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

Particle Model of Matter

Astrophysics

Famous Physicists

Conservation of Energy and Momentum

Scientific Method Physics

Wave Optics

Study anywhere. Anytime. Across all devices.

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