Special Functions

Special functions are important objects in both mathematics and physics. First we introduce the gamma function Γ(z) as a continuous generalization of n! and prove the beta function identity, Euler’s reflection formula, and the product formula of the gamma function. Then we introduce the Gauss hypergeometric function as the power series solution of the Gauss hypergeometric equation and prove Euler’s integral representation. Moreover, Jacobi polynomials are introduced from the finite-sum cases of the Gauss hypergeometric function and their orthogonality is proved. Legendre orthogonal polynomials are discussed in detail. Weierstrass’s elliptic function ℘(z) is a doubly periodic function with second-order poles, which will be used later in solving nonlinear partial differential equations. Weierstrass’s zeta function ζ(z) is an integral of −℘(z), that is, ζ′(z)=−℘(z). Moreover, Weierstrass’s sigma function σ(z) satisfies σ′(z)/σ(z)=ζ(z). We discuss these functions and their properties in this chapter to a certain depth. Finally, we present Jacobi’s elliptic functions $\operatorname{sn}(z\mid m), \operatorname{cn}(z\mid m)$ , and $\operatorname {dn}(z\mid m)$ , and we derive the nonlinear ordinary differential equations that they satisfy. These functions are also very useful in solving nonlinear partial differential equations.