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===== Statement =====
We give a short proof of the expression for the characteristic function of a centered standard gaussian random variable. 
Let $X\sim N(0,1)$. The aim is to show that $\varphi(u)=\PE[\rme^{\rmi u X}]=\rme^{-u^2/2}$ for all $u \in \rset$. 

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$\bproof$
First write 
$$
\varphi(u)=\int \rme^{\rmi u x}  \frac{\rme^{-x^2/2}}{\sqrt{2\pi}} \rmd x=\rme^{-u^2/2} \int \underbrace{\frac{\rme^{-(x-\rmi u)^2/2}} {\sqrt{2\pi}}}_{\psi(u,x)} \rmd x 
$$
Denote $A(u)=\int \psi(u,x)\rmd x$. Since $A(0)=1$, we only need to show that $A$ is constant. But, note that $\frac{\partial \psi(u,x)}{\partial u} =-\rmi \frac{\partial \psi(u,x)}{\partial x} $ so that 
$$
A'(u)\stackrel{(\star)}{=}\int \frac{\partial \psi(u,x)}{\partial u} \rmd x=-\int \rmi\frac{\partial \psi(u,x)}{\partial x}\rmd x=\left[-\rmi \psi(u,\cdot)\right]_{-\infty}^{+\infty}=-\rmi \left[\rme^{-(x-\rmi u)^2/2}/\sqrt{2\pi}\right]_{x \to -\infty}^{x \to \infty}=0
$$
Note that $(\star)$ is true since for all $u\in \rset$ and all bounded neighborhood $V$ of $u$, 
$$
\int \sup_{u'\in V} \left|\frac{\partial \psi(u',x)}{\partial x}\right|  \rmd x\leq \int [|x|+\sup_{u'\in V} |u'|] \rme^{-x^2/2+\sup_{u'\in V}|u'|^2/2} \rmd x<\infty
$$ 
The proof is completed. 
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