Let $ x>0 $ and $a \in {\mathbb R}$, $b \in {\mathbb R} $. We consider a following improper integral. We have:
\begin{equation} {\mathfrak J}_{a,b}(x):= \int\limits_0^\infty e^{-2 x \sinh(\omega)} \cdot K_a(2 x \cosh(\omega))\cdot [\cosh(\omega)]^b d\omega \end{equation}
Below we evaluated the integral in question in the special case $b = 1-a$. We have:
\begin{eqnarray} &&{\mathfrak J}_{a,b}(x) = \tag{1} \\ &&\int\limits_{-\infty}^0 e^{2 \xi x} \left(\frac{1}{1+\xi^2}\right)^{a/2} \cdot K_a(2 x \sqrt{1+\xi^2}) d\xi \tag{1a} \\ &&\frac{\sqrt{\pi}}{4 x^a} \cdot \int\limits_0^\infty \xi^{a-3/2} e^{-\xi} \left(1+ erf[-\frac{x}{\sqrt{\xi}}] \right) d\xi \tag{2} \\ && \frac{\sqrt{\pi}}{4 x^a} \cdot \left( (-\frac{3}{2} + a)! - \frac{4}{\sqrt{\pi}}\int\limits_0^x \xi ^{-1+a} K_{-1+a}(2 \xi) d\xi \right) \tag{3} \\ &&\frac{x^{-a}}{4} \left( \sqrt{\pi} (-\frac{3}{2}+a)! -2 x \Gamma(-1+a) \cdot F_{1,2}\left[ \frac{1}{2}; \frac{3}{2},2-a; x^2\right] -\frac{2x^{-1+2 a}}{-1+2 a} \cdot \Gamma(1-a) \cdot F_{1,2}\left[ -\frac{1}{2}+a ; a, \frac{1}{2}+a; x^2\right] \right) \tag{4} \end{eqnarray}
In $(1a)$ we substituted for $\xi = \sinh(\omega)$.
In $(2)$ we used a following trick. In the error function in the integrand below we added a term $A \sqrt{\xi} $
then we differentiated the whole integral by $A$ then we did the
resulting integral using the third identity from the top in here and finally we changed $A$ to $\xi$ and integrated the result with respect to later variable from minus infinity to zero.In $(3)$ we split the integral into two parts, evaluated the easy part and
then did the same trick with differentiating by the integral by with respect to a parameter-- in this case by $x$-- doing integral
over $\xi$ then using the identity above and then finally integrating the result over $x$. Finally in $(4)$ we used the definition of the modified Bessel function and the
series expansion of it and we integrated term by term.
In[512]:= a = RandomReal[{0, 2}];
x = RandomReal[{0, 2}];
NIntegrate[
E^(-2 x Sinh[omega]) BesselK[a, 2 x Cosh[omega]] Cosh[omega]^(
1 - a), {omega, 0, Infinity}]
NIntegrate[
E^(2 xi x) (1/(1 + xi^2))^(a/2)
BesselK[a, 2 Sqrt[(1 + xi^2) x^2]], {xi, -Infinity, 0}]
(Sqrt[[Pi]] /(4 (x)^ (a))) NIntegrate[
xi^(a - 3/2) E^- xi (1 + Erf[-(x/ Sqrt[xi])]), {xi, 0, Infinity}]
(Sqrt[[Pi]] /(
4 (x)^ (a))) ((-(3/2) + a)! -
NIntegrate[(4 (xi^2)^(1/2 (-1 + a)) BesselK[-1 + a, 2 xi])/
Sqrt[[Pi]], {xi, 0, x}])
1/4 x^-a (Sqrt[[Pi]] (-(3/2) + a)! -
2 x Gamma[-1 + a] HypergeometricPFQ[{1/2}, {3/2, 2 - a}, x^2] - (
2 x^(-1 + 2 a)
Gamma[1 - a] HypergeometricPFQ[{-(1/2) + a}, {a, 1/2 + a},
x^2])/(-1 + 2 a))
Out[514]= 0.787477
Out[515]= 0.787477
Out[516]= 0.787477
Out[517]= 0.787477
Out[518]= 0.787477
Having said all this my question would be how do we get the result for arbitrary values of the parameter $b$ ?
