Kelvin functions

In applied mathematics, the Kelvin functions ber_ν(x) and bei_ν(x) are the real and imaginary parts, respectively, of

$J_\nu \left (x e^{\frac{3 \pi i}{4}} \right ),\,$

where x is real, and J_ν(z), 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

$K_\nu \left (x e^{\frac{\pi i}{4}} \right ),\,$

where K_ν(z) 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 xe^iφ, 0 ≤ φ < 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.

Below, Γ(z) is the Gamma function and ψ(z) is the Digamma function.

ber(x)

File:KelvinFunctionBer.png

ber(x) for x between 0 and 10.

File:KelvinFunctionBerNorm.png

$\mathrm{ber}(x) / e^{x/\sqrt{2}}$ for x between 0 and 100.

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 Γ(z) is the Gamma function. The special case ber₀(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)

File:KelvinFunctionBei.png

bei(x) for x between 0 and 10.

File:KelvinFunctionBeiNorm.png

$\mathrm{bei}(x) / e^{x/\sqrt{2}}$ for x between 0 and 100.

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 bei₀(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)

File:KelvinFunctionKer.png

ker(x) for x between 0 and 10.

File:KelvinFunctionKerNorm.png

$\mathrm{ker}(x) e^{x/\sqrt{2}}$ for x between 0 and 100.

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

$\begin{align} \mathrm{ker}_n(x) &= - \ln\left(\frac{x}{2}\right) \mathrm{ber}_n(x) + \frac{\pi}{4}\mathrm{bei}_n(x) \\ &\qquad + \frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k \\ &\qquad \qquad + \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k \end{align}$

The special case ker₀(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)

File:KelvinFunctionKei.png

kei(x) for x between 0 and 10.

File:KelvinFunctionKeiNorm.png

$\mathrm{kei}(x) e^{x/\sqrt{2}}$ for x between 0 and 100.

For n an integer kei_n(x) has the (complicated) series expansion

$\begin{align} \mathrm{kei}_n(x) &= - \ln\left(\frac{x}{2}\right) \mathrm{bei}_n(x) - \frac{\pi}{4}\mathrm{ber}_n(x) \\ &\qquad -\frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k \\ &\qquad \qquad + \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k \end{align}$

The special case kei₀(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).

See also

• Bessel function

External links

• Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource. [1]
• GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com: [2]