Question

Find one or more values of each of the following complex expressions and compare with acomputer solution.

${\mathbf{(}\mathbf{-}\mathbf{e}\mathbf{)}}^{\mathbf{x}\mathbf{i}}$

Short answer

The value of ${\left(-e\right)}^{xi}$is $-e^{\left(-\left[x^2\pm 2\eta {\mathrm{\pi }}^2\right]\right)}$ where n = o,1,2,3,.....

Step-by-step solution

Given Information.

The given complex number is ${\left(-e\right)}^{\mathrm{\pi i}}$.

Definition of Complex Number.

The numbers that are presented in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$where, 'x' is real numbers and 'iy' is an imaginary number, those numbers are referred to as called Complex numbers.

Find the value of (-eπi).

Let,

$$\begin{gathered} u=\left(-e^{i\mathrm{\pi }}\right) \\ z=In\left(-e\right) \\ u=\left(-e^{i\mathrm{\pi }}\right) \\ =e^{In{\left(-e\right)}^{i\mathrm{\pi }}} \\ =e^{i\mathrm{\pi }\left(\mathrm{In}\left(-\mathrm{e}\right)\right)}.......\left(1\right) \end{gathered}$$

$$\begin{gathered} z=In\left(-e\right) \\ =i\left(\mathrm{\pi }\pm 2\mathrm{n\pi }\right) \\ =i\left(\mathrm{\pi }\pm 2\mathrm{n\pi }\right).....\left(2\right) \\ n=0,1,2,3,..... \end{gathered}$$

Put the value of equation (2) in equation (1).

$$\begin{gathered} u=e^{i\left(\mathrm{\pi }\pm 2\mathrm{n\pi }\right)\left(i\mathrm{\pi }\right)} \\ =e^{-\left({\mathrm{\pi }}^2\pm 2{\mathrm{n\pi }}^2\right)} \\ n=0,1,2,3,..... \end{gathered}$$

Hence, the value of ${\left(-e\right)}^{\mathrm{\pi i}}$is $-e^{-\left({\mathrm{\pi }}^2\pm 2{\mathrm{n\pi }}^2\right)}$ where n = 0,1,2,3,.....