Let $D_{\nu}(.)$ is the parabolic cylinder function


And $\Gamma(.)$ is the Gamma function.


$s_y(\mu,\nu,t,z)=2^{\nu}\displaystyle\sum_{k=0}^{\infty}\frac{\Gamma(\nu+k)e^{-(\nu t+z+2kt)^2/(4y)}}{\sqrt{2\pi}y^{1+\mu/2}\Gamma(\nu) k!} D_{\nu+1}(\frac{\nu t+z+2kt}{\sqrt{y}})$

and let

$$es_y(\mu,\nu,t,x,z)=\displaystyle\sum_{k=0}^{\infty}\frac{(-z)^k}{k!} s_y(\mu+k,\nu+k,t,x+z+kt) $$

and let

$$is_y(\nu,t,r,z,x)=\displaystyle\sum_{l=0}^{\infty}\frac{x^{\nu+2l}}{\Gamma(\nu+l+1)l!}es_y(1+\nu+2l,1+\nu+2l,t,r,z) $$

Assume that $u,w$ are variables, the other variables are know. I would like to compute an integral like this $$I=\int_{0}^{\infty}\int_0^{\infty}f(u,w) is_a(2\nu,cu,0,w+d,e\sqrt{w}) dudw$$

the integral $I$ can be approximated by truncating the series at large $n$. When i did so, the result converges very slowly, (not very accurate).

My questions: is there a better way to approximate the above double integral ? Do we have a program language that can use to find $s_y(),es_y(.),is_y(.)$ exactly ( I mean without truncating the series)?


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