Fourier Series: A Method to Solve Problems
Fourier Series
Fourier series started life as a method to solve problems about the flow of heat through ordinary materials. It has grown so far that if you search our library’s data base for the keyword “Fourier” you will find 425 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics and radio communication and . . . . People have even tried to use it to analyze the stock market. (It didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible example. It provides an indispensible tool in solving partial differential equations, and a later chapter will show some of these tools at work. 5.1 Examples The power series or Taylor series is based on the idea that you can write a general function as an infinite series of powers. The idea of Fourier series is that you can write a function as an infinite series of sines and cosines. You can also use functions other than trigonometric ones, but I’ll leave that generalization aside for now. Legendre polynomials are an important example of functions used for such expansions. An example: On the interval 0 < x < L the function x2 varies from 0 to L2 . It can be written as the series of cosines L2 4L2 + 2 x = 3 π
2 ∞ 1
(−1)n nπx cos 2 n L 2πx 1 3πx πx 1 − cos + cos − ··· L 4 L 9 L (1)
=
L2 3
−
4L2 π2
cos
To see if this is even plausible, examine successive partial sums of the series, taking one term, then two terms, etc. Sketch the graphs of these partial sums to see if they start to look like the function they are supposed to represent (left graph). The graphs of the series, using terms up to n = 5 does pretty well at representing the given function.
5 3 1
5 3
1
highest harmonic: 5
highest harmonic: 5
The same function can be written in terms of sines with another series: 2L2 x = π
2 ∞ 1
(−1)n+1 2 − 2 3 1 − (−1)n ) n π n
1
sin
nπx L
(2) jnearing@miami.edu James Nearing, University of Miami
Fourier series started life as a method to solve problems about the flow of heat through ordinary materials. It has grown so far that if you search our library’s data base for the keyword “Fourier” you will find 425 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics and radio communication and . . . . People have even tried to use it to analyze the stock market. (It didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible example. It provides an indispensible tool in solving partial differential equations, and a later chapter will show some of these tools at work. 5.1 Examples The power series or Taylor series is based on the idea that you can write a general function as an infinite series of powers. The idea of Fourier series is that you can write a function as an infinite series of sines and cosines. You can also use functions other than trigonometric ones, but I’ll leave that generalization aside for now. Legendre polynomials are an important example of functions used for such expansions. An example: On the interval 0 < x < L the function x2 varies from 0 to L2 . It can be written as the series of cosines L2 4L2 + 2 x = 3 π
2 ∞ 1
(−1)n nπx cos 2 n L 2πx 1 3πx πx 1 − cos + cos − ··· L 4 L 9 L (1)
=
L2 3
−
4L2 π2
cos
To see if this is even plausible, examine successive partial sums of the series, taking one term, then two terms, etc. Sketch the graphs of these partial sums to see if they start to look like the function they are supposed to represent (left graph). The graphs of the series, using terms up to n = 5 does pretty well at representing the given function.
5 3 1
5 3
1
highest harmonic: 5
highest harmonic: 5
The same function can be written in terms of sines with another series: 2L2 x = π
2 ∞ 1
(−1)n+1 2 − 2 3 1 − (−1)n ) n π n
1
sin
nπx L
(2) jnearing@miami.edu James Nearing, University of Miami