Chapter 2: Q6P (page 74)

Question: Evaluate each of the following in x+yform, and compare with a computer solution.\
$\ln (\frac{1 - i}{\sqrt{2}})$

Short Answer

Expert verified

The x+y form of the given equation $\ln (\frac{1 - i}{\sqrt{2}})$ is $(\frac{7 π}{4} + 2 n π) i .$

Step by step solution

Given Information.

The given expression is, $\ln (\frac{1 - i}{\sqrt{2}})$

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.

Step 3:Convert in polar form.

Consider $z = (\frac{1 - i}{\sqrt{2}})$

Write the polar form of the number.

$z = r e^{(i θ)}$

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

$r = |\frac{1 - i}{\sqrt{2}}| r = 1 θ = 2 π - \frac{π}{4} θ = \frac{7 π}{4}$

Put the values in the polar form.

$z = e^{(\frac{7 π}{4} i)}$

Write in the form of x+y .

Convert the polar form into the rectangular form.

$w = \ln (r e^{(θ i)}) w = \ln (r) + (\frac{7 π}{4} + 2 n π) i w = \ln (1) + (\frac{7 π}{4} + 2 n π) i w = \ln (1) + (\frac{7 π}{4} + 2 n π) i w h e r e n = 0, \pm 1, \pm 2, \pm 3, \dots$

Therefore, the form of the given equation $\ln (\frac{1 - i}{\sqrt{2}})$ is, $(\frac{7 π}{4} + 2 n π) i$ .

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Question: Evaluate each of the following in x+yform, and compare with a computer solution.\
$\ln (\frac{1 - i}{\sqrt{2}})$

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Step 3:Convert in polar form.

Write in the form of x+y .

One App. One Place for Learning.

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Question: Evaluate each of the following in x+yform, and compare with a computer solution.\ln1-i2

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Step 3:Convert in polar form.

Write in the form of x+y .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Electromagnetism

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Study anywhere. Anytime. Across all devices.

Question: Evaluate each of the following in x+yform, and compare with a computer solution.\
$\ln (\frac{1 - i}{\sqrt{2}})$