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

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Homework Help Overview

The discussion revolves around simplifying a double integral involving exponential and power functions. The integral in question includes variables X and Y, constants a, b, c, and non-negative integers d and e, with a function F that is one-to-one.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the potential substitution of variables, particularly suggesting using X + Y + c as a new variable. There is also mention of the Jacobian matrix in relation to the transformation of variables.

Discussion Status

The conversation is ongoing, with participants exploring various approaches to simplify the integral. Some guidance has been offered regarding variable substitution, but there is no consensus on a definitive method or outcome yet.

Contextual Notes

Participants note the challenge posed by the function F, as its specific form is unknown, which complicates the simplification process. There is also an emphasis on leaving the function F unchanged in the final expression.

vineel49
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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
 
Last edited:
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vineel49 said:
I am new to this forum, so I am not able to convert it to equation
 
Last edited:
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.
 
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.
 
hi vineel49! :smile:
vineel49 said:
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 :wink:
 
tiny-tim said:
hi vineel49! :smile:


well, the obvious way is to make X + Y + c one of two new variables, and then integrate wrt the other :wink:
It is not that simple, I am trying since morning on this one.
 
what did you get when you tried it? :smile:
 

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