Simplifying the Pauli matrix expression $e^{-i\sigma_x\phi/2}$

Question

As in the title, the expression is:

$$e^{-i \sigma_x \phi/2}$$

Where $\sigma_x$ is:

$$\left\{ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right\}$$

Where would I even begin in simplifying this?

Answer 1

We have: $\sigma_{x}^2 = I$, so $\sigma_{x}^{2n} = I$ and $\sigma_{x}^{2n+1} = \sigma_{x}$. Then power series: \begin{align} \exp(-i\phi\sigma_{x}/2)) &= \sum_{k = 0}^{\infty} \frac{(-i\phi/2)^{k} \sigma_{x}^{k}}{k!} \\ &= \sum_{k = 0}^{\infty} \frac{(-i\phi/2)^{2k} \sigma_{x}^{2k}}{(2k)!} + \sum_{k = 0}^{\infty} \frac{(-i\phi/2)^{2k+1} \sigma_{x}^{2k+1}}{(2k+1)!} \\ &= \sum_{k = 0}^{\infty} \frac{(-1)^{k}(\phi/2)^{2k}}{(2k)!} I - i\sum_{k = 0}^{\infty} \frac{(-1)^{k}(\phi/2)^{2k+1}}{(2k+1)!} \sigma_{x} \end{align} These power series are pretty well known . . .