I try to calculate the integral \begin{align} f(y,z):=\int_{\mathbb R} \frac{\exp\left(-y(x^2+z\cdot x\sqrt{x^2+1} ) \right)}{\sqrt{x^2+1}}\mathrm dx \end{align} on the set $y>0$ and $0<z<1$. I could show by some calculation that in the limit $z\to0$ holds \begin{align} f(y,0)=e^{y/2}K_0(y/2), \end{align} where $K_0$ denotes the modified Bessel function of the second kind and \begin{align} \lim_{z\to1}f(y,z)=f(y,1)=\infty,\\ \lim_{y\to0}f(y,z)=f(0,z)=\infty. \end{align} But I can't evaluate the integral in the interior of the defined set. Any help, ideas or hints how to solve this would be appreciated.
Here is a selection of what I've tried:
- several substitutions (e.g. $\sinh$,$\dots$),
- integration by parts,
- tried to evaluate the integral $$\partial_zf(y,z)=\int_{\mathbb R} -yx\cdot \exp\left(-y(x^2+z\cdot x\sqrt{x^2+1} ) \right)\mathrm dx, $$
- tried to evaluate the integrals $$\partial_yf(y,z)=\int_{\mathbb R} -(x^2+zx\sqrt{x^2+1})\cdot \frac{\exp\left(-y(x^2+z\cdot x\sqrt{x^2+1} ) \right)}{\sqrt{x^2+1}}\mathrm dx $$ and $$f(y,z)-\partial_yf(y,z)=\int_{\mathbb R} (\sqrt{x^2+1}+zx)\cdot \exp\left(-y(x^2+z\cdot x\sqrt{x^2+1} ) \right)\mathrm dx. $$