Kelvin functions

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The Kelvin functions Ber_ν(x) and Bei_ν(x) are the real and imaginary parts, respectively, of

J ν (x e 3π i / 4),

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 $K ν (x e 3π i / 4)$ , where $K ν (z)$ is the ν^th order modified Bessel function of the second kind.

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.

1 Ber(x)
2 Bei(x)
3 Ker(x)
4 Kei(x)
5 See also
6 References
7 External links

[edit] Ber(x)

Ber(x) for

x

between 0 and 10.

$\mathrm{Ber}(x) / e^{x/\sqrt{2}}$ 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 Ber $0 (x)$ , commonly denoted as just Ber $(x)$ , has the series expansion

$\mathrm{Ber}(x) = 1 + \sum_{k \geq 1} \frac{(-1)^k (x/2)^{4k}}{[(2k)!]^2}$

and asymptotic series

$\mathrm{Ber}(x) \sim \frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2 \pi x}} [f_1(x) \cos \alpha + g_1(x) \sin \alpha] - \frac{\mathrm{Kei}(x)}{\pi}$ ,

where $\alpha = x/\sqrt{2} - \pi/8$ , and

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

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

[edit] Bei(x)

Bei(x) for

x

between 0 and 10.

$\mathrm{Bei}(x) / e^{x/\sqrt{2}}$ for

x

between 0 and 100.

For integers $n$ , Bei $n (x)$ has the series expansion

$\mathrm{Bei}_n(x) = \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \frac{\sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right]}{k! \Gamma(n + k + 1)} \left(\frac{x^2}{4}\right)^k$

where $Γ(z)$ is the Gamma function. The special case Bei $0 (x)$ , commonly denoted as just Bei $(x)$ , has the series expansion

$\mathrm{Bei}(x) = \sum_{k \geq 0} \frac{(-1)^k (x/2)^{4k+2}}{[(2k+1)!]^2}$

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 $(x)$ .

[edit] Ker(x)

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

$\mathrm{Ker}_n(x) = \frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k - \ln\left(\frac{x}{2}\right) \mathrm{Ber}_n(x) + \frac{\pi}{4}\mathrm{Bei}_n(x) + \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k$

Ker(x) for

x

between 0 and 10.

$\mathrm{Ker}(x) e^{x/\sqrt{2}}$ for

x

between 0 and 100.

where $ψ(z)$ is the Digamma function. The special case Ker $0 (x)$ , commonly denoted as just Ker $(x)$ , has the series expansion

$\mathrm{Ker}(x) = -\ln\left(\frac{x}{2}\right) \mathrm{Ber}_n(x) + \frac{\pi}{4}\mathrm{Bei}_n(x) + \sum_{k \geq 0} (-1)^k \frac{\psi(2k + 1)}{[(2k)!]^2} \left(\frac{x^2}{4}\right)^{2k}$

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 $\beta = x/\sqrt{2} + \pi/8$ , and

$f_2(x) = 1 + \sum_{k \geq 1} (-1)^k \frac{\cos(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2$

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

[edit] Kei(x)

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

$\mathrm{Kei}_n(x) = \frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k - \ln\left(\frac{x}{2}\right) \mathrm{Bei}_n(x) - \frac{\pi}{4}\mathrm{Ber}_n(x) + \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k$

Kei(x) for

x

between 0 and 10.

$\mathrm{Kei}(x) e^{x/\sqrt{2}}$ for

x

between 0 and 100.

where $ψ(z)$ is the Digamma function. The special case Kei $0 (x)$ , commonly denoted as just Kei $(x)$ , has the series expansion

$\mathrm{Kei}(x) = -\ln\left(\frac{x}{2}\right) \mathrm{Bei}_n(x) - \frac{\pi}{4}\mathrm{Ber}_n(x) + \sum_{k \geq 0} (-1)^k \frac{\psi(2k + 2)}{[(2k+1)!]^2} \left(\frac{x^2}{4}\right)^{2k+1}$

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 $(x)$ .

[edit] See also

Bessel function

[edit] References

Abramowitz, Milton & Stegun, Irene A., eds. (1965), “Chapter 9”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .

[edit] 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]

Categories: Special functions | Special hypergeometric functions

Kelvin functions

From Wikipedia, the free encyclopedia

Contents

[edit] Ber(x)

[edit] Bei(x)

[edit] Ker(x)

[edit] Kei(x)

[edit] See also

[edit] References

[edit] External links

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