Electromagnetic Boundary Problems

B    Special Functions

In this appendix, we will summarize some useful properties of special functions that are of use in applications. We make no attempt at completeness, nor even coherence. For complete collections of such properties, the reader should consult the literature.¹

B.1    GAMMA FUNCTION

The Gamma function Γ(z) is defined for any z with positive real part by the integral

$\Gamma \left(z\right)={\displaystyle {\int }_0^{\infty }{t}^{z-1}{e}^{-t}}dt$

(B.1)

When ν = m, an integer, we can evaluate the integral explicitly to get

$\Gamma \left(m+1\right)=m!$

(B.2)

Various identities follow from its definition, notably

$\Gamma \left(z+1\right)=z\Gamma \left(z\right)$

(B.3)

$\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$

(B.4)

$\Gamma \left(z\right)\Gamma \left(1-z\right)=\frac{\pi }{\sin \pi z}$

(B.5)

B.2    BESSEL FUNCTIONS

Bessel’s differential equation

$z^2\frac{d^{2}f}{d{z}^2}+z\frac{df}{dz}+\left(z^2-{\nu }^2\right)f=0$

(B.6)

is a linear, second-order ...