Chapter 7: Q-7-13-23MP (page 389)

Use Parseval’s theorem and Problem 12.11 to evaluate.
$\frac{c o s^{2} (\frac{α π}{2})}{{(1 - α^{2})}^{2}} d α$

Short Answer

Expert verified

The evaluation by Parseval’s theorem is $\int_{0}^{\infty} \frac{\cos^{2} (\frac{α π}{2})}{{(1 - α^{2})}^{2}} d α = \frac{π^{2}}{8}$

Step by step solution

Given Information

Given integralis $\int_{0}^{\infty} \frac{\cos^{2} (\frac{α π}{2})}{{(1 - α^{2})}^{2}} d α$

Definition of Fourier Transform

A function's Fourier transform is a complex-valued function that represents the complex sinusoids that make up the original function.A Fourier transform is a mathematical transformation that decomposes functions that are spatially or temporally dependent into functions that are spatially or temporally dependent.

Find average of the square of solutions of Problem 12.11

Find the Fourier transform in Problem 12.11, that is, $g (α) = \frac{1}{π} \frac{\cos (\frac{π α}{2})}{1 - α^{2}}$ .

Find the average of the square of function over the interval.

$<{|f (x)|}^{2}> = \frac{1}{2 π} \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} x d x = \frac{1}{4 π} \int_{- \frac{π}{2}}^{\frac{π}{2}} (1 + \cos (2 x)) d x = \frac{1}{4 π} {[π + \frac{\sin (2 x)}{2}]}_{- \frac{π}{2}}^{\frac{π}{2}} = \frac{1}{4}$

Use Parseval theorem

By Parseval’s theorem,

$\frac{1}{4} = \int_{- \infty}^{\infty} {|g (α)|}^{2} g α \frac{1}{4} = \frac{1}{π^{2}} \int_{- \infty}^{\infty} \frac{\cos^{2} (\frac{π α}{2})}{{(1 - α^{2})}^{2}} d α \frac{1}{4} = \frac{2}{π^{2}} \int_{0}^{\infty} \frac{\cos^{2} (\frac{π α}{2})}{{(1 - α^{2})}^{2}} d α \int_{0}^{\infty} \frac{\cos^{2} (\frac{π α}{2})}{{(1 - α^{2})}^{2}} d α = \frac{π^{2}}{8}$

Therefore, from the Parseval theorem, $\int_{0}^{\infty} \frac{\cos^{2} (\frac{α π}{2})}{{(1 - α^{2})}^{2}} d α = \frac{π^{2}}{8}$ .

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Use Parseval’s theorem and Problem 12.11 to evaluate.
$\frac{c o s^{2} (\frac{α π}{2})}{{(1 - α^{2})}^{2}} d α$

Short Answer

Step by step solution

Given Information

Definition of Fourier Transform

Find average of the square of solutions of Problem 12.11

Use Parseval theorem

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Use Parseval’s theorem and Problem 12.11 to evaluate.cos2(απ2)(1-α2)2dα

Short Answer

Step by step solution

Given Information

Definition of Fourier Transform

Find average of the square of solutions of Problem 12.11

Use Parseval theorem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Measurements

Thermodynamics

Geometrical and Physical Optics

Work Energy and Power

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Use Parseval’s theorem and Problem 12.11 to evaluate.
$\frac{c o s^{2} (\frac{α π}{2})}{{(1 - α^{2})}^{2}} d α$