Chapter 7: Q5P (page 355)

Sketch several periods of the corresponding periodic function of period $2 τ τ$ . Expand the periodic function in a sine-cosine Fourier series.
role="math" localid="1659236419546" $f (x) = {\begin{matrix} 0, & - π < x < 0 \\ - 1, & 0 < x < \frac{π}{2} \\ 1, & \frac{π}{2} < x < π \end{matrix}$

Short Answer

Expert verified

The answer of the given function is

$f (x) = - \frac{2}{π} [\sum_{n = 1 + 4 m}^{\infty} \frac{\cos n x}{n} - \sum_{n = 1 + 4 m}^{\infty} \frac{\cos n x}{n}] - \frac{4}{π} [\sum_{n = 1 + 4 m}^{\infty} \frac{\sin n x}{n}] w h e r e m = 0, 1, 2 . . . .$

Step by step solution

Given

The given function is $f (x) = {\begin{matrix} 0, & - π < x < 0 \\ - 1, & 0 < x < \frac{π}{2} \\ 1, & \frac{π}{2} < x < π \end{matrix}$

The concept of the Fourier series for the function f(x)

The Fourier series for the function $f (x)$ :

$f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos nx + b_{n} \sin nx) a_{0} = \frac{1}{π} \int_{- π}^{π} f (x) dx a_{0} = \frac{1}{π} \int_{- π}^{π} f (x) \cos nxdx b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \sin nxdx If f (x) is an even function : b_{n} = 0 f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} a_{n} \cos nx If f (x) is an odd function : a_{0} = a_{n} = 0 f (x) = \sum_{n = 1}^{\infty} b_{n} \sin nx$

From the given information

The sketch of the given function shown below.

Coefficients of $a_{n}$ :

$a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \cos nxdx = \frac{1}{π} [- \int_{0}^{\frac{π}{2}} cosnxdx + \int_{\frac{π}{2}}^{π} cosnxdx] = \frac{1}{nπ} [- {[\sin nx]}^{\frac{π}{2}} 0 + {[\sin nx]}_{\frac{π}{2}}^{π}] = - \frac{2}{nπ} \sin (\frac{nπ}{2})$

Calculate Coefficients of bn

Coefficients of $b_{n}$ are given below.

$b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \sin n x d x = \frac{1}{π} [- \int_{0}^{\frac{π}{2}} s in n x d x + \int_{\frac{π}{2}}^{π} \sin n x d x] = \frac{1}{nπ} [{[\cos n x]}_{0}^{\frac{π}{2}} - {[\cos n x]}_{\frac{π}{2}}^{π}] = \frac{1}{nπ} [2 \cos (\frac{n π}{2}) - \{1 + {(- 1)}^{n}\}] H e n c e t h e e x p a n s i o n i s f (x) = - \frac{2}{π} [\sum_{n = 1 + 4 m}^{\infty} \frac{\cos n x}{n} - \sum_{n = 3 + 4 m}^{\infty} \frac{\cos n x}{n}] - \frac{4}{π} [\sum_{n = 2 + 4 m}^{\infty} \frac{\sin n x}{n}] w h e r e m = 0, 1, 2 - -$

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Sketch several periods of the corresponding periodic function of period $2 τ τ$ . Expand the periodic function in a sine-cosine Fourier series.
role="math" localid="1659236419546" $f (x) = {\begin{matrix} 0, & - π < x < 0 \\ - 1, & 0 < x < \frac{π}{2} \\ 1, & \frac{π}{2} < x < π \end{matrix}$

Short Answer

Step by step solution

Given

The concept of the Fourier series for the function f(x)

From the given information

Calculate Coefficients of bn

One App. One Place for Learning.

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Sketch several periods of the corresponding periodic function of period 2ττ . Expand the periodic function in a sine-cosine Fourier series.role="math" localid="1659236419546" f(x)={0,-π<x<0-1,0<x<π21,π2<x<π

Short Answer

Step by step solution

Given

The concept of the Fourier series for the function f(x)

From the given information

Calculate Coefficients of bn

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Turning Points in Physics

Modern Physics

Particle Model of Matter

Scientific Method Physics

Fields in Physics

Study anywhere. Anytime. Across all devices.

Sketch several periods of the corresponding periodic function of period $2 τ τ$ . Expand the periodic function in a sine-cosine Fourier series.
role="math" localid="1659236419546" $f (x) = {\begin{matrix} 0, & - π < x < 0 \\ - 1, & 0 < x < \frac{π}{2} \\ 1, & \frac{π}{2} < x < π \end{matrix}$