How should I calculate $\int_{0}^{\infty}\exp(-x-a\cdot(1/x)^b)dx$

Question

$$\int_{0}^{\infty}\exp(-x-a\cdot(1/x)^b)\,\mathrm{d}x$$

A similiar question solved. I even cannot get an answer from wolfram. Is there any thought on this problem?

Answer 1

Assuming that $b$ is an integer and $a>0$, if we exclude the simple cases $b=0$ and $b=1$, we face Meijer G functions.

$$\color{blue}{I_b=\int_0^\infty \exp\Big[ x-\frac a {x^b}\Big]\,dx}$$

$$I_2=\frac {1} {\pi^{1/2}}G_{0,3}^{3,0}\left(\frac{a}{2 ^2}| \begin{array}{c} 0,\frac{1}{2},1 \end{array} \right)$$ $$I_3=\frac {\sqrt 3} {2\pi}G_{0,4}^{4,0}\left(\frac{a}{3^3}| \begin{array}{c} 0,\frac{1}{3},\frac{2}{3},1 \end{array} \right)$$ $$I_4=\frac {1} {\sqrt{2} \pi ^{3/2}}G_{0,5}^{5,0}\left(\frac{a}{4^4}| \begin{array}{c} 0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1 \end{array} \right)$$ $$I_5=\frac {\sqrt{5}} {4 \pi ^{2}}G_{0,6}^{6,0}\left(\frac{a}{5^5}| \begin{array}{c} 0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1 \end{array} \right)$$ $$I_6=\frac {\sqrt{3}} {4 \pi ^{5/2}}G_{0,7}^{7,0}\left(\frac{a}{6^6}| \begin{array}{c} 0,\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6},1 \end{array} \right)$$ I suppose that we can conjecture that $$\color{red}{I_b=\sqrt{b \,(2 \pi )^{1-b}}\,\,G_{0,(b+1)}^{(b+1),0}\left(\frac{a}{b^b}| \begin{array}{c} 0,\frac{1}{b},\frac{2}{b},\cdots,\frac{b-1}{b},1 \end{array} \right)}$$