Integral involving Bessel function, Struve Function and Exponential Function.

Question

I am trying to solve solve an integral of the form:
$$\int_{-\infty}^{+\infty}e^{-2x/a}[H_0(x/b)-Y_0(x/b)]dx$$ $$\int_{-\infty}^{+\infty} e^{-2x/a}H_o(x/b)dx-\int_{-\infty}^{+\infty} e^{-2x/a}Y_o(x/b)dx$$
$Y_n$ is Bessel function with $n=0$ and $H_n$ is Struve function with $n=0$ How can I solve these integral? Any hints please.

Can these be solved using MATLAB?

Answer 1

As it currently stands, the integral diverges due to the behavior of the Struve H function on $\mathbb R^-.~$ However, on $\mathbb R^+$ we have :

$$\int_0^\infty\exp(-x\text{ sinh }u)~Y_0(x)~dx~=~-\frac2\pi~\frac{u}{\cosh u}$$

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$$\int_0^\infty\exp(-x\text{ csch }u)~H_0(x)~dx~=~+\frac2\pi~\frac{u}{\coth u}$$