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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

Given information

Derivative formulas are given.

Concept of Fourier series

The Fourier expansion of function is

Apply integration

Integrate above function.

Integrate further as shown below.

Evaluate the complex version

The complex version is as follows:

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Most popular questions from this chapter

In each of the following problems you are given a function on the interval $- π < x < π$ .Sketch several periods of the corresponding periodic function of period $2 τ τ$ . Expand the periodic function in a sine-cosine Fourier series,

$f (x) = {\begin{cases} - x, - π < x < 0 \\ x, 0 < x < π \end{cases}$

From the fact you know, find in your head the average value of

a) $x^{3} - 3sinh2x + {sin}^{2} πx + cos3πx$ on $(- 5,5)$

b) ${2sin}^{2} 3x - 4cosx + 5xcosh2x - {xcos}^{2} x$ on $(- π,π)$

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) $I n | 1 - x |$ (b) $(1 + x) (s i n x + c o s x)$

$f (x) = \frac{1}{2} a_{0} + \sum_{1}^{\infty} a_{n} c o s n x + \sum_{1}^{\infty} b_{n} s i n n x = \sum_{- \infty}^{\infty} c_{n} e^{i n x}$ , use Euler's formula to find $a_{n}$ and $b_{n}$ in terms of $c_{n}$ and $- c_{n}$ , and to find $c_{n}$ and $- c_{n}$ in terms of $a_{n}$ and $b_{n}$ a.

In each case, show that a particle whose coordinate is (a) x = Re z , (b) y =Im z , is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion.

$z = 5 e^{i t}$

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

Short Answer

Step by step solution

Given information

Concept of Fourier series

Apply integration

Evaluate the complex version

One App. One Place for Learning.

Most popular questions from this chapter

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