Kelvin functions explained

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 ^{(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

K _{\nu} \left (x e ^{(\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

ber _{n(x)} = \left( (x)/(2) \right) ^{n} \sum _{k} (\cos\left[\left( (3 n)/(+ (k)/(2) \right)\pi\right] 4))/(\left( k! \Gamma(n + k +1 )) (x ^{2})/(4) \right) ^{k} ,

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

ber (x) = 1 + \sum _{k} (( -1 ) ^{k})/(\left( [( 2 k)!] ^{2}) (x)/(2) \right ) ^{4k}

and asymptotic series

ber (x) \sim (e ^{(x)/(\sqrt{ 2)})/() } \left (f_1(x) \cos \alpha + g_1(x) \sin \alpha \right) - \frac,

where

\alpha = (x)/(\sqrt{ 2) } - \frac,f _{1(x)} = 1 + \sum _{k} (\cos(k \pi / 4 ))/(k! ( 8 x) ^{k}) \prod ^{k} _{l =1} ( 2 l - 1 ) ^{2} g _{1(x)} = \sum _{k} (\sin(k \pi / 4 ))/(k! ( 8 x) ^{k}) \prod ^{k} _{l =1} ( 2 l - 1 ) ^{2} .

bei(x)

For integers n, bei_n(x) has the series expansion

bei _{n(x)} = \left( (x)/(2) \right) ^{n} \sum _{k} (\sin\left[\left( (3 n)/(+ (k)/(2) \right)\pi\right] 4))/(\left( k! \Gamma(n + k +1 )) (x ^{2})/(4) \right) ^{k} .

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

bei (x) = \sum _{k} (( -1 ) ^{k})/(\left( [( 2 k +1 )!] ^{2}) (x)/(2) \right ) ^{4k+2}

and asymptotic series

bei (x) \sim (e ^{(x)/(\sqrt{ 2)})/() } [f_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{align} & ker _{n(x)} = -ln \left( (x)/(2) \right) ber _{n(x)} + (\pi)/(4) bei _{n(x)} \\ & + (1)/(\left( 2) (x)/(2) \right) ^{-n} \sum ^{n -1} _{k =0} \cos\left[\left( (3 n)/(4) + (k)/(2) \right)\pi\right] ((n - k -1 )!)/(\left( k!) (x ^{2})/(4) \right) ^{k} \\ & + (1)/(\left( 2) (x)/(2) \right) ^{n} \sum _{k} \cos\left[\left( (3 n)/(4) + (k)/(2) \right)\pi\right] (\psi(k +1 ) + \psi(n + k +1 ))/(\left( k! (n + k)!) (x ^{2})/(4) \right) ^{k} . \end{align}

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

ker (x) =-ln \left( (x)/(2) \right) ber (x) + (\pi)/(4) bei (x) + \sum _{k} ( -1 ) ^{k} (\psi( 2 k +1 ))/(\left( [( 2 k)!] ^{2}) (x ^{2})/(4) \right) ^{2k}

and the asymptotic series

ker (x) \sim \sqrt{ (\pi)/(2 x) } e^ [f_2(x) \cos \beta + g_2(x) \sin \beta],

where

\beta = (x)/(\sqrt{ 2) } + \frac,f _{2(x)} = 1 + \sum _{k} ( -1 ) ^{k} (\cos(k \pi / 4 ))/(k! ( 8 x) ^{k}) \prod ^{k} _{l =1} ( 2 l - 1 ) ^{2} g _{2(x)} = \sum _{k} ( -1 ) ^{k} (\sin(k \pi / 4 ))/(k! ( 8 x) ^{k}) \prod ^{k} _{l =1} ( 2 l - 1 ) ^{2.}

kei(x)

For integer n, kei_n(x) has the series expansion

\begin{align} & kei _{n(x)} = -ln \left( (x)/(2) \right) bei _{n(x)} - (\pi)/(4) ber _{n(x)} \\ & - (1)/(\left( 2) (x)/(2) \right) ^{-n} \sum ^{n -1} _{k =0} \sin\left[\left( (3 n)/(4) + (k)/(2) \right)\pi\right] ((n - k -1 )!)/(\left( k!) (x ^{2})/(4) \right) ^{k} \\ & + (1)/(\left( 2) (x)/(2) \right) ^{n} \sum _{k} \sin\left[\left( (3 n)/(4) + (k)/(2) \right)\pi\right] (\psi(k +1 ) + \psi(n + k +1 ))/(\left( k! (n + k)!) (x ^{2})/(4) \right) ^{k} . \end{align}

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

kei (x) =-ln \left( (x)/(2) \right) bei (x) - (\pi)/(4) ber (x) + \sum _{k} ( -1 ) ^{k} (\psi( 2 k +2 ))/(\left( [( 2 k +1 )!] ^{2}) (x ^{2})/(4) \right) ^{2k+1}

and the asymptotic series

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

where β, f₂(x), and g₂(x) are defined as for ker(x).

See also

• Bessel function

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: https://web.archive.org/web/20070407195618/http://www.codecogs.com/d-ox/maths/special/bessel/kelvin.php