Special Functions

Gamma, Bessel, Legendre and friends. The named functions that solve the equations of engineering: gamma, beta and error functions; Bessel functions for circular geometry; and Legendre, Chebyshev, Hermite and Laguerre orthogonal polynomials with Gaussian quadrature.

• Gamma, Beta & Error Functions — The gamma function extends the factorial to real and complex numbers, the beta function packages a family of definite integrals, and the error function is the integral of the bell curve. Learn their definitions, key identities (Γ(n+1) = n!, Γ(½) = √π, B = ΓΓ/Γ), how to evaluate nasty integrals with them, and where they appear in heat conduction, diffusion, probability and reliability engineering.
• Bessel Functions — Bessel functions are the natural 'sines and cosines' of problems with circular or cylindrical geometry. Learn Bessel's equation, the series for Jₙ(x), the second-kind Yₙ and modified Iₙ, Kₙ functions, zeros and orthogonality, and apply them to drum vibration, heat flow in cylinders and fins, and FM spectra.
• Legendre & Orthogonal Polynomials — Legendre, Chebyshev, Hermite and Laguerre polynomials are families that are mutually orthogonal under a weighted inner product. Learn their generating equations, Rodrigues' formula, recurrences, orthogonality integrals, and why they power Gaussian quadrature, least-squares fitting, spectral methods, spherical potentials and quantum mechanics.