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Chapter 7: Q18P (page 364)

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series.
f(x) = x² , 0 < x < 10

Short Answer

Expert verified

The sketch for the function is shown below.

Step by step solution

Given information

The function, f(x) = x² , 0 < x < 10 .

Concept of Complex coefficients

The complex coefficients are:

Obtain values as shown below.

Obtain further as follows:

Then function is,

Draw the graph

The sketch for the function is shown below.

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Most popular questions from this chapter

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) $\int_{0}^{4 π / 3} s i n^{2} (\frac{3 x}{2}) d x$

(b) $\int_{- π / 2}^{3 π / 2} c o s^{2} (\frac{x}{2}) d x$

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) $e^{i n x}$

(b) $x e^{x}$

For each of the periodic functions in Problems 5.1to 5.11.use Dirichlet's theorem to find the value to which the Fourier series converges at $x = 0, \pm π / 2, \pm π, \pm 2 π$ .

Normalize $f (x)$ in Problem 21; that is find the factor Nso that $\int_{- \infty}^{\infty} {| N f (x) |}^{2} = 1$ .Let $ψ (x) = N f (x)$ , and find $ϕ (p)$ as given in Problem 35. Verify Parseval’s theorem, that is, show that $\int_{- \infty}^{\infty} {| ϕ (p) |}^{2} d p = 1$ .

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Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series.f(x) = x2 , 0 < x < 10

Short Answer

Step by step solution

Given information

Concept of Complex coefficients

Draw the graph

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Most popular questions from this chapter

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Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series.
f(x) = x² , 0 < x < 10