Module complex_analysis 

Complex analysis, special functions, and conformal mappings.

Overview

Provides:

• Complex number type with full arithmetic and transcendental functions
• Cauchy integral formula and residue computation
• Conformal mappings (Joukowski, Möbius transformation)
• Contour integration and winding numbers
• Gamma function (Lanczos approximation), Beta function
• Riemann zeta function (Euler–Maclaurin)
• Complete elliptic integrals K(k) and E(k)
• Bessel functions J_n(x) and Y_n(x)
• Legendre polynomials P_n(x) and Hermite polynomials H_n(x)

Structs

Complex

A complex number z = re + i·im.

ContourIntegral

A polygonal contour in the complex plane.

Functions

bessel_j

Bessel function of the first kind J_n(x).

bessel_y

Bessel function of the second kind Y_n(x) (Neumann function).

beta_function

Compute the Beta function B(a, b) = Γ(a)·Γ(b) / Γ(a+b).

cauchy_integral

Compute the Cauchy contour integral ∮ f(z) dz along a polygonal contour.

conformal_map_joukowski

Joukowski conformal mapping: w = z + c²/z.

conformal_map_mobius

Möbius (linear fractional) transformation: w = (a·z + b) / (c·z + d).

elliptic_e

Complete elliptic integral of the second kind E(k).

elliptic_k

Complete elliptic integral of the first kind K(k).

gamma_lanczos

Compute the Gamma function Γ(z) using the Lanczos approximation.

hermite_h

Hermite polynomial H_n(x) (physicists’ convention).

legendre_p

Legendre polynomial P_n(x) evaluated at x ∈ [−1, 1].

residue

Compute the residue of f at a (simple or higher-order) pole.

zeta_euler_maclaurin

Approximate the Riemann zeta function ζ(s) using the Euler–Maclaurin formula.