Special Functions

Mathematical Principles & Theorems

Investigates classical transcendental and special functions: (1) Gamma Function \(\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt\), extending factorials \(\Gamma(n+1) = n!\) and satisfying reflection formula \(\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}\); (2) Bessel Functions of the first kind \(J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\Gamma(m+\alpha+1)}(\frac{x}{2})^{2m+\alpha}\); (3) Legendre Polynomials \(P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n\); (4) Error function \(\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt\).

Operating Instructions

• Select special function: Gamma \(\Gamma(x)\), Bessel \(J_n(x)\), Legendre \(P_n(x)\), or Error Function \(\text{erf}(x)\).
• Set evaluation range \([x_{\min}, x_{\max}]\) and function order parameter \(n\) using numeric inputs.
• Inspect the high-precision 2D curve plot showing asymptotic trends, zeros, and singularity poles.
• Review numerical tables of evaluated values, series approximations, and orthogonality relations.