Integrating a Bessel Function $K_0(ax)$ without the constant 'a'.

Question

Thanks for reading !

I am having a problem with the numerical integration of the Bessel function

$$y = K_0(ax)$$

Since my constant is too large ($a = 6800$) I am getting the large arguments approximation for the Bessel K0 function, making the integration difficult.

My question is: Is there some mathematical method or manipulation to remove the constant 'a' from the integration like

$$y = K_0(x)$$

to make me integrate it like small arguments and then insert the constant 'a' in it somehow after ?

Thank you very much !

Answer 1

Use limit behaviour for this Bessel function $$K_n(x)\sim \sqrt{\dfrac{\pi}{2x}}e^{-x}$$ as $x\to\infty$.

Answer 2

What is the exact task that you want to solve?

The obviuos way is to change the variable $u=ax$. There is even a closed form $$\int K_0(ax) \mathrm{d}x = \frac{a x K_0(a x) + \frac {1}{2} \pi a x \mathbf{L}_1(ax) K_0(ax) + \mathbf{L}_0(ax)K_1(ax)}{a}$$ with the modified Struve functions $\mathbf{L}_n$. For more info about the $\mathbf{L}_n$ (e.g. asymptotic expansions) see https://dlmf.nist.gov/11 .

The first two high order terms for $a\to \infty$ are $$\frac{1}{2a}\left(\pi + \sqrt{\frac{2\pi}{ax}} e^{-ax}\right)$$