Lecture 3.6: Real vs. complex Fourier series Matthew Macauley - PowerPoint PPT Presentation

Lecture 3.6: Real vs. complex Fourier series Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 1 / 6

Overview Last time, we derived formulas for the complex Fourier series of a function. Complex Fourier series If f ( x ) is a piecewise continuous 2 L -periodic function, then we can write ∞ ∞ i π nx i π nx + c − n e − i π nx � � � L � f ( x ) = c n e = c 0 + c n e L L n = −∞ n =1 where � L � L = 1 = 1 i π nx f ( x ) e − i π nx � � � L � c 0 = f , 1 f ( x ) dx , c n = f , e dx . L 2 L 2 L − L − L Here, we will see how to go between the real and complex versions of a Fourier series. It’s just a simple application of the following identities that we’ve already seen: Euler’s formula (and consequences) e i θ = cos θ + i sin θ , e − i θ = cos θ − i sin θ , cos θ = e i θ + e − i θ sin θ = e i θ − e − i θ , . 2 2 i M. Macauley (Clemson) Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 2 / 6

From the real to the complex Fourier series Proposition ∞ The complex Fourier coefficients of f ( x ) = a 0 � a n cos n π x + b n sin n π x 2 + are L L n =1 c n = a n − ib n c − n = a n + ib n , . 2 2 M. Macauley (Clemson) Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 3 / 6

From the complex to the real Fourier series Proposition ∞ i π nx � The real Fourier coefficients of f ( x ) = c n e are L n = −∞ a n = c n + c − n , b n = i ( c n − c − n ) . M. Macauley (Clemson) Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 4 / 6

Computations Example 1: square wave � 1 , 0 < x < π Find the complex Fourier series of f ( x ) = − 1 , π < x < 2 π. M. Macauley (Clemson) Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 5 / 6

Computations Example 2 Compute the real Fourier series of the 2 π -periodic extension of the function e x defined on − π < 0 < π . M. Macauley (Clemson) Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 6 / 6