Chapter 14: Q7P (page 667)

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

Short Answer

Expert verified

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

Step by step solution

Definition of complex number and formula of hyperbolic function

Complex numberis represented by $a + i b$ , where a is the real number and b is the imaginary number.
Formula of hyperbolic cosine function: $c o s h x = \frac{e^{x} + e^{- x}}{2}$ .
Formula of hyperbolic sine function: $s i n h x = \frac{e^{x} - e^{- x}}{2}$ .

Solve complex number

Given the function is $\cosh z$ written as $\frac{e^{z} + e^{- z}}{2}$ .

The complex number z can be written as $z = x + i y$ , where x is a real part and y is an imaginary part.

Therefore $\cosh z$ is simplified as $\frac{e^{x + i y} + e^{- (x + i y)}}{2}$ .

Find real and imaginary parts.

Simplify the expression future.

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

Use Euler’s formula into the obtained expression

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

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

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

Short Answer

Step by step solution

Definition of complex number and formula of hyperbolic function

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.coshz

Short Answer

Step by step solution

Definition of complex number and formula of hyperbolic function

Solve complex number

Find real and imaginary parts.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

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Astrophysics

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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.
$c o s h z$