- Kelvin functions
The Kelvin functions Berν("x") and Beiν("x") are the real and
imaginary part s, respectively, of:J_ u(x e^{3 pi i/4}),,
where "x" is real, and J_ u(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_ u(x e^{3 pi i/4}),, where K_ u(z), is the νth 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 Ber0(x), commonly denoted as just Berx), 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, Bein(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 Bei0(x), commonly denoted as just Beix), 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 Berx).
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 Ker0(x), commonly denoted as just Kerx), 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
:mathrm{Ker}(x) sim sqrt{frac{pi}{2x e^{-frac{x}{sqrt{2} [f_2(x) cos eta + g_2(x) sin eta] ,
where eta = 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
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 Kei0(x), commonly denoted as just Keix), 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
:mathrm{Kei}(x) sim -sqrt{frac{pi}{2x e^{-frac{x}{sqrt{2} [f_2(x) sin eta + g_2(x) cos eta] ,
where eta, f_2(x), and g_2(x) are defined as for Kerx).
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]
Wikimedia Foundation. 2010.