric)
_rφ_s with base q; reduces to hypergeometric as q → 1. Heine series (Heine 1846), Gasper-Rahman (2004).
4.7 Appell, Kampé de Fériet, Horn functions
Multi-variable hypergeometric: Appell F_1, F_2, F_3, F_4; Kampé de Fériet two-variable; Horn series; Lauricella F_A, F_B, F_C, F_D in n variables.
4.8 Mathieu functions
Mathieu’s ODE: y” + (a - 2q cos 2x) y = 0. Solutions: angular ce_n(x, q), se_n(x, q) (periodic for special “characteristic” values a_n(q), b_n(q)) and radial Mc, Ms. Originate from elliptical drum (Mathieu 1868); used in waveguides, Paul traps, RF mass spectrometers, parametric oscillators.
4.9 Heun’s equation
z(z-1)(z-a) y” + [γ(z-1)(z-a) + δ z(z-a) + ε z(z-1)] y’ + (αβ z - q) y = 0; four regular singular points 0, 1, a, ∞. General Heun (confluent, biconfluent, doubly confluent, triconfluent) with explicit local Frobenius series; Heun functions Hn. The most general second-order Fuchsian ODE with four regular singular points (Heun 1889). Appears in black-hole perturbation theory (Mano-Suzuki-Takasugi 1996), quantum dots, mathematical physics.
5. Orthogonal polynomials (classical)
Generic three-term recurrence: x p_n(x) = a_n p_{n+1}(x) + b_n p_n(x) + c_n p_{n-1}(x). Pearson ODE: σ(x) y” + τ(x) y’ + λ y = 0 with σ polynomial degree ≤ 2.
5.1 Legendre P_n(x)
(1-x²) y” - 2x y’ + n(n+1) y = 0; weight w(x) = 1 on [-1, 1]; ‖P_n‖² = 2/(2n+1).
Rodrigues: P_n(x) = (1/(2^n n!)) (d/dx)^n (x² - 1)^n.
- P_0 = 1, P_1 = x, P_2 = (3x² - 1)/2, P_3 = (5x³ - 3x)/2
- Recurrence: (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)
- Generating: 1/sqrt(1 - 2xt + t²) = Σ_n P_n(x) t
Associated Legendre P_l^m(x): (1-x²) y” - 2x y’ + [l(l+1) - m²/(1-x²)] y = 0.
5.2 Spherical harmonics Y_l^m(θ, φ)
Y_l^m(θ, φ) = sqrt((2l+1)/(4π) · (l-m)!/(l+m)!) P_l^m(cos θ) e^{i m φ}
Eigenfunctions of L² and L_z on the unit sphere; orthonormal on S². Foundations: Laplace (1782) on equipotentials.
- Addition theorem: P_l(cos γ) = (4π/(2l+1)) Σ_m Y_l^m*(θ_1, φ_1) Y_l^m(θ_2, φ_2)
- Plane wave: e^{i k · r} = 4π Σ_l Σ_m i^l j_l(kr) Y_l^m*(k̂) Y_l^m(r̂)
5.3 Hermite
Physicists’ H_n(x): y” - 2x y’ + 2n y = 0, weight e^{-x²} on R; ‖H_n‖² = sqrt(π) 2^n n!.
H_n(x) = (-1)^n e^{x²} (d/dx)^n e^{-x²}.
- H_0 = 1, H_1 = 2x, H_2 = 4x² - 2, H_3 = 8x³ - 12x
- Recurrence: H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)
- Generating: e^{2xt - t²} = Σ_n H_n(x) t^n / n!
Probabilists’ He_n(x) = 2^{-n/2} H_n(x/sqrt 2); weight e^{-x²/2}/sqrt(2π); Charlier-Hermite expansion of densities.
Eigenfunctions of quantum harmonic oscillator (Hermite 1864).
5.4 Laguerre
y” + ((1-x)/x) y’ + (n/x) y = 0 on [0, ∞), weight e^{-x}.
L_n(x) = (1/n!) e^x (d/dx)^n (x^n e^{-x})
- L_0 = 1, L_1 = 1 - x, L_2 = (1/2)(x² - 4x + 2)
- Recurrence: (n+1) L_{n+1}(x) = (2n+1-x) L_n(x) - n L_{n-1}(x)
Associated L_n^α(x): weight x^α e^{-x}; ‖L_n^α‖² = Γ(n + α + 1)/n!. Radial part of hydrogen atom (Laguerre 1879).
5.5 Chebyshev T_n, U_n
First kind T_n(cos θ) = cos(n θ); weight 1/sqrt(1 - x²); ‖T_n‖² = π/2 (π for n=0).
- T_0 = 1, T_1 = x, T_2 = 2x² - 1, T_3 = 4x³ - 3x
- T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x)
Second kind U_n(cos θ) = sin((n+1)θ)/sin θ; weight sqrt(1 - x²); ‖U_n‖² = π/2.
Third T̄_n weight sqrt((1-x)/(1+x)); Fourth Ū_n weight sqrt((1+x)/(1-x)).
Minimax approximation property: T_n / 2^{n-1} minimizes sup-norm on [-1, 1] among monic polynomials of degree n. Chebyshev nodes give near-optimal interpolation (Chebyshev 1854).
5.6 Jacobi P_n^{(α, β)}(x)
Weight (1-x)^α (1+x)^β on [-1, 1]; Legendre (α = β = 0), Chebyshev T (α = β = -1/2), U (α = β = 1/2), Gegenbauer (α = β).
P_n^{(α, β)}(x) = (Γ(α + n + 1)/(n! Γ(α + β + n + 1))) Σ_m (n choose m) [Γ(α + β + n + m + 1)/Γ(α + m + 1)] ((x-1)/2)^m.
5.7 Gegenbauer / ultraspherical C_n^λ
Weight (1 - x²)^{λ - 1/2}. C_n^{1/2} = P_n (Legendre), C_n^1 = U_n. Generating: (1 - 2 x t + t²)^{-λ} = Σ_n C_n^λ(x) t^n. Eigenfunctions on hyperspheres.
5.8 Zernike Z_n^m(ρ, φ)
Orthogonal polynomials on unit disk: Z_n^m(ρ, φ) = R_n^m(ρ) e^{i m φ} with n - |m| even. Used to express optical aberrations (Zernike 1934).
5.9 Charlier, Krawtchouk, Meixner, Hahn
Discrete orthogonal polynomial families on N or {0, …, N} (Askey-Wilson scheme):
- Charlier (Poisson weight)
- Krawtchouk (binomial)
- Meixner (negative binomial)
- Hahn / dual Hahn (hypergeometric)
- Continuous Hahn / Wilson
- Askey-Wilson (q-analog, top of Askey scheme; Askey-Wilson 1985)
5.10 Wigner-d, 3j, 6j, 9j symbols
Angular-momentum coupling:
- Wigner-d small d^j_{m’m}(β) — rotation matrix elements about y-axis
- Big D-function: D^j_{m’m}(α, β, γ) = e^{-i m’ α} d^j_{m’m}(β) e^{-i m γ}
- 3j: Wigner-Eckart coupling C(j_1 j_2 j_3 | m_1 m_2 m_3) = (-1)^{j_1 - j_2 - m_3} sqrt(2j_3 + 1) (j_1 j_2 j_3; m_1 m_2 -m_3)
- 6j: Racah W-coefficients; recoupling of three angular momenta
- 9j: recoupling of four angular momenta
Reference: Edmonds, A.R. (1957). Angular Momentum in Quantum Mechanics, Princeton; Varshalovich-Moskalev-Khersonskii (1988).
5.11 Sphere harmonics and tensor harmonics generalizations
Wigner D and spin-weighted spherical harmonics _sY_lm (Newman-Penrose 1966) for gravitational waves and CMB.
6. Elliptic integrals and functions
6.1 Legendre normal forms of elliptic integrals
First kind: F(φ, k) = ∫_0^φ dθ / sqrt(1 - k² sin² θ); complete K(k) = F(π/2, k). Second kind: E(φ, k) = ∫_0^φ sqrt(1 - k² sin² θ) dθ; complete E(k). Third kind: Π(n; φ, k) = ∫_0^φ dθ / [(1 - n sin² θ) sqrt(1 - k² sin² θ)]; complete Π(n, k).
Modulus k ∈ [0, 1]; complementary k’ = sqrt(1 - k²).
Legendre’s relation: E(k) K(k’) + E(k’) K(k) - K(k) K(k’) = π/2.
Arithmetic-geometric mean (Gauss 1799): AGM(a, b) iteration; K(k) = π/(2 AGM(1, k’)). Fast evaluation.
6.2 Carlson symmetric forms
R_F, R_J, R_D, R_C, R_G (Carlson 1979): symmetric in arguments; better numerical properties.
6.3 Jacobi elliptic functions sn, cn, dn
φ = am(u, k); sn(u, k) = sin φ; cn(u, k) = cos φ; dn(u, k) = sqrt(1 - k² sin² φ).
- Doubly periodic: real period 4 K(k), imaginary period 2 i K’(k’) (K’ = K(k’))
- Identities: sn² + cn² = 1; k² sn² + dn² = 1
- (d/du) sn(u) = cn(u) dn(u)
- Twelve Glaisher functions ns, nc, nd, sc, sd, cs, cd, dc, ds, cn, sn, dn
Pendulum motion, mathematical biology (Lotka-Volterra elliptic invariants), KdV solitons.
6.4 Weierstrass ℘-function
℘(z; ω_1, ω_2) = 1/z² + Σ_{(m,n)≠(0,0)} [1/(z - 2 m ω_1 - 2 n ω_2)² - 1/(2 m ω_1 + 2 n ω_2)²]
Doubly periodic; ℘‘(z)² = 4 ℘(z)³ - g_2 ℘(z) - g_3; invariants g_2 = 60 Σ’ 1/Ω^4, g_3 = 140 Σ’ 1/Ω^6.
Universal cover of complex tori; modular forms.
6.5 Theta functions
Jacobi θ_1, θ_2, θ_3, θ_4 (and θ_{00}, θ_{01}, θ_{10}, θ_{11}):
θ_3(z, q) = Σ_{n=-∞}^∞ q^{n²} e^{2 i n z}, |q| < 1.
Jacobi identity: θ_3(0|τ)⁴ = θ_2(0|τ)⁴ + θ_4(0|τ)⁴.
Riemann theta Θ(z, Ω) for g-dimensional torus, Ω positive-imaginary symmetric g×g; foundation of abelian function theory (Riemann 1857).
6.6 Modular forms and Eisenstein series
E_k(τ) = 1 - (2k/B_k) Σ_n σ_{k-1}(n) q^n with q = e^{2πi τ}, k ≥ 4 even. Modular discriminant Δ(τ) = (2π)^{12} η(τ)^{24}; j-invariant j(τ) = E_4³/Δ.
Dedekind eta η(τ) = q^{1/24} Π_n (1 - q^n).
Foundation of class-field theory, monstrous moonshine (Conway-Norton 1979, Borcherds 1992).
7. Riemann zeta and L-functions
7.1 Riemann zeta ζ(s)
ζ(s) = Σ_{n=1}^∞ 1/n^s, Re s > 1.
Euler product: ζ(s) = Π_p (1 - p^{-s})^{-1} (Euler 1737). Functional equation (Riemann 1859):
- ξ(s) = (1/2) s(s-1) π^{-s/2} Γ(s/2) ζ(s) = ξ(1-s)
Trivial zeros at s = -2, -4, …; non-trivial zeros in critical strip 0 < Re s < 1. Riemann hypothesis: all non-trivial zeros have Re s = 1/2 (Clay Millennium Problem, $1M prize).
Values:
- ζ(2) = π²/6 (Basel, Euler 1735)
- ζ(4) = π⁴/90; ζ(2k) = (-1)^{k+1} (2π)^{2k} B_{2k}/(2 (2k)!)
- ζ(0) = -1/2; ζ(-1) = -1/12; ζ(-(2k-1)) = -B_{2k}/(2k)
- ζ(3) Apéry’s constant, irrational (Apéry 1978), ≈ 1.2020569
Riemann-Siegel formula (Siegel 1932 reading Riemann’s Nachlass): efficient evaluation on critical line via ζ(1/2 + it) = Z(t) e^{-i θ(t)} with Z(t) computable in O(sqrt(t)) terms.
Zero counting: N(T) = (T/(2π)) log(T/(2π e)) + (7/8) + O(log T) (Riemann-von Mangoldt).
7.2 Hurwitz zeta ζ(s, a)
ζ(s, a) = Σ_{n=0}^∞ 1/(n + a)^s, Re s > 1, a > 0. ζ(s, 1) = ζ(s); ζ(s, 1/2) = (2^s - 1) ζ(s).
7.3 Dirichlet L-functions
L(s, χ) = Σ_n χ(n)/n^s = Π_p (1 - χ(p) p^{-s})^{-1}, χ a Dirichlet character mod q. L(1, χ) ≠ 0 for non-trivial χ → Dirichlet’s theorem on primes in arithmetic progressions (Dirichlet 1837). Generalized Riemann Hypothesis (GRH): all non-trivial zeros of all L(s, χ) on Re s = 1/2.
7.4 Lerch zeta and Lerch transcendent
Φ(z, s, a) = Σ_{n=0}^∞ z^n / (n + a)^s; contains Hurwitz (z = 1), polylogarithm.
7.5 Polylogarithm Li_s(z)
Li_s(z) = Σ_{n=1}^∞ z^n / n^s, |z| < 1.
- Li_1(z) = -log(1 - z)
- Li_2(z) — dilogarithm (Euler-Spence)
- Li_2(1) = ζ(2); Li_2(1/2) = π²/12 - (log 2)²/2; Li_2(-1) = -π²/12
- Li_3(1) = ζ(3); higher Li_s(1) = ζ(s) for integer s
Five-term identity (Abel 1827): Li_2 satisfies a functional equation in five points. Generalized to motivic / single-valued polylogs (Zagier, Goncharov).
8. Painlevé transcendents
Six Painlevé equations P_I-P_VI (Painlevé 1900, Gambier 1910) — nonlinear second-order ODEs whose solutions have no movable singularities other than poles. Solutions are new transcendental functions (“Painlevé transcendents”):
- P_I: y” = 6 y² + x (no parameters)
- P_II: y” = 2 y³ + x y + α
- P_III: y” = y’²/y - y’/x + (α y² + β)/x + γ y³ + δ/y
- P_IV: y” = y’²/(2 y) + (3/2) y³ + 4 x y² + 2(x² - α) y + β/y
- P_V: rational in y, y’
- P_VI: most general; isomonodromic deformation of Fuchsian system
Applications: random matrix theory (Tracy-Widom 1994: largest eigenvalue distribution F_1, F_2, F_4 in terms of P_II), KdV solitons, 2D Ising correlation functions (Wu-McCoy-Tracy-Barouch 1976), integrable QFT.
Reference: Fokas-Its-Kapaev-Novokshenov (2006). Painlevé Transcendents: The Riemann-Hilbert Approach, AMS.
9. Fractional-calculus and Mittag-Leffler
9.1 Mittag-Leffler
E_α(z) = Σ_{n=0}^∞ z^n / Γ(α n + 1); generalizes exp = E_1.
Two-parameter: E_{α, β}(z) = Σ_n z^n / Γ(α n + β).
Solves fractional-order ODE D^α y = λ y (Caputo or Riemann-Liouville derivative); relaxation in viscoelasticity, anomalous diffusion.
Reference: Mittag-Leffler, M.G. (1903). “Sur la nouvelle fonction E_α(x),” Comptes Rendus, 137, 554-558.
9.2 Wright function W_{α, β}
W_{α, β}(z) = Σ_n z^n /(n! Γ(α n + β)). Mainardi function M_α(z) = W_{-α, 1-α}(-z): Green’s function for fractional diffusion equation.
9.3 Fox H and generalized Mittag-Leffler
Prabhakar function E_{α, β}^γ(z) = Σ_n (γ)_n z^n / (n! Γ(α n + β)); fractional Cattaneo, generalized Maxwell viscoelasticity.
10. Other functions and constants
10.1 Lambert W (product log)
W(z) e^{W(z)} = z. Two real branches W_0 (principal, ≥ -1) and W_{-1} (≤ -1) for z ∈ [-1/e, 0]; complex branches W_k.
- W(1) = Ω = 0.5671432904… (Omega constant)
- W(-1/e) = -1; W(0) = 0
- Series: W(z) = Σ_{n=1}^∞ (-n)^{n-1} z^n / n! (radius of convergence 1/e)
Originated in Lambert’s transcendental equation (1758), Euler (1783). Applications: delay differential equations, enzyme kinetics, quantum chemistry (jellium), generalized Wien displacement law (Corless-Gonnet-Hare-Jeffrey-Knuth 1996).
10.2 Parabolic cylinder D_ν(z)
y” + (ν + 1/2 - z²/4) y = 0; Hermite at non-negative integer ν.
10.3 Whittaker M_{κ, μ}, W_{κ, μ}
y” + (-1/4 + κ/z + (1/4 - μ²)/z²) y = 0; reduces to Kummer.
10.4 Coulomb wave functions F_l, G_l, H_l^{±}
Used in nuclear and atomic scattering; bound and continuum hydrogenic states.
10.5 Tricomi confluent hypergeometric U
Second linearly-independent solution of Kummer’s ODE; log singularity at 0.
10.6 Spheroidal wave functions
Prolate Sl_n^m and oblate; eigenfunctions of D’Alembert operator on prolate/oblate ellipsoids. Used in optics (bandlimited functions, Slepian 1961), antenna theory.
10.7 Mathieu (see 4.8)
10.8 Lamé functions
Solutions of Lamé’s equation (ellipsoidal harmonics), eigenfunctions in ellipsoidal coordinates. Lamé 1837.
10.9 Heun (see 4.9)
10.10 Resurgent functions and trans-series
Écalle’s theory (1981): functions with factorially-divergent asymptotic series whose Borel transforms are resurgent — endless analytic continuation along restricted paths. Stokes phenomena, exponentially-small corrections (Berry-Howls, Costin). Applied in QFT, instantons, asymptotics of partition functions.
11. Quadrature rules and numerical evaluation
11.1 Newton-Cotes
Closed: nodes equispaced including endpoints. Trapezoidal (linear, error O(h²)); Simpson (quadratic, O(h^4)); Boole/Bode (degree 4, O(h^6)). Open variants exclude endpoints.
11.2 Romberg integration (1955)
Richardson extrapolation of trapezoidal rule; T_{n, k} = (4^k T_{n, k-1} - T_{n-1, k-1})/(4^k - 1); converges fast for smooth integrands.
11.3 Gauss-Legendre
n points exact for polynomial degree 2n - 1 on [-1, 1]; nodes are zeros of P_n(x); weights w_i = 2/[(1 - x_i²)(P_n’(x_i))²].
11.4 Gauss-Hermite
∫_{-∞}^∞ e^{-x²} f(x) dx ≈ Σ w_i f(x_i), x_i zeros of H_n. Variant for normal expectations after substitution.
11.5 Gauss-Laguerre
∫_0^∞ e^{-x} f(x) dx ≈ Σ w_i f(x_i), x_i zeros of L_n.
11.6 Gauss-Chebyshev
First-kind (weight 1/sqrt(1-x²)): nodes x_i = cos((2i-1)π/(2n)), weights π/n. Second-kind (weight sqrt(1-x²)): nodes cos(iπ/(n+1)).
11.7 Gauss-Jacobi, Gauss-Gegenbauer
Generalize to weight (1-x)^α (1+x)^β on [-1, 1] or (1-x²)^{λ-1/2}.
11.8 Gauss-Kronrod, Patterson
Embedded rules (n + n+1 point) for error estimation; QUADPACK (Piessens et al. 1983).
11.9 Clenshaw-Curtis
Chebyshev nodes; nested (n → 2n); FFT-evaluable; comparable accuracy to Gauss-Legendre per node for smooth functions, often better in practice (Trefethen 2008).
11.10 Tanh-sinh (double exponential)
Substitution x = tanh((π/2) sinh t); rapid convergence for integrals with endpoint singularities, near-machine-precision in few dozen nodes (Takahasi-Mori 1974).
11.11 Sparse grids (Smolyak 1963)
Tensor-product reduction giving error O(N^{-r} (log N)^{(r-1)(d-1)}) instead of O(N^{-r/d}) in d dims for smooth integrands.
11.12 Adaptive quadrature
Locally refine where estimated error is largest (DEQuad, QUADPACK, MATLAB integral).
12. Asymptotic methods
12.1 Laplace method
∫_a^b e^{N φ(x)} f(x) dx ~ sqrt(2π/(N|φ”(x_0)|)) f(x_0) e^{N φ(x_0)}, with x_0 the maximizer.
12.2 Stationary phase
∫ e^{i N ψ(x)} f(x) dx ~ Σ_k sqrt(2π/(N|ψ”(x_k)|)) e^{i (N ψ(x_k) + π/4 sgn ψ”(x_k))} f(x_k).
12.3 Saddle-point / steepest descent
Deform contour through saddle of analytic phase; quadratic Gaussian approximation gives leading order.
12.4 Watson’s lemma
Asymptotic of Laplace transform ∫_0^∞ e^{-st} f(t) dt for s → ∞: F(s) ~ Σ_n Γ(α + n + 1) a_n / s^{α + n + 1}.
12.5 WKB
Asymptotic of y” + Q(x) y/ε² = 0 in ε → 0; y ~ Q^{-1/4} exp(±(i/ε) ∫ sqrt(Q) dx); turning points handled by Airy connection.
13. Software / Implementations
- DLMF interactive (dlmf.nist.gov)
- SciPy scipy.special — vast coverage of erf, gamma, Bessel, hypergeometric, elliptic, Mathieu, spheroidal, Airy
- mpmath / SymPy — arbitrary-precision evaluation
- Mathematica / Wolfram Language —
MeijerG,Hypergeometric*,EllipticTheta,WeierstrassP,PainleveT* - Boost.Math (C++) — Cephes-derived; certified accuracy bounds
- Maple, Maxima symbolic
- GSL GNU Scientific Library
- AMOS (Amos 1986) Bessel for complex arguments
- Faddeeva package (Johnson 2012) — w(z), erf, erfc, erfcx, erfi, Dawson at machine precision throughout C
- Arb / FLINT rigorous arbitrary-precision ball arithmetic (Johansson 2017)
14. Identities and connection table
- Half-integer Bessel reduces to elementary: J_{n+1/2}, K_{1/2} closed form
- ₂F₁(1, 1; 2; -z) = log(1+z)/z; ₂F₁(1/2, 1/2; 3/2; z²) = arcsin(z)/z; ₂F₁(1/2, 1; 3/2; z²) = arctanh(z)/z
- Legendre: P_l(x) = ₂F₁(-l, l+1; 1; (1-x)/2)
- Bessel-Hermite-Laguerre via Kummer/Whittaker