Q7P Find the real and imaginary part... [FREE SOLUTION]

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.

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

Use Euler鈥檚 formula into the obtained expression

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

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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91影视

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.

One App. One Place for Learning.

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91影视

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

Medical Physics

Waves Physics

Dynamics

Fundamentals of Physics

Electric Charge Field and Potential

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

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