I have this integral that comes from my research with some Fourier Transforms of spectrum functions:


$$ G(\tau) = \int\_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c\_1 - c\_2 e^{-c\_3 x} ) } dx $$


where $c\_1, c\_2, c\_3, \Lambda, n > 0$.


The only way for me now is to use a series expansion of the term $e^{i \tau ( c\_1 - c\_2 e^{-c\_3 x} ) }$ and take the first few terms. However, I was wondering if there is a smarter way to do it.


====== EDIT =========


Here, $i = \sqrt{-1}$.



Is this integral solvable analytically?
there is a closed form solution for
$$I\_n = \int\_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c\_1 - c\_2 e^{-c\_3 x} ) } dx$$
for integer $n$, for example, for $n=0$:


$$I\_0=\frac{1}{\lambda (c\_3+\lambda)}e^{i c\_1 \tau}$$
$$\qquad\times \left[(c\_3+\lambda) \, \_1F\_2\left(\frac{\lambda}{2 c\_3};\tfrac{1}{2},\tfrac{\lambda}{2 c\_3}+1;-\tfrac{1}{4} c\_2^2 \tau^2\right)-i c\_2 \lambda \tau \, \_1F\_2\left(\tfrac{\lambda}{2 c\_3}+\tfrac{1}{2};\tfrac{3}{2},\tfrac{\lambda}{2 c\_3}+\tfrac{3}{2};-\tfrac{1}{4} c\_2^2 \tau^2\right)\right].$$


for larger integer $n$ the expressions are similar, in terms of hypergeometric functions, but much longer.


