Characteristic Function of Gaussian Distribution/Lemma 1

Lemma for Characteristic Function of Gaussian Distribution
Let $\map \phi t$ denote the characteristic function of the Gaussian distribution with mean $\mu$ and variance $\sigma^2$.

Let:


 * $k = \mu + i t \sigma^2$
 * $c = e^{\mu i t - \frac 1 2 t^2 \sigma^2}$

Then:


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

Proof
The characteristic function is defined as

Begin by verifying that:


 * $i t x - \dfrac {\paren {x - \mu}^2} {2 \sigma^2} = -\dfrac {\paren {x - k}^2 + 2 \mu i t \sigma^2 - t^2 \sigma^4} {2 \sigma^2}$

We can then simplify the integral in $(1)$: