Question

Evaluate each of the following in $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$form, and compare with a computer solution.

$\mathbf{\left(}\mathit{i}\mathbf{\right)}\mathit{l}{\mathit{n}}^{\mathbf{i}}$

Short answer

The $\mathbf{x}+\mathbf{iy}$form of the given equation $i^{\ln i}$ is $e^{\left(-{\left(\frac{\pi }{2}+2n\pi \right)}^2\right)}$

Step-by-step solution

Given Information.

The given expression is $\left(i\right){\ln }^i$.

Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$in which $\mathit{x}$is the real part and $\mathit{y}$is the imaginary part.

Convert in rectangular form.

Consider $w=(i{)}^{\ln i}$.

Write the given number in polar form.

$w=e^{\left[\ln {\left(i\right)}^{\ln \left(i\right)}\right]}$

Use the logarithm property.

$$\begin{gathered} w=e^{\left[\ln \right(i\left)\ln \right(i\left)\right]} \\ w=e^{\left(\left(\ln \left(1\right)+\left(\frac{\pi }{2}+2n\pi \right)i\right)\left(\ln \left(1\right)+\left(\frac{\pi }{2}+2n\pi \right)i\right)\right)} \\ w=e^{\left({\left(\frac{\pi }{2}+2n\pi \right)}^2{i}^2\right)} \end{gathered}$$

$w=e^{\left(-{\left(\frac{\pi }{2}+2n\pi \right)}^2\right)}$

Where $n=0,\pm 1,\pm 2,\pm 3,\dots$

Therefore, $x+iy$form of the given equation $i^{\ln i}$ is $e^{\left(-{\left(\frac{\pi }{2}+2n\pi \right)}^2\right)}$.