Summary of Fourier Series
Suppose f is a piecewise continuous periodic function of period 2L, then f
has a Fourier series representation
a0 ∞ n x nπx
f x) = +∑ ancos + bnsin .
2 n=1 L L
Where the coefficients a’s and b’s are given by the Euler-Fourier formulas:
1 L m x
am= ∫ f x)cos dx , m = 0, 1, 2, 3, …
L −L L
L
b = 1 f x)sinnπx dx
n L −L L , n = 1, 2, 3, …
The Fourier Convergence Theorem
Theorem: Suppose f and f ′ are piecewise continuous on the interval
−L ≤ x ≤ L. Further, suppose that f is defined elsewhere so that it is periodic
with period 2L. Then f has a Fourier series as stated above whose
coefficients are given by the Euler-Fourier formulas. The Fourier series
converge to f (x) at all points where f is continuous, and to
x→c−f (x)+lx→c+ (x)/2
at every point c where f is discontinuous. Fourier Cosine and Sine Series
If f is an even periodic function of period 2L, then its Fourier series contains
only cosine (include, possibly, the constant term) terms. It will not have any
sine term. That is, its Fourier series is of the form
a0 ∞ nπx
f ( )= + ∑ an cos .
2 n 1 L
Its Fourier coefficients are determined by:
L
2 mπx
am = ∫ f x)cos dx , m = 0, 1, 2, 3, …
L 0 L
b n= 0, n = 1, 2, 3, …
If f is an odd periodic funct

More
Less