Chapter 14: Q8P (page 667)

Find the real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$s i n z$

Short Answer

Expert verified

The real part $u (x, y)$ of the function is $\frac{1}{2} \cos x (e^{- y} + e^{y})$ and the imaginary part $v (x, y)$ of the function is $\frac{1}{2} \sin x (e^{- y} - e^{y})$ .

Step by step solution

Definition of complex number

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

Solve complex number

Given the function is $\sin z$ .

From the Euler’s formula, $e^{i x} = \cos x + i \sin x$ ,

$\cos x = \frac{e^{i x} + e^{- i x}}{2}$ , $\sin x = \frac{e^{i x} - e^{- i x}}{2 i}$

Therefore $\sin z$ can be written as:

$\sin z = \frac{e^{i z} + e^{- i z}}{2}$

The complex number can be written as:

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

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

Find real and imaginary parts

Simplify the expression future.

$\frac{e^{i (x + i y)} + e^{- i (x + i y)}}{2} = \frac{e^{i x} \cdot e^{i^{2} y} + e^{- i x} \cdot e^{- i^{2} y}}{2} = \frac{e^{i x} \cdot e^{- y} + e^{- i x} \cdot e^{y}}{2}$

Use Euler’s formula into the obtained expression

$\frac{e^{i x} \cdot e^{- y} + e^{- i x} \cdot e^{y}}{2} = \frac{e^{- y} (\cos x + i \sin x) + e^{y} (\cos x - i \sin x)}{2} = \frac{\cos x (e^{- y} + e^{y}) + i \sin x (e^{- y} - e^{y})}{2} = \frac{1}{2} \cos x (e^{- y} + e^{y}) + i \frac{1}{2} \sin x (e^{- y} - e^{y})$

Hence the real part is $\frac{1}{2} \cos x (e^{- y} + e^{y})$ and imaginary part is $\frac{1}{2} \sin x (e^{- y} - e^{y})$ .

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Find the real and imaginary parts $u (x, y)$ and $v (x, y)$ of the following functions.
$s i n z$

Short Answer

Step by step solution

Definition of complex number

Solve complex number

Find real and imaginary parts

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

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Step by step solution

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Solve complex number

Find real and imaginary parts

One App. One Place for Learning.

Most popular questions from this chapter

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