Question

Evaluate each of the following in x + iyform, and compare with a computer solution.

ln ( i - 1 )

Short answer

The x+iy form of the given equation $\ln (i-1)$ is, $\ln \left(\sqrt{2}\right)+\left(\frac{3\mathrm{\pi }}{4}+2n\mathrm{\pi }\right)i$.

Step-by-step solution

Given Information.

The given expression is $(-e)$.

Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x+iy in which x is the real part and y is the imaginary part.

Convert in polar form.

Consider, $z=i-1$.

Write the polar form of the number.

$z=e^{\left(i\theta \right)}$

The angle is located in second quadrant so the angle must be accordingly.

$$\begin{gathered} r=\left|i-1\right| \\ r=\sqrt{2} \\ \theta =\mathrm{\pi }-\frac{\mathrm{\pi }}{4} \\ \mathrm{\theta }=\frac{3\mathrm{\pi }}{4} \end{gathered}$$

Put the values in the polar form.

$z=\sqrt{2}{e}^{\left(\frac{3\mathrm{\pi }}{4}i\right)}$

Write in the form of x+iy .

Convert the polar form into the rectangular form.

$$\begin{gathered} w=\ln \left(r{e}^{\left(\theta i\right)}\right) \\ w=\ln \left(r\right)+\left(\frac{3\mathrm{\pi }}{4}+2n\mathrm{\pi }\right)i \\ w=\ln \left(\sqrt{2}\right)+\left(\frac{3\mathrm{\pi }}{4}+2n\mathrm{\pi }\right)i \\ w=\ln \left(\sqrt{2}\right)+\left(\frac{3\mathrm{\pi }}{4}+2n\mathrm{\pi }\right)iwheren=0\pm ,1\pm 2,\pm 3,....... \end{gathered}$$

Therefore, the x+iy form of the given equation $\ln (i-1)$ is $\ln \left(\sqrt{2}\right)+\left(\frac{3\mathrm{\pi }}{4}+2n\mathrm{\pi }\right)i$.