How to Simplify a Double Integral with Exponential and Power Functions?

[vineel49]

Homework Statement

$$\left[\int\limits_0^{Inf} {\int\limits_0^{Inf} {e^{ - aX - bY} \cdot F(X + Y + c)} }\cdot X^d \cdot Y^e \cdot dX \cdot dY\right]$$

Homework Equations

a,b,c are constants; d & e are non negative integers; X and Y are variables.
F is a one to one function. Please simplify. The answer is in single Integrals. Leave the Function F as it is.

The Attempt at a Solution

put X+Y=V, Y=U

 

[vineel49]

I am new to this forum, so I am not able to convert it to equation

 

[HallsofIvy]

Try two dollar signs, $ $ without the space, at both ends:
$$\left[\int\limits_0^{Inf} {\int\limits_0^{Inf} {e^{ - aX - bY} \cdot F(X + Y + c)} }\cdot X^d \cdot Y^e \cdot dX \cdot dY\right]$$
I also changed "\[" and "\]" to "\left[" and "\right]",.

Without knowing the function F, I don't see any way to simplify that.

 

[vineel49]

F is a one to one function. Please simplify in such a way that the answer is left out with only a single Integral. Please simplify as much as possible. Leave the Function F as it is.

 

[tiny-tim]

hi vineel49!

Please simplify in such a way that the answer is left out with only a single Integral.

well, the obvious way is to make X + Y + c one of two new variables, and then integrate wrt the other

 

[vineel49]

hi vineel49!


well, the obvious way is to make X + Y + c one of two new variables, and then integrate wrt the other

It is not that simple, I am trying since morning on this one.

 

[tiny-tim]

what did you get when you tried it?

 

[vineel49]

what did you get when you tried it?

You have to use Jacobian Matrix and then solve. Here is the wikipedia link


http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant