Overview
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^ \frac 3 \pi i 4 \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 : e^ -\nu\pi i/2 K_\nu \left (x e^ \frac \pi i 4 \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 ber _n(x) = \left(\frac x 2 \right)^n \sum_ k \geq 0 \frac \cos\left \left(\frac 3n 4 + \frac k 2 \right)\pi\right k! \Gamma(n + k + 1) \left(\frac x^2 4 \right)^k , where is the gamma function. The special case ber 0 (x), commonly denoted as just ber(x), has the series expansion : \mathrm ber (x) = 1 + \sum_ k \geq 1 \frac (-1)^k (2k)! ^2 \left(\frac x 2 \right )^ 4k and asymptotic series : \mathrm ber (x) \sim \frac e^ \frac x \sqrt 2 \sqrt 2 \pi x \left (f_1(x) \cos \alpha + g_1(x) \sin \alpha \right ) - \frac \mathrm kei (x) \pi , where : \alpha = \frac x \sqrt 2 - \frac \pi 8 , : f_1(x) = 1 + \sum_ k \geq 1 \frac \cos(k \pi / 4) k! (8x)^k \prod_ l = 1 ^k (2l - 1)^2 : g_1(x) = \sum_ k \geq 1 \frac \sin(k \pi / 4) k! (8x)^k \prod_ l = 1 ^k (2l - 1)^2 .
bei(x)
For integers n, bei n (x) has the series expansion : \mathrm bei _n(x) = \left(\frac x 2 \right)^n \sum_ k \geq 0 \frac \sin\left \left(\frac 3n 4 + \frac k 2 \right)\pi\right k! \Gamma(n + k + 1) \left(\frac x^2 4 \right)^k . The special case bei 0 (x), commonly denoted as just bei(x), has the series expansion : \mathrm bei (x) = \sum_ k \geq 0 \frac (-1)^k (2k+1)! ^2 \left(\frac x 2 \right )^ 4k+2 and asymptotic series : \mathrm bei (x) \sim \frac e^ \frac x \sqrt 2 \sqrt 2 \pi x f_1(x) \sin \alpha - g_1(x) \cos \alpha - \frac \mathrm ker (x) \pi , 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 align &\mathrm ker _n(x) = - \ln\left(\frac x 2 \right) \mathrm ber _n(x) + \frac \pi 4 \mathrm bei _n(x) \\ &+ \frac 1 2 \left(\frac x 2 \right)^ -n \sum_ k=0 ^ n-1 \cos\left \left(\frac 3n 4 + \frac k 2 \right)\pi\right \frac (n-k-1)! k! \left(\frac x^2 4 \right)^k \\ &+ \frac 1 2 \left(\frac x 2 \right)^n \sum_ k \geq 0 \cos\left \left(\frac 3n 4 + \frac k 2 \right)\pi\right \frac \psi(k+1) + \psi(n + k + 1) k!
(n+k)! \left(\frac x^2 4 \right)^k . \end align The special case ker 0 (x), commonly denoted as just ker(x), has the series expansion : \mathrm ker (x) = -\ln\left(\frac x 2 \right) \mathrm ber (x) + \frac \pi 4 \mathrm bei (x) + \sum_ k \geq 0 (-1)^k \frac \psi(2k + 1) (2k)! ^2 \left(\frac x^2 4 \right)^ 2k and the asymptotic series : \mathrm ker (x) \sim \sqrt \frac \pi 2x e^ -\frac x \sqrt 2 f_2(x) \cos \beta + g_2(x) \sin \beta , where : \beta = \frac x \sqrt 2 + \frac \pi 8 , : f_2(x) = 1 + \sum_ k \geq 1 (-1)^k \frac \cos(k \pi / 4) k! (8x)^k \prod_ l = 1 ^k (2l - 1)^2 : g_2(x) = \sum_ k \geq 1 (-1)^k \frac \sin(k \pi / 4) k! (8x)^k \prod_ l = 1 ^k (2l - 1)^2.
kei(x)
For integer n, kei n (x) has the series expansion : \begin align &\mathrm kei _n(x) = - \ln\left(\frac x 2 \right) \mathrm bei _n(x) - \frac \pi 4 \mathrm ber _n(x) \\ &-\frac 1 2 \left(\frac x 2 \right)^ -n \sum_ k=0 ^ n-1 \sin\left \left(\frac 3n 4 + \frac k 2 \right)\pi\right \frac (n-k-1)! k! \left(\frac x^2 4 \right)^k \\ &+ \frac 1 2 \left(\frac x 2 \right)^n \sum_ k \geq 0 \sin\left \left(\frac 3n 4 + \frac k 2 \right)\pi\right \frac \psi(k+1) + \psi(n + k + 1) k!
(n+k)! \left(\frac x^2 4 \right)^k . \end align The special case kei 0 (x), commonly denoted as just kei(x), has the series expansion : \mathrm kei (x) = -\ln\left(\frac x 2 \right) \mathrm bei (x) - \frac \pi 4 \mathrm ber (x) + \sum_ k \geq 0 (-1)^k \frac \psi(2k + 2) (2k+1)! ^2 \left(\frac x^2 4 \right)^ 2k+1 and the asymptotic series : \mathrm kei (x) \sim -\sqrt \frac \pi 2x e^ -\frac x \sqrt 2 f_2(x) \sin \beta + g_2(x) \cos \beta , where β, f 2 (x), and g 2 (x) are defined as for ker(x).
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: https://web.archive.org/web/20070407195618/http://www.codecogs.com/d-ox/maths/special/bessel/kelvin.php