Kelvin functions


In applied mathematics, the Kelvin functions berν and beiν are the real and imaginary parts, respectively, of
where x is real, and, is the νth order Bessel function of the first kind. Similarly, the functions kerν and keiν are the real and imaginary parts, respectively, of
where is the νth order modified Bessel function of the second kind.
These functions are named after William Thomson, 1st Baron Kelvin.
While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments With the exception of bern and bein for integral n, the Kelvin functions have a branch point at x = 0.
Below, is the gamma function and is the digamma function.

ber(''x'')

For integers n, bern has the series expansion
where is the gamma function. The special case ber0, commonly denoted as just ber, has the series expansion
and asymptotic series
where

bei(''x'')

For integers n, bein has the series expansion
The special case bei0, commonly denoted as just bei, has the series expansion
and asymptotic series
where α,, and are defined as for ber.

ker(''x'')

For integers n, kern has the series expansion
The special case ker0, commonly denoted as just ker, has the series expansion
and the asymptotic series
where

kei(''x'')

For integer n, kein has the series expansion
The special case kei0, commonly denoted as just kei, has the series expansion
and the asymptotic series
where β, f2, and g2 are defined as for ker.