Kelvin functions

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In applied mathematics, the Kelvin functions berν(x) and beiν(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 Jν(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

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

where Kν(z) is the νth 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, 0 ≤ φ < 2π. With the exception of Bern(x) and Bein(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)[edit]

ber(x) for x between 0 and 10.
\mathrm{ber}(x) / e^{x/\sqrt{2}} for x between 0 and 100.

For integers n, bern(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)[edit]

bei(x) for x between 0 and 10.
\mathrm{bei}(x) / e^{x/\sqrt{2}} for x between 0 and 100.

For integers n, bein(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)[edit]

ker(x) for x between 0 and 10.
\mathrm{ker}(x) e^{x/\sqrt{2}} for x between 0 and 100.

For integers n, kern(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)[edit]

kei(x) for x between 0 and 10.
\mathrm{kei}(x) e^{x/\sqrt{2}} for x between 0 and 100.

For n an integer kein(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[edit]

References[edit]

External links[edit]

  • 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]