Question

Question: Solve the system

$\frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\overrightarrow{\mathbf{x}}$

for a 2x2 matrix A with complex eigenvalues$\mathit{p}\mathbf{\pm }\mathit{i}\mathit{q}$.

2. Find${\mathit{e}}^{\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\pi }\mathbf{i}}$.

Short answer

Answer:

The solution is i.

Step-by-step solution

Euler’s formula 

$e^{it}=cost+isint$

Explanation of the solution

To find the value of $e^{\frac{1}{2}\pi i}$.

By using the Euler’s formula as follows.

$$\begin{gathered} e^{it}=cost+isint \\ {e}^{\frac{1}{2}\pi i}=cos\left(\frac{\pi }{2}\right)+isin\left(\frac{\pi }{2}\right) \\ {e}^{i\frac{1}{2}\pi }=0+i \\ {e}^{i\frac{1}{2}\pi }=i \end{gathered}$$

Hence, the value of$e^{\frac{1}{2}\pi i}$ is i .