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Chapter 7: Q21P (page 364)

Write out the details of the derivation of the formulas (8.3)

Short Answer

Expert verified

The value of coefficient is .

Step by step solution

01

Given information

Derivative formulas are given.

02

Concept of Fourier series

The Fourier expansion of function is

03

Apply integration

Integrate above function.

Integrate further as shown below.

04

Evaluate the complex version

The complex version is as follows:

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Most popular questions from this chapter

Sketch several periods of the corresponding periodic function of period 2蟺 . Expand the periodic function in a sine-cosine Fourier series.

f(x)={-1,-蟺<x<蟺21,蟺2<x<蟺,

Use the results 鈭�absin2kxdx=鈭�abcos2kxdx=12(b-a) to evaluate the following integrals without calculation.

(a)鈭�04蟺/3sin2(3x2)dx

(b)鈭�-蟺/23蟺/2cos2(x2)dx

For each of the periodic functions in Problems 5.1 to 5.11 , use Dirichlet's theorem to find the value to which the Fourier series converges at虫=0,卤蟺/2,卤蟺,卤2蟺 .

The functions in Problems 1 to 3 are neither even nor odd. Write each of them as the sum of an even function and an odd function.

(a) einx

(b)xex

The velocity of sound in sea water is about 1530m/sec. Write an equation for a sinusoidal sound wave in the ocean, of amplitude 1 and frequency1000hertz .
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