Q12P - Mathematical Methods in Physical Sciences

Question

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

${\mathbf{\left(}\mathbf{i}\mathbf{\right)}}^{\mathbf{3}\mathbf{+}\mathbf{i}}$

Step-by-Step Solution

Answer

The $\mathrm{x}+\mathrm{iy}$form of the given equation $i^{\left(3+i\right)}$is, $-i{e}^{\left(-\frac{\mathrm{\pi }}{2}\overline{+}2n\mathrm{\pi }\right)}$

Step 1: Given Information.

The given expression is ${\left(i\right)}^{3+i}$.

Step 2: 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.

Step 3: Convert in rectangular form.

Consider, $w=i^{\left(3+i\right)}$.

 

Write the given number in polar form.

$w=e^{\left[\left(3+i\right)\mathrm{In}\left(i\right)\right]}$

 

Use the logarithm property.

$$\begin{gathered} w=e^{\left[\left(3+i\right)\mathrm{In}\left(i\right)\right]} \\ w=e^{\left[\left(3+i\right)(\mathrm{In}\left(1\right)+i\left(\frac{\mathrm{\pi }}{2}\overline{+}2\mathrm{n\pi }\right)\right]} \\ w=e^{\left[\left(3+i\right)i\left(\frac{\mathrm{\pi }}{2}\underline{+}2\mathrm{n\pi }\right)\right]} \\ w=e^{\left(-\frac{\mathrm{\pi }}{2}\overline{+}2n\mathrm{\pi }+\mathrm{i}\left(\frac{3\mathrm{\pi }}{2}\underline{+}6\mathrm{n\pi }\right)\right)} \\ w=e^{\left(-\frac{\mathrm{\pi }}{2}\overline{+}2n\mathrm{\pi }\right)}{e}^{\left(i\left(\frac{3\mathrm{\pi }}{2}\underline{+}6\mathrm{n\pi }\right)\right)} \\ w=-i{e}^{\left(-\frac{\mathrm{\pi }}{2}\overline{+}2n\mathrm{\pi }\right)} \end{gathered}$$

Where $n=0,\underline{+}1,\underline{+}2,\underline{+}3,...$

 

Therefore, $x=iy$form of the given equation $i^{\left(3+i\right)}$is $-i{e}^{\left(\frac{\mathrm{\pi }}{2}\overline{+}2n\mathrm{\pi }\right)}$