Exercise 4.2.6 : Suppose f ( t ) is defined on ( - ⇡ , ⇡ ] as f ( t ) = 8 > > < > > : - 1 if - ⇡ < t  0 , 1 if 0 < t  ⇡ . Extend periodically and compute the Fourier series of f ( t ) . Exercise 4.2.7 : Suppose f ( t ) is defined on ( - ⇡ , ⇡ ] as t 3 . Extend periodically and compute the Fourier series of f ( t ) . Exercise 4.2.8 : Suppose f ( t ) is defined on [ - ⇡ , ⇡ ] as t 2 . Extend periodically and compute the Fourier series of f ( t ) . There is another form of the Fourier series using complex exponentials that is sometimes easier to work with. Exercise 4.2.9 : Let f ( t ) = a 0 2 + 1 X n = 1 a n cos( nt ) + b n sin( nt ) . Use Euler’s formula e i ✓ = cos( ✓ ) + i sin( ✓ ) to show that there exist complex numbers c m such that f ( t ) = 1 X m = -1 c m e imt . Note that the sum now ranges over all the integers including negative ones. Do not worry about convergence in this calculation. Hint: It may be better to start from the complex exponential form and write the series as c 0 + 1 X m = 1 c m e imt + c - m e - imt .

166 CHAPTER 4. FOURIER SERIES AND PDES Exercise 4.2.101 : Suppose f ( t ) is defined on [ - ⇡ , ⇡ ] as f ( t ) = sin ( t ) . Extend periodically and compute the Fourier series. Exercise 4.2.102 : Suppose f ( t ) is defined on ( - ⇡ , ⇡ ] as f ( t ) = sin ( ⇡ t ) . Extend periodically and compute the Fourier series. Exercise 4.2.103 : Suppose f ( t ) is defined on ( - ⇡ , ⇡ ] as f ( t ) = sin 2 ( t ) . Extend periodically and compute the Fourier series. Exercise 4.2.104 : Suppose f ( t ) is defined on ( - ⇡ , ⇡ ] as f ( t ) = t 4 . Extend periodically and compute the Fourier series.

4.3. MORE ON THE FOURIER SERIES 167 4.3 More on the Fourier series Note: 2 lectures, §9.2 – §9.3 in [EP], §10.3 in [BD] Before reading the lecture, it may be good to first try Project IV (Fourier series) from the IODE website: . After reading the lecture it may be good to continue with Project V (Fourier series again). 4.3.1 2 L -periodic functions We have computed the Fourier series for a 2 ⇡ -periodic function, but what about functions of di ↵ erent periods. Well, fear not, the computation is a simple case of change of variables. We can just rescale the independent axis. Suppose that we have a 2 L -periodic function f ( t ) ( L is called the half period ). Let s = ⇡ L t . Then the function g ( s ) = f ✓ L ⇡ s ◆ is 2 ⇡ -periodic. We want to also rescale all our sines and cosines. We want to write f ( t ) = a 0 2 + 1 X n = 1 a n cos ✓ n ⇡ L t ◆ + b n sin ✓ n ⇡ L t ◆ . If we change variables to s we see that g ( s ) = a 0 2 + 1 X n = 1 a n cos( ns ) + b n sin( ns ) . We compute a n and b n as before. After we write down the integrals we change variables from s back to t . a 0 = 1 ⇡ Z ⇡ - ⇡ g ( s ) ds = 1 L Z L - L f ( t ) dt , a n = 1 ⇡ Z ⇡ - ⇡ g ( s ) cos( ns ) ds = 1 L Z L - L f ( t ) cos ✓ n ⇡ L t ◆ dt , b n = 1 ⇡ Z ⇡ - ⇡ g ( s ) sin( ns ) ds = 1 L Z L - L f ( t ) sin ✓ n ⇡ L t ◆ dt . The two most common half periods that show up in examples are ⇡ and 1 because of the simplicity. We should stress that we have done no new mathematics, we have only changed variables. If you understand the Fourier series for 2 ⇡ -periodic functions, you understand it for 2 L - periodic functions. All that we are doing is moving some constants around, but all the mathematics is the same.

168 CHAPTER 4. FOURIER SERIES AND PDES Example 4.3.1: Let f ( t ) = | t | for - 1 < t  1, extended periodically. The plot of the periodic extension is given in Figure 4.8. Compute the Fourier