I am trying to implement the following equation from this paper and having some troubles in the interpretation of $bei'$ and $ber'$. I understand from the definition of Kelvin functions that for integers n, $ber_n(x)$ can be defied, and usually, $bei(x)$ means $bei_0(x)$. I can compute it using Matlab function.
What is the usual interpretation of $bei'$ and $ber'$ ? is it $bei_1(x)$ and $ber_1(x)$ or is it the derivative? It appears that it is very common notation and not explained in the paper (I could not find anywhere else)
First of all: $\operatorname{bei}(x)$ means $\operatorname{bei}_0(x)$ and not $\operatorname{ber}_0(x)$.
Second: $\operatorname{bei}'(x)$ is the derivative of $\operatorname{bei}(x)$ and not $\operatorname{bei}_1(x)$ but the derivatives can be expressed with the first order function, see https://dlmf.nist.gov/10.63.E3
$$\operatorname{ber}'(x) = \frac{1}{\sqrt{2}}\Big( \operatorname{ber}_1(x) + \operatorname{bei}_1(x)\Big)$$ $$\operatorname{bei}'(x) = \frac{1}{\sqrt{2}}\Big( \operatorname{bei}_1(x) - \operatorname{ber}_1(x)\Big)$$