Is there Any way to determine solution for this integration?

[Walter Roberson]

Is K(x) the Jacobi K function?

[Bajdar Nouredine]

dear @Walter Roberson thank you for reply,

suppose K is constant (I edited the equation)

[Paul]

Is the dtheta * dy backwards in the lower integral? It kind of looks like the inner integral is over y and the outer integral over theta.

[Bajdar Nouredine]

dear @Paul yes

Answer 1

[Walter Roberson]

The answer to your question seems to be NO, that there is no way to get the solution.

If you integrate by y in the inner integral and theta in the outer integral, then you can convert two layers of integration to get polylog and psi function calls together with one-layer-deep integrations. Each integration can be expressed as the sum of three terms, and if you break that into the sum of three integrals, two of the pieces integrate easily.

So you can get down to expressions involving where j is an integer.

Although that involves division by sin(0) as long as K is positive that would give exp(-infinity) which goes to zero faster than sin(theta)^3 does so it appears to be convergent. But there does not appear to be a closed form for it.

For any given K value, the integrals can be evaluated numerically, and combined numerically with all of the polylogs, and you can come up with approximate formulas involving p. Just not a symbolic form.