Kelvin functions

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

:$J\_\; u(x\; e^\{3\; pi\; i/4\}),,$

where "x" is real, and $J\_\; u(z),$ 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 $K\_\; u(x\; e^\{3\; pi\; i/4\}),$, where $K\_\; u(z),$ 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.

Ber(x)

For integers "n", Ber_"n"("x") has the series expansion

:$mathrm\{Ber\}\_n(x)\; =\; left(frac\{x\}\{2\}\; ight)^n\; sum\_\{k\; geq\; 0\}\; frac\{cosleft\; [left(frac\{3n\}\{4\}\; +\; frac\{k\}\{2\}\; ight)pi\; ight]\; \}\{k!\; Gamma(n\; +\; k\; +\; 1)\}\; left(frac\{x^2\}\{4\}\; ight)^k$

where $Gamma(z)$ 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

:$f\_1(x)\; =\; 1\; +\; sum\_\{k\; geq\; 1\}\; frac\{cos(k\; pi\; /\; 4)\}\{k!\; (8x)^k\}\; prod\_\{l\; =\; 1\}^k\; (2l\; -\; 1)^2$

:$g\_1(x)\; =\; sum\_\{k\; geq\; 1\}\; frac\{sin(k\; pi\; /\; 4)\}\{k!\; (8x)^k\}\; prod\_\{l\; =\; 1\}^k\; (2l\; -\; 1)^2$

Bei(x)

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

:$mathrm\{Bei\}\_n(x)\; =\; left(frac\{x\}\{2\}\; ight)^n\; sum\_\{k\; geq\; 0\}\; frac\{sinleft\; [left(frac\{3n\}\{4\}\; +\; frac\{k\}\{2\}\; ight)pi\; ight]\; \}\{k!\; Gamma(n\; +\; k\; +\; 1)\}\; left(frac\{x^2\}\{4\}\; ight)^k$

where $Gamma(z)$ 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 $alpha$, $f\_1(x)$, and $g\_1(x)$ are defined as for Ber$(x)$.

Ker(x)

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

:$mathrm\{Ker\}\_n(x)\; =\; frac\{1\}\{2\}\; left(frac\{x\}\{2\}\; ight)^\{-n\}\; sum\_\{k=0\}^\{n-1\}\; cosleft\; [left(frac\{3n\}\{4\}\; +\; frac\{k\}\{2\}\; ight)pi\; ight]\; frac\{(n-k-1)!\}\{k!\}\; left(frac\{x^2\}\{4\}\; ight)^k\; -\; lnleft(frac\{x\}\{2\}\; ight)\; mathrm\{Ber\}\_n(x)\; +\; frac\{pi\}\{4\}mathrm\{Bei\}\_n(x)\; +\; frac\{1\}\{2\}\; left(frac\{x\}\{2\}\; ight)^n\; sum\_\{k\; geq\; 0\}\; cosleft\; [left(frac\{3n\}\{4\}\; +\; frac\{k\}\{2\}\; ight)pi\; ight]\; frac\{psi(k+1)\; +\; psi(n\; +\; k\; +\; 1)\}\{k!\; (n+k)!\}\; left(frac\{x^2\}\{4\}\; ight)^k$

where $psi(z)$ is the Digamma function. The special case Ker$\_0(x)$, commonly denoted as just Ker$(x)$, has the series expansion

:$mathrm\{Ker\}(x)\; =\; -lnleft(frac\{x\}\{2\}\; ight)\; mathrm\{Ber\}\_n(x)\; +\; frac\{pi\}\{4\}mathrm\{Bei\}\_n(x)\; +\; sum\_\{k\; geq\; 0\}\; (-1)^k\; frac\{psi(2k\; +\; 1)\}\{\; [(2k)!]\; ^2\}\; left(frac\{x^2\}\{4\}\; ight)^\{2k\}$

and the asymptotic series

:

where , and

:$f\_2(x)\; =\; 1\; +\; sum\_\{k\; geq\; 1\}\; (-1)^k\; frac\{cos(k\; pi\; /\; 4)\}\{k!\; (8x)^k\}\; prod\_\{l\; =\; 1\}^k\; (2l\; -\; 1)^2$

:$g\_2(x)\; =\; sum\_\{k\; geq\; 1\}\; (-1)^k\; frac\{sin(k\; pi\; /\; 4)\}\{k!\; (8x)^k\}\; prod\_\{l\; =\; 1\}^k\; (2l\; -\; 1)^2$

Kei(x)

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

:$mathrm\{Kei\}\_n(x)\; =\; frac\{1\}\{2\}\; left(frac\{x\}\{2\}\; ight)^\{-n\}\; sum\_\{k=0\}^\{n-1\}\; sinleft\; [left(frac\{3n\}\{4\}\; +\; frac\{k\}\{2\}\; ight)pi\; ight]\; frac\{(n-k-1)!\}\{k!\}\; left(frac\{x^2\}\{4\}\; ight)^k\; -\; lnleft(frac\{x\}\{2\}\; ight)\; mathrm\{Bei\}\_n(x)\; -\; frac\{pi\}\{4\}mathrm\{Ber\}\_n(x)\; +\; frac\{1\}\{2\}\; left(frac\{x\}\{2\}\; ight)^n\; sum\_\{k\; geq\; 0\}\; sinleft\; [left(frac\{3n\}\{4\}\; +\; frac\{k\}\{2\}\; ight)pi\; ight]\; frac\{psi(k+1)\; +\; psi(n\; +\; k\; +\; 1)\}\{k!\; (n+k)!\}\; left(frac\{x^2\}\{4\}\; ight)^k$

where $psi(z)$ is the Digamma function. The special case Kei$\_0(x)$, commonly denoted as just Kei$(x)$, has the series expansion

:$mathrm\{Kei\}(x)\; =\; -lnleft(frac\{x\}\{2\}\; ight)\; mathrm\{Bei\}\_n(x)\; -\; frac\{pi\}\{4\}mathrm\{Ber\}\_n(x)\; +\; sum\_\{k\; geq\; 0\}\; (-1)^k\; frac\{psi(2k\; +\; 2)\}\{\; [(2k+1)!]\; ^2\}\; left(frac\{x^2\}\{4\}\; ight)^\{2k+1\}$

and the asymptotic series

:

where , $f\_2(x)$, and $g\_2(x)$ are defined as for Ker$(x)$.

See also

* Bessel function

References

*

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]