In applied mathematics, the Kelvin functions ber<sub>ý</sub>(x) and bei<sub>ý</sub>(x) are the real and imaginary parts, respectively, of
where x is real, and , is the ý<sup>th</sup> order Bessel function of the first kind. Similarly, the functions ker<sub>ý</sub>(x) and kei<sub>ý</sub>(x) are the real and imaginary parts, respectively, of
where is the ý<sup>th</sup> 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 ber<sub>n</sub>(x) and bei<sub>n</sub>(x) for integral n, the Kelvin functions have a branch point at x = 0.
Below, is the gamma function and is the digamma function.
For integers n, ber<sub>n</sub>(x) has the series expansion
where is the gamma function. The special case ber<sub>0</sub>(x), commonly denoted as just ber(x), has the series expansion
where
For integers n, bei<sub>n</sub>(x) has the series expansion
The special case bei<sub>0</sub>(x), commonly denoted as just bei(x), has the series expansion
and asymptotic series
where ñ, , and are defined as for ber(x).
For integers n, ker<sub>n</sub>(x) has the (complicated) series expansion
The special case ker<sub>0</sub>(x), commonly denoted as just ker(x), has the series expansion
and the asymptotic series
where
For integer n, kei<sub>n</sub>(x) has the series expansion
The special case kei<sub>0</sub>(x), commonly denoted as just kei(x), has the series expansion
and the asymptotic series
where ò, f<sub>2</sub>(x), and g<sub>2</sub>(x) are defined as for ker(x).