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Assume I have a function:
f[x,y]
This function has two real values (a,b) that are very small O(0).
My end result should be:
[tex] F[a,b] = \int_{y_a}^{y_b} \int_{x_a}^{x_b} dx dy f[x,y] [/tex]
Whereupon I would then expand F about a=0 and b=0.
Now my problem is that the integration is very difficult. So much so that mathematica takes about 20 minutes to do the full dual integral, and I really end up doing a numerical integration. But I need an analytic form as well.
My question is, can I somehow taylor expand the original function f[x,y] in (a,b) FIRST, then integrate that? What would I have to do about the limits? xa and xb, the limits of integration for x, are actually functions of (a,b,y) and you and yb are functions of (a,b).
These functions of (a,b) are also non-trivial, may include powers of inverse order.
Is that a clear question? In the end I know (a,b) leading order contributions are of the order [tex]a^{-2}, b^{0}[/tex]
tl;dr : How do I Taylor expand a dual integral whose limits are also functions of the expansion parameter.
f[x,y]
This function has two real values (a,b) that are very small O(0).
My end result should be:
[tex] F[a,b] = \int_{y_a}^{y_b} \int_{x_a}^{x_b} dx dy f[x,y] [/tex]
Whereupon I would then expand F about a=0 and b=0.
Now my problem is that the integration is very difficult. So much so that mathematica takes about 20 minutes to do the full dual integral, and I really end up doing a numerical integration. But I need an analytic form as well.
My question is, can I somehow taylor expand the original function f[x,y] in (a,b) FIRST, then integrate that? What would I have to do about the limits? xa and xb, the limits of integration for x, are actually functions of (a,b,y) and you and yb are functions of (a,b).
These functions of (a,b) are also non-trivial, may include powers of inverse order.
Is that a clear question? In the end I know (a,b) leading order contributions are of the order [tex]a^{-2}, b^{0}[/tex]
tl;dr : How do I Taylor expand a dual integral whose limits are also functions of the expansion parameter.