Lecture 3.2: Computing Fourier series and exploiting symmetry Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 1 / 8
Exploiting symmetry ∞ � There are many shortcuts to computing Fourier series: f ( x ) = a 0 a n cos n π x L + b n sin n π x 2 + L . n =1 Definition A function f : R → R is even if f ( x ) = f ( − x ) for all x ∈ R , odd if f ( x ) = − f ( − x ) for all x ∈ R . even odd neither x n (even n ) x n (odd n ) x 2 + x 3 . e inx (= cos nx + i sin nx ) cos nx sin nx symmetric about y -axis symmetric about origin neither Why we care � L � L If f is even, then f ( x ) dx = 2 f ( x ) dx . − L 0 � L If f is odd, then f ( x ) dx = 0. − L M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 2 / 8
2 + � a n cos n π x Exploiting symmetry: f ( x ) = a 0 L + b n sin n π x L Big shortcut If f is even, then every b n = 0: � L = 1 � � f , sin n π x f ( x ) sin n π x b n = dx = 0 . L L L − L � �� � even · odd = odd If f is odd, then every a n = 0: � L = 1 � � f , cos n π x f ( x ) cos n π x a n = dx = 0 . L L L − L � �� � odd · even = odd Small shortcut If f is even, then � L � L = 1 dx = 2 � � f , cos n π x f ( x ) cos n π x f ( x ) cos n π x a n = dx . L L L L L − L 0 � �� � even · even = even If f is odd, then � L � L = 1 dx = 2 � � f , sin n π x f ( x ) sin n π x f ( x ) sin n π x b n = dx . L L L L L 0 − L � �� � odd · odd = even M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 3 / 8
An odd square wave Example 1 � 1 0 < x < 1 Consider the square wave of period 2 defined by f ( x ) = − 1 − 1 < x < 0 M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 4 / 8
A sawtooth wave Example 2 Consider the sawtooth wave defined by f ( x ) = x on ( − L , L ) and extended to be periodic. M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 5 / 8
An even function Example 3 Consider the function defined by f ( x ) = x 2 on [ − 1 , 1] and extended to be periodic. M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 6 / 8
The average value of a Fourier series Proposition For any Fourier series ∞ f ( x ) = a 0 � a n cos n π x + b n sin n π x 2 + L , L n =1 the average value of f ( x ) is a 0 2 . M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 7 / 8
Exercise Consider a Fourier series ∞ f ( x ) = a 0 � 2 + a n cos nx + b n sin nx . n =1 What is the Fourier series of the function obtained by (i) reflecting f across the y -axis? (ii) reflecting f across the x -axis? (iii) reflecting f across the origin? M. Macauley (Clemson) Lecture 3.2: Computing Fourier series & symmetry Advanced Engineering Mathematics 8 / 8
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