Kelvin functions

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

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

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 $K_\nu(x e^{\pi i/4})\,$, where $K_\nu(z)\,$ 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 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)

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 $\Gamma(z)$ is the Gamma function. The special case Ber₀(x), commonly denoted as just Ber(x), has the series expansion

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

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)

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 $\Gamma(z)$ is the Gamma function. The special case Bei₀(x), commonly denoted as just Bei(x), has the series expansion

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

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

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

Ker(x) for $x$ between 0 and 10.

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

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

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

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

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

Kei(x) for $x$ between 0 and 10.

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

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

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

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 $\beta$, $f_2(x)$, and $g_2(x)$ are defined as for Ker$(x)$.

See also

• Bessel function

References

• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 379, ISBN 978-0486612720, MR0167642, http://www.math.sfu.ca/~cbm/aands/page_379.htm .
• Olver, F. W. J.; Maximon, L. C. (2010), "Bessel functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/10 

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]