Q9P Find the real and imaginary part... [FREE SOLUTION]

Chapter 14: Q9P (page 667)

Find the real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$\frac{1}{z}$

Short Answer

Expert verified

The real part $u (x, y)$ of the function is $\frac{x}{x^{2} + y^{2}}$ and the imaginary part $v (x, y)$ of the function is $\frac{y}{x^{2} + y^{2}}$ .

Step by step solution

Definition of complex number

Complex number is represented by, where $a + i b$ is the a real number and bis the imaginary number.

Solve complex number

Given the function is $\frac{1}{z}$ .

The complex number z can be written as: $\frac{1}{x + i y}$

$z = x + i y$ , where x is a real part and y is an imaginary part.

The function $\frac{1}{z}$ can be written as

Multiply with complex conjugate of $z = x + i y$ into denominator and numerator to separate real and imaginary parts.

$\frac{1}{z} = \frac{(x - i y)}{(x + i y) (x - i y)} = \frac{(x - i y)}{x^{2} - i^{2} y^{2}} = \frac{(x + i y)}{x^{2} + y^{2}}$

Find real and imaginary parts

Simplify the expression future

\frac{(x + i y)}{x^{2} + y^{2}} = \frac{x}{x^{2} + y^{2}} + i \frac{y}{x^{2} + y^{2}}

Hence the real part is $\frac{x}{x^{2} + y^{2}}$ and imaginary part is $\frac{y}{x^{2} + y^{2}}$ .

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91影视

Find the real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$\frac{1}{z}$

Short Answer

Step by step solution

Definition of complex number

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Find the real and imaginary parts u(x,y)and v(x,y)of the following functions.1z

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Step by step solution

Definition of complex number

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Find real and imaginary parts

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Measurements

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Study anywhere. Anytime. Across all devices.

Find the real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$\frac{1}{z}$