Why Is the Rayleigh Sommerfeld Integral Challenging to Compute Directly?

[chiraganand]

Hi, I have read about the rayleigh sommerfeld integral and its a surface integral. Why is it difficult to calculate the integral directly?

 

[Ssnow]

Can you give a reference for this question ?
Ssnow

 

[berkeman]

@chiraganand was last at PF in Sept. 2020, but may still be receiving e-mail notifications of updates to his threads. Hopefully he will be able to respond.

 

[chiraganand]

Can you give a reference for this question ?
Ssnow

Oh wow I had moved on from this one, but thanks for replying! This is the the reference. so there are a lot of different ways to make it easier to calculate but my main question was how would one go about solving it without any simplifications and at which step would one get stuck

 

[Ssnow]

Ok, now I understand the question, the principal problem is that the surface integral is not a simple surface integral but it has an oscillating part given by the oscillatory complex factor $e^{ik|\vec{R}_{2}-\vec{r}_{1}|}$. This is the reason you can try to have an asymptotic expansion using different ways as the stationary phase expansion or others ... In addition you have the variable up to the exponent and also in the denominator- You can try to express it in terms of special functions as $\int_{0}^{t}\frac{e^{x}}{x^{\alpha}}dx$ but to have an exact primitive I don't think it is simple ...
Ssnow