Kelvin Functions

In applied mathematics, the Kelvin functions ber_''ν''(''x'') and bei_''ν''(''x'') are the real and imaginary parts, respectively, of

:$J_\nu \left (x e^ \right ),\,$

where ''x'' is real, and , is the ''ν''^th order Bessel function of the first kind. Similarly, the functions ker_ν(''x'') and kei_ν(''x'') are the real and imaginary parts, respectively, of

:$K_\nu \left (x e^ \right ),\,$

where is the ''ν''^th 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_''n''(''x'') and bei_''n''(''x'') 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'', ber_''n''(''x'') has the series expansion

:$\mathrm_n(x) = \left(\frac\right)^n \sum_ \frac \left(\frac\right)^k ,$

where is the gamma function. The special case ber₀(''x''), commonly denoted as just ber(''x''), has the series expansion

:$\mathrm(x) = 1 + \sum_ \frac \left(\frac \right )^$

and asymptotic series

:$\mathrm(x) \sim \frac \left (f_1(x) \cos \alpha + g_1(x) \sin \alpha \right ) - \frac$,

where

:$\alpha = \frac - \frac,$
:$f_1(x) = 1 + \sum_ \frac \prod_^k (2l - 1)^2$
:$g_1(x) = \sum_ \frac \prod_^k (2l - 1)^2 .$

bei(''x'')

For integers ''n'', bei_''n''(''x'') has the series expansion

:$\mathrm_n(x) = \left(\frac\right)^n \sum_ \frac \left(\frac\right)^k .$

The special case bei₀(''x''), commonly denoted as just bei(''x''), has the series expansion

:$\mathrm(x) = \sum_ \frac \left(\frac \right )^$

and asymptotic series

:\mathrm(x) \sim \frac _1(x) \sin \alpha - g_1(x) \cos \alpha- \frac,

where α, $f_1(x)$, and $g_1(x)$ are defined as for ber(''x'').

ker(''x'')

For integers ''n'', ker_''n''(''x'') has the (complicated) series expansion

:\begin
&\mathrm_n(x) = - \ln\left(\frac\right) \mathrm_n(x) + \frac\mathrm_n(x) \\
&+ \frac \left(\frac\right)^ \sum_^ \cos\left left(\frac + \frac\right)\pi\right\frac \left(\frac\right)^k \\
&+ \frac \left(\frac\right)^n \sum_ \cos\left left(\frac + \frac\right)\pi\right\frac \left(\frac\right)^k .
\end

The special case ker₀(''x''), commonly denoted as just ker(''x''), has the series expansion

:$\mathrm(x) = -\ln\left(\frac\right) \mathrm(x) + \frac\mathrm(x) + \sum_ (-1)^k \frac \left(\frac\right)^$

and the asymptotic series

:\mathrm(x) \sim \sqrt e^ _2(x) \cos \beta + g_2(x) \sin \beta

where

:$\beta = \frac + \frac,$
:$f_2(x) = 1 + \sum_ (-1)^k \frac \prod_^k (2l - 1)^2$
:$g_2(x) = \sum_ (-1)^k \frac \prod_^k (2l - 1)^2.$

kei(''x'')

For integer ''n'', kei_''n''(''x'') has the series expansion

:\begin
&\mathrm_n(x) = - \ln\left(\frac\right) \mathrm_n(x) - \frac\mathrm_n(x) \\
&-\frac \left(\frac\right)^ \sum_^ \sin\left left(\frac + \frac\right)\pi\right\frac \left(\frac\right)^k \\
&+ \frac \left(\frac\right)^n \sum_ \sin\left left(\frac + \frac\right)\pi\right\frac \left(\frac\right)^k .
\end

The special case kei₀(''x''), commonly denoted as just kei(''x''), has the series expansion

:$\mathrm(x) = -\ln\left(\frac\right) \mathrm(x) - \frac\mathrm(x) + \sum_ (-1)^k \frac \left(\frac\right)^$

and the asymptotic series

:\mathrm(x) \sim -\sqrt e^ _2(x) \sin \beta + g_2(x) \cos \beta

where ''β'', ''f''₂(''x''), and ''g''₂(''x'') are defined as for ker(''x'').

See also

* Bessel function

References

*
*{{dlmf, first=F. W. J. , last=Olver, first2=L. C. , last2=Maximon, id=10, title=Bessel functions

External links

* Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource

* GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com


Special hypergeometric functions
Functions