Question $\ln (-1+j)$

Studdy Solution

STEP 1

1. The logarithm function $\ln(z)$ is defined for complex numbers $z$ .
2. The complex logarithm is a multi-valued function.
3. The principal value of the complex logarithm is typically considered.

STEP 2

1. Express the complex number in polar form.
2. Determine the magnitude of the complex number.
3. Determine the argument (angle) of the complex number.
4. Use the polar form to find the logarithm.

STEP 3

Express the complex number $-1 + j$ in polar form. A complex number $z = a + bj$ can be expressed in polar form as $z = r(\cos \theta + j\sin \theta)$ or $z = re^{j\theta}$ , where $r$ is the magnitude and $\theta$ is the argument.

STEP 4

Determine the magnitude $r$ of the complex number $-1 + j$ .
$r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}$

STEP 5

Determine the argument $\theta$ of the complex number $-1 + j$ . The argument is the angle the complex number makes with the positive real axis, measured counterclockwise.
$\theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1)$
Since $-1 + j$ is in the second quadrant, $\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$ .

STEP 6

Use the polar form to find the logarithm. The complex logarithm is given by:
$\ln(z) = \ln(r) + j\theta$
Substitute the values of $r$ and $\theta$ :
$\ln(-1 + j) = \ln(\sqrt{2}) + j\frac{3\pi}{4}$
$= \frac{1}{2}\ln(2) + j\frac{3\pi}{4}$
The principal value of $\ln(-1 + j)$ is $\boxed{\frac{1}{2}\ln(2) + j\frac{3\pi}{4}}$ .

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