I have the following integral.
$$ G(q) = \int_{0}^{\infty} y^\alpha \exp(-y) \exp(-j b q\exp(-c y)) dy $$
with $$ \alpha \in \mathbb{R}^+, \quad q, b, c \in \mathbb{R}, \quad j = \sqrt{-1}$$
I have asked this before in different forms, but I have settled down with the following approximation using Gauss-Laguerre Polynomial.
$$ G(q) = \sum_{i = 1}^n w_i f(y_i), $$
where, $$ w_i = \frac{\Gamma(\alpha + n + 1)y_i}{n! (n+1)^2 [L^{\alpha}_{n+1}(y_i)]^2} $$ $$ f(y) = \exp(-j b q\exp(-c y)) $$
And $y_i$ are the zeros of the Laguerre polynomial $L_n^{\alpha}(x)$. To compute this faster (as I need to compute it for several $q$, $\alpha$, and $c$ combinations), I have stored the zeros of $L_n^{\alpha}(x)$ in a matrix form for several $\alpha$ values.
The questions I am asking myself:
- Can this still be reduced to a nicer form if not a closed form?
- Can the original integral be computed in the complex plane?
- I want to create an array of $G(q)$ with $q$. And I have seen the following: $$ G(-q) = G^*(q), $$ where * represents conjugate. And $G(q)$ with $q$ increasing from $0$ behaves like a decaying complex exponential. Can something be done to approximate this with this information? Can a decaying complex exponential function be formulated to approximate this function but still have all its parameters, such as $b, c, \alpha$?
We can express by infinte series:
$$\int_0^{\infty } y^{\alpha } \exp (-y) \exp (-i b q \exp (-c y)) \, dy=\sum _{m=0}^{\infty } \frac{(1+c m)^{-1-\alpha } (-i b q)^m \Gamma (1+\alpha )}{m!}$$ with 10 terms we have 10 correct digits.
If $\alpha \in \mathbb{Z}_{>\, 0} $ then: