Kelvin Functions

The Kelvin functions ber, bei, ker, and kei have complex extensions specified by the following series on the principal branch and :

			(1)
			(2)
			(3)
			(4)

Here is the gamma function and is the Pochhammer symbol. The first two series hold for all complex numbers , with reciprocal gamma factors interpreted as ; the last two are stated for noninteger , with integer orders obtained by taking the limit of the complete expression. All powers of use the principal branch of the natural logarithm. These series converge for every finite off the branch cut and define analytic functions there.

For real and , the bei and ber functions satisfy the relations introduced by Kelvin:

			(5)
			(6)
			(7)
			(8)

where is a Bessel function of the first kind and is a modified Bessel function of the first kind. These functions satisfy the Kelvin differential equation.

For real and , the functions kei and ker are similarly defined by

(9)

where is a modified Bessel function of the second kind. For the special case ,

			(10)
			(11)