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Sunday, August 16, 2015

On a Question in Fourier Transform

Someone asked about the second last line below:
Where A,ωR. And I want to record the explanation:

Fact. For any yR, we define g(y):RR by g(y)=eA(x+iy)2dx, then g(y)=0, and therefore eA(x+iy)2dx=g(y)=g(0)=eAx2dx, and the last one is well-known that can be converted to the standard normal density function.

Proof. Just note that g(y)=g(y) and therefore g(y)(1)=g(y), next by the formula of g(y) we can show that g(y)=g(y), therefore g(y)=g(y)=g(y)g(y)=0.

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