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In applied mathematics, the '''Kelvin functions''' ''ber''''ν''(''x'') and ''bei''''ν''(''x'') are the [[real part|real]] and [[imaginary part]]s, respectively, of

:<math>J_\nu(x e^{3 \pi i/4}),\,</math>

where ''x'' is real, and ''J''''ν''(''z''), 
is the νth order [[Bessel function]] of the first kind. Similarly, the functions Kerν(''x'') and Keiν(''x'') are the real and imaginary parts, respectively, of
<math>K_\nu(x e^{\pi i/4})\,</math>, 
where <math>K_\nu(z)\,</math> is the νth order [[Bessel function#Modified Bessel functions|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 ''x e''''i'' φ, φ ∈ [0, 2π). With the exception of Ber''n''(''x'') and Bei''n''(''x'') for integral ''n'', the Kelvin functions have a [[branch point]] at ''x'' = 0.

== ber(''x'') ==

[[Image:KelvinFunctionBer.png|thumb|right|ber(''x'') for ''x'' between 0 and 10.]]
[[Image:KelvinFunctionBerNorm.png|thumb|right|<math>\mathrm{ber}(x) / e^{x/\sqrt{2}}</math> for <math>x</math> between 0 and 100.]]

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

:<math>\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</math>

where Γ(''z'') is the [[Gamma function]]. The special case ''ber''''0''(''x''), commonly denoted as just ''ber''(''x''), has the series expansion

:<math>\mathrm{ber}(x) = 1 + \sum_{k \geq 1} \frac{(-1)^k (x/2)^{4k}}{[(2k)!]^2}</math>

and [[asymptotic series]]

:<math>\mathrm{ber}(x) \sim \frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2 \pi x}} [f_1(x) \cos \alpha + g_1(x) \sin \alpha] - \frac{\mathrm{kei}(x)}{\pi}</math>,

where <math>\alpha = x/\sqrt{2} - \pi/8</math>, and

:<math>f_1(x) = 1 + \sum_{k \geq 1} \frac{\cos(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</math>

:<math>g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</math>

== bei(''x'') ==

[[Image:KelvinFunctionBei.png|thumb|right|bei(x) for <math>x</math> between 0 and 10.]]
[[Image:KelvinFunctionBeiNorm.png|thumb|right|<math>\mathrm{bei}(x) / e^{x/\sqrt{2}}</math> for <math>x</math> between 0 and 100.]]

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

:<math>\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</math>

where Γ(''z'') is the [[Gamma function]]. The special case bei0(x), commonly denoted as just bei(x), has the series expansion

:<math>\mathrm{bei}(x) = \sum_{k \geq 0} \frac{(-1)^k (x/2)^{4k+2}}{[(2k+1)!]^2}</math>

and asymptotic series

:<math>\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}</math>,

where <math>\alpha</math>, <math>f_1(x)</math>, and <math>g_1(x)</math> are defined as for ber<math>(x)</math>.

 

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

:<math>
\begin{align}
\mathrm{ker}_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 - \ln\left(\frac{x}{2}\right) \mathrm{ber}_n(x) + \frac{\pi}{4}\mathrm{bei}_n(x) \\
& {} \quad + \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}
</math>

[[Image:KelvinFunctionKer.png|thumb|right|ker(x) for <math>x</math> between 0 and 10.]]
[[Image:KelvinFunctionKerNorm.png|thumb|right|<math>\mathrm{ker}(x) e^{x/\sqrt{2}}</math> for ''x'' between 0 and 100.]]

where <math>\psi(z)</math> is the [[Digamma function]]. The special case ker<math>_0(x)</math>, commonly denoted as just ker<math>(x)</math>, has the series expansion

:<math>\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}</math>

and the asymptotic series

:<math>\mathrm{ker}(x) \sim \sqrt{\frac{\pi}{2x}} e^{-\frac{x}{\sqrt{2}}} [f_2(x) \cos \beta + g_2(x) \sin \beta],</math>

where <math>\beta = x/\sqrt{2} + \pi/8</math>, and

:<math>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</math>

:<math>g_2(x) = \sum_{k \geq 1} (-1)^k \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2.</math>

 

== kei(''x'') ==

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

:<math>\mathrm{kei}_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 - \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 \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</math>

[[Image:KelvinFunctionKei.png|thumb|right|kei(x) for <math>x</math> between 0 and 10.]]
[[Image:KelvinFunctionKeiNorm.png|thumb|right|<math>\mathrm{kei}(x) e^{x/\sqrt{2}}</math> for <math>x</math> between 0 and 100.]]

where <math>\psi(z)</math> is the [[Digamma function]]. The special case kei<math>_0(x)</math>, commonly denoted as just kei<math>(x)</math>, has the series expansion

:<math>\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}</math>

and the asymptotic series

:<math>\mathrm{kei}(x) \sim -\sqrt{\frac{\pi}{2x}} e^{-\frac{x}{\sqrt{2}}} [f_2(x) \sin \beta + g_2(x) \cos \beta],</math>

where <math>\beta</math>, <math>f_2(x)</math>, and <math>g_2(x)</math> are defined as for ker<math>(x)</math>.

 

== See also ==

* [[Bessel function]]

== References ==
* {{Abramowitz_Stegun_ref|9|379}}
*{{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. [http://mathworld.wolfram.com/KelvinFunctions.html]
* GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com: [http://www.codecogs.com/d-ox/maths/special/bessel/kelvin.php]

[[Category:Special hypergeometric functions]]

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