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Summary of Fourier Series

Supposefis a piecewise continuous periodic function of period 2L, then f

has a Fourier series representation

=

++=

1

0 sincos2

)(n

nnL

xnb

L

xna

axf

.

Where the coefficients as and bs are given by the Euler-Fourier formulas:

=L

L

m dx

L

xmxf

L

a

cos)(1

, m= 0, 1, 2, 3,

=L

L

n dxL

xnxf

Lb

sin)(1

, n= 1, 2, 3,

The Fourier Convergence Theorem

Theorem: Suppose fand f are piecewise continuous on the intervalL x L. Further, suppose that fis defined elsewhere so that it is periodic

with period 2L. Then f has a Fourier series as stated above whosecoefficients are given by the Euler-Fourier formulas. The Fourier series

converge to f(x) at all points where fis continuous, and to

2/)(lim)(lim

+

+ xfxf

cxcx

at every point cwhere f is discontinuous.

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Fourier Cosine and Sine Series

If fis an even periodic function of period 2L, then its Fourier series containsonly cosine (include, possibly, the constant term) terms. It will not have any

sine term. That is, its Fourier series is of the form

=

+=1

0 cos2

)(n

nL

xna

axf

.

Its Fourier coefficients are determined by:

=L

m dx

L

xmxf

L

a

0

cos)(2

, m= 0, 1, 2, 3,

bn= 0, n= 1, 2, 3,

If fis an odd periodic function of period 2L, then its Fourier series containsonly sine terms. It will not have any cosine term. That is, its Fourier series

is of the form

=

=1

sin)(n

nL

xnbxf

.

Its Fourier coefficients are determined by:

am= 0, m= 0, 1, 2, 3,

=L

n dxL

xnxf

Lb

0

sin)(2

, n= 1, 2, 3,

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The Cosine and Sine Series Extensions

If fand f are piecewise continuous functions defined on the interval0 t L, then fcan be extended into an even periodic function of period 2L,

such that f(x) =F(x) on the interval [0, L], and whose Fourier series is,therefore, a cosine series: