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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

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 $x=0,±π/2,±π,±2π$ .

Repeat the example using the same Fourier series but at $x = π / 2$ .

In Problems 3 to 12, find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the x axis are the same.

${cos}^{2} \frac{x}{2} on (0, \frac{π}{2})$

Use the results $\int_{a}^{b} s i n^{2} k x d x = \int_{a}^{b} c o s^{2} k x d x = \frac{1}{2} (b - a)$ to evaluate the following integrals without calculation.

(a) $\int_{0}^{2 π / ω} s i n^{2} ω t d t$

(b) $\int_{0}^{2} c o s^{2} 2 π t d t$

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) $(1 + x) (s i n x + c o s x)$

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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

Recommended explanations on Physics Textbooks

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