## Kelvin Functions

Denoting x = r/i, the four yn(x) functions in (6.15) are Kelvin functions. These are real and imaginary parts of zero-order Bessel functions I0(%/ix) and K0(\fix), defined from Dwight  and Abramovitz & Stegun  by

I0(Vix) = ber x + i bei x, Ko(Vix) = ker x + i kei x.

The Kelvin functions are represented by the following series b — 1 _ (x/2)4 + (x/2)8 ber x — 1 (2!)2 + (4!)2

where y is the Euler constant y = 0.57721 56649 01532 ...

Denoting A2 a Laplacian operator with respect to x, i.e. A2 = iV2, these functions have the following properties

A2ber x — - bei x, A2 bei x — ber x, A2 ker x — — kei x, A2 kei x — ker x, and satisfy independently

For x < 10, the four Kelvin functions are plotted in Fig. 6.2. For large values of the argument, the determination of the Kelvin functions require use of asymptotic relations as follows berx bei x

V2nx

1 ex

\j2kx

L0(x)cos(--f) -M0(x) sm(--?). L0(x) sin( - - f ) + M0(x) cos( - -§ ) L0 (-x) cos( —+f) + M0(-x) sin( —+f -Lq(-x) sin( — + D + M0(-x) cos( -fa+f

 ^^ ker bei ^
0 0