How Can Cylindrical Coordinates Simplify Complex Number Integration?

[tanaygupta2000]

Homework Statement

Show that (1/π) ∫∫d(Re{a})d(Im{a}) |a><a| = I

where |a> is a coherent state = exp(-a*a/2) (a^n)/√n! |n>
and I is identity operator

Relevant Equations

|a> = exp(-a*a/2) Σ(a^n)/√n! |n>
<a| = exp(-a*a/2) Σ(a*^n)/√n! <n|
|n><n| = I

I began this solution by assuming a = x+iy since a is a complex number.
So I wrote expressions of <a| and |a> in which |n><n| = I.
I got the following integral:

Σ 1/πn! ∫∫ dx dy exp[-(x^2 + y^2)] (x^2 + y^2)^n I

I tried solving it using Integration by Parts but got stuck in the (x^2 + y^2)^n part.
Please help how can I evaluate this integral in an easier way.
Thank You !

 

[anuttarasammyak]

Why don't you regard x and y are coordinates of xy plane and achieve integration by cylindrical coordinates i.e.
\int dx \int dy = \int_0^{2\pi} d \phi \int_0^\infty r dr

 

[tanaygupta2000]

Why don't you regard x and y are coordinates of xy plane and achieve integration by cylindrical coordinates i.e.
\int dx \int dy = \int_0^{2\pi} d \phi \int_0^\infty r dr

Yess! It very well worked! Lots of Thanks.