Kelvin functions ker kei functions evaluation

Hi,

I am working with Kelvin functions and in the specific with ker(x) and kei(x). I build a short function to evaluate them using the Matlab build in besselk function but when I test it I am not obtaining the expected results.

The values should be:

kei(0)=-0.78540; ker(0)=-inf

Instead I get kei(0)=0

Can anyone help?

Thank you.

% Define a range of x values
x = linspace(0, 6, 100);

% Calculate ber and bei
[ker_vals, kei_vals] = kelvink(0, x);

% Plot the results
figure;
plot(x, ker_vals, 'b-', x, kei_vals, 'r--');
title('Kelvin KER and KEI Functions');
xlabel('x');
ylabel('Function Value');
legend('ker(x)', 'kei(x)');

function [ker, kei] = kelvink(n, x)
  a = exp(pi*1i/4);
  ke = besselk(n,a*x);
  ker = real(ke);
  kei = imag(ke);
end

Answer 1

[Torsten]

The value kei(0) = -pi/4 only exists in the limit :

syms x
a = exp(pi*1i/4);
limit(imag(besselk(0,a*x)),x,0,"right")

ans = 

limit(imag(besselk(0,x)),x,0,"right")

ans = 

0

imag(besselk(0,0))

ans = 0

The value ker(0) = + Inf agrees with MATHEMATICA:

https://www.wolframalpha.com/input?i=KelvinKer%5B0%2C0%5D