Characteristic Function of Gaussian Distribution

Theorem
The characteristic function of the Gaussian distribution with mean $\mu$ and variance $\sigma^2$ is


 * $\map \phi t = e^{i t \mu - \frac 1 2 t^2 \sigma^2}$

Lemma 2
By Lemma 1:


 * $\map \phi t = c \dfrac 1 {\sqrt {2 \pi \sigma^2} } \ds \int_{x \mathop \in \R} e^{-\paren {\frac {x - k} {\sqrt 2 \sigma} }^2} \rd x$

Let $z = \paren {\dfrac {x - k} {\sqrt 2 \sigma} }$.

Then: