Kelvin functions

J_{ν} (x e^{\frac{3 π 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^{\frac{π i}{4}}),

where K_ν(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 xe^iφ, 0 ≤ φ < 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.

Below, Γ(z) is the gamma function and ψ(z) is the digamma function.

ber(x)

For integers n, ber_n(x) has the series expansion

{b e r}_{n} (x) = {(\frac{x}{2})}^{n} \sum_{k \geq 0} \frac{\cos [(\frac{3 n}{4} + \frac{k}{2}) π]}{k! Γ (n + k + 1)} {(\frac{x^{2}}{4})}^{k},

where Γ(z) is the gamma function. The special case ber₀(x), commonly denoted as just ber(x), has the series expansion

b e r (x) = 1 + \sum_{k \geq 1} \frac{(- 1)^{k}}{[(2 k)!]^{2}} {(\frac{x}{2})}^{4 k}

and asymptotic series

b e r (x) \sim \frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2 π x}} (f_{1} (x) \cos α + g_{1} (x) \sin α) - \frac{k e i (x)}{π}

where

α = \frac{x}{\sqrt{2}} - \frac{π}{8},

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

g_{1} (x) = \sum_{k \geq 1} \frac{\sin (k π / 4)}{k! (8 x)^{k}} \prod_{l = 1}^{k} (2 l - 1)^{2} .

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ber(x)

bei(x)

ker(x)

kei(x)

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