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Fourier Series in Periodic Functions: Adv. Eng. Math HW 6, Assignments of Mathematics

Colorado School of MinesMathematics

A math homework assignment from the advanced engineering mathematics course (math 348) for the summer term of 2009. The assignment focuses on determining the fourier series representations of various periodic functions. It includes five problems, each involving the graph, symmetry analysis, and fourier coefficient calculation for different functions.

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Uploaded on 08/19/2009

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MATH 348 - Advanced Engineering Mathematics July 10, 2009
Homework 6, Summer 2009 Due: July 14, 2009
Fourier Series Representations of Periodic Functions
1. Consider the shifted function,
f(x) = x+α, α ∈R,(1)
which is 2π-periodic on the interval −π < x < π.
(a) Graph fon (−2π, 2π).
(b) Is the function even, odd or neither?
(c) Determine the Fourier coefficients a0, an, bnof f.
(d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa-
tion of fassuming α= 1.
Hint: First reference theorem 11.3.2 on page 492 and then recall that we found the Fourier Series of f(x) = x, x ∈(−π, π)
in class.
2. Consider the function,
f(x) = x2, x ∈[−π, π],(2)
which is 2π-periodic on the stated interval.
(a) Graph fon (−2π, 2π).
(b) Is the function even, odd or neither?
(c) Determine the Fourier coefficients a0, an, bnof f.
(d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa-
tion of f.
Hint: Use symmetry arguments to simplify your calculations.
3. Consider the function,
f(x) = x2, x ∈[0,2π],(3)
which is 2π-periodic on the stated interval.
(a) Graph fon (−4π, 4π).
(b) Is the function even, odd or neither?
(c) Determine the Fourier coefficients a0, an, bnof f.
(d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa-
tion of f.
Hint: This problem can be dealt with one of two ways. One method is to write the function fdown as a piecewise function
from −πto πand proceed as usual. This is cumbersome and a simpler method is availble. First note that the orthogonality
relations, 11.1.1 page 482, HW5.P3, hold for any 2πinterval of integration.1If this is true then the coefficients of the
Fourier series representation of fcan be redeveloped for any 2πinterval. 2In this way it is possible to shift the Fourier
series to use the principle domain, x∈[0,2π], of the periodic function instead of the domain [−π, π ].
1Can you show this?
2What would the formulas look like in this case?
1
pf2

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MATH 348 - Advanced Engineering Mathematics July 10, 2009 Homework 6, Summer 2009 Due: July 14, 2009 Fourier Series Representations of Periodic Functions

  1. Consider the shifted function, f (x) = x + α, α ∈ R, (1) which is 2π-periodic on the interval −π < x < π. (a) Graph f on (− 2 π, 2 π). (b) Is the function even, odd or neither? (c) Determine the Fourier coefficients a 0 , an, bn of f. (d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa- tion of f assuming α = 1. Hint: First reference theorem 11.3.2 on page 492 and then recall that we found the Fourier Series of f (x) = x, x ∈ (−π, π) in class.
  2. Consider the function, f (x) = x^2 , x ∈ [−π, π], (2) which is 2π-periodic on the stated interval. (a) Graph f on (− 2 π, 2 π). (b) Is the function even, odd or neither? (c) Determine the Fourier coefficients a 0 , an, bn of f. (d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa- tion of f. Hint: Use symmetry arguments to simplify your calculations.
  3. Consider the function, f (x) = x^2 , x ∈ [0, 2 π], (3) which is 2π-periodic on the stated interval. (a) Graph f on (− 4 π, 4 π). (b) Is the function even, odd or neither? (c) Determine the Fourier coefficients a 0 , an, bn of f. (d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa- tion of f. Hint: This problem can be dealt with one of two ways. One method is to write the function f down as a piecewise function from −π to π and proceed as usual. This is cumbersome and a simpler method is availble. First note that the orthogonality relations, 11.1.1 page 482, HW5.P3, hold for any 2 π interval of integration.^1 If this is true then the coefficients of the Fourier series representation of f can be redeveloped for any 2π interval. 2 In this way it is possible to shift the Fourier series to use the principle domain, x ∈ [0, 2 π], of the periodic function instead of the domain [−π, π].

(^1) Can you show this? (^2) What would the formulas look like in this case?

1

  1. Let

f (x) =

{ (^0) , − 2 < x < 0 x, 0 < x < 2 be a 4-periodic function. That is f (x + p) = f (x) where p = 4. (a) Graph f on (− 4 , 4). (b) Is the function even, odd or neither? (c) Determine the Fourier coefficients a 0 , an, bn of f. (d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa- tion of f.

  1. Let

u(t) =

{ (^0) , −L < t < 0 Esin(ωt), 0 < t < L where E represents the amplitude and ω the frequency of the output of the half-wave rectifier. Assume that u(t) is a 2L-periodic function. That is u(t + p) = u(t) where p = 2L.

(a) Graph u on (− 2 L, 2 L). (b) Is the function even, odd or neither? (c) Determine the Fourier coefficients a 0 , an, bn of u. (d) Using http://www.tutor-homework.com/grapher.html graph the first five terms of your Fourier Series Representa- tion of u, assuming that E = ω = 1 and L = π. Hint: This problem is an example from section 11.2 of the book. Many steps have been omitted and the goal is to reproduce the work. You should use the example in the text and our example from class to guide your steps.