Chapter 7: Q2P (page 354)

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

Short Answer

Expert verified

The expansion is $f (x) = \frac{1}{4} + \frac{1}{π} [\sum_{n = 1 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 3 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 1 + 2 m}^{\infty} \frac{\sin n x}{n} + 2 \sum_{n = 2 + 4 m}^{\infty} \frac{\sin n x}{n}] w h e r e m = 0, 1, 2, - - .$

Step by step solution

Given

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

The concept of the Fourier series for the function

The Fourier series for the function f (x):

$f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos nx + b_{n} \sin nx) a_{0} = \frac{1}{π} \int_{- π}^{π} f (x) d x a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) c o s n x d x b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) s i n n x d x$

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:

$a_{0} = a_{a} = 0 f (x) = \sum_{n = 1}^{\infty} b_{n} \sin nx$

From the given information

Coefficients of :

$a_{0} = \frac{1}{π} \int_{- π}^{π} f (x) d x = \frac{1}{π} \int_{- π}^{\frac{π}{2}} d x = \frac{1}{π} {[x]}_{0}^{\frac{π}{2}} = \frac{1}{2}$

Calculate Coefficients of an

Coefficients of :

$a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) s i n n x d x = \frac{1}{π} \int_{0}^{\frac{π}{2}} f (x) s i n d x = \frac{1}{π} {[\cos d x]}_{0}^{\frac{π}{2}} = \frac{1}{π} [1 - \cos (\frac{n π}{2})]$

Calculate Coefficients of bn

Coefficients of ;

$\frac{1}{π}, \frac{2}{2 π}, \frac{1}{3 π}, 0, \frac{1}{5 π}, \frac{2}{6 π}, \frac{1}{7 π}$

Hence the expansion is

$[\sum_{n = 1 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 3 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 1 + 2 m}^{\infty} \frac{\sin n x}{n} + 2 \sum_{n = 2 + 4 m}^{\infty} \frac{\sin n x}{n}] w h e r e m = 0, 1, 2, - - .$

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

Short Answer

Step by step solution

Given

The concept of the Fourier series for the function

From the given information

Calculate Coefficients of an

Calculate Coefficients of bn

One App. One Place for Learning.

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Sketch several periods of the corresponding periodic function of period 2π. Expand the periodic function in a sine-cosine Fourier series.f(x)=0,-π<x<01,0<x<π2,0,π2<x<π.

Short Answer

Step by step solution

Given

The concept of the Fourier series for the function

From the given information

Calculate Coefficients of an

Calculate Coefficients of bn

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Mechanics and Materials

Conservation of Energy and Momentum

Torque and Rotational Motion

Physical Quantities And Units

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

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