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.

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