Where A,ω∈R. And I want to record the explanation:
Fact. For any y∈R, we define g(y):R→R by g(y)=∫∞−∞e−A(x+iy)2dx, then g′(y)=0, and therefore ∫∞−∞e−A(x+iy)2dx=g(y)=g(0)=∫∞−∞e−Ax2dx, 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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