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University of San Carlos

College of EngineeringNasipit, Talamban, Cebu City 6000

MEP 201 Advanced Engineering Mathematics

NOTES: FOURIER SERIES

o Definition:PERIODIC FUNCTIONThe function xf is said to be periodic of period p2 if there exists a number p such that

p2xfxf for all x . p must be the smallest number for which p2xfxf .

o Definition:FOURIER SERIES OF xf Expansion of a function xf that is periodic of period p2 into a series of the form:

1n pxnsinnbpxncosna2

axf 0

The Euler-Fourier coefficients are obtained using the following formulas:

p2d

d dxxf

p1

0a

p2d

d dx

pxncosxf

p1

na

p2d

d dx

pxnsinxf

p1

nb

o DIRICHLET CONDITIONS: conditions that xf must satisfy so that it can be expanded into aFourier series.

If xf is single-valued, of period p2 , and if at any one period it has a finite number of maximaand minima and a finite number of finite discontinuities, then the Fourier series of xf convergesto xf at all points where xf is continuous. At each point of discontinuity of xf , the seriesconverges to the average of the values approached by xf from the right and from the left.

o A very important first step in the Fourier series expansion of a periodic function is the determinationof the period p2 of the function because all formulas are expressed in terms of p . If the value used

for p is wrong, EVERYTHING ELSE IS WRONG!

o When the graph of a periodic function is given, the period is determined by looking for the segmentof the curve whose SHAPE IS REPEATED along the abscissa. The length of this segment (whose

shape is repeated) is the period p2 of the function.

o Definition:EVEN AND ODD FUNCTIONS xf is said to be an even function of x , if for every x , xfxf .

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xf is said to be an odd function of x , if for every x , xfxf .

The graph of an even function is symmetrical about the y -axis. The graph of an odd function is

symmetrical about the origin.

The product of two even functions is an even function.The product of two odd functions is an even function.

The product of an odd and an even function is an odd function.

o HALF-RANGE FOURIER SERIES EXPANSIONSFourier Cosine Series

If xf is an even function of period p2 which satisfies the Dirichlet conditions, then the Fourierseries of xf is

1n

pxncosna2

axf 0

where

p

0 dxxf

p2

0a and p

0 dx

pxncosxf

p2

na

Fourier Sine Series

If xf is an odd function of period p2 which satisfies the Dirichlet conditions, then the Fourierseries of xf is

1n

pxnsinnbxf

where

p

0 dx

pxnsinxf

p2

nb

o FUNCTIONS DEFINED ONLY IN bxa If a function xf is defined only in the interval bxa , i.e., its value outside of the given intervalis not prescribed, then the function is not periodic and thus cannot in principle be expanded into a

Fourier series. However, many periodic functions xF can be created whose definition in bxa is identical to that of xf . Hence, xF can be expanded into a Fourier series that will also becomethe Fourier series representation of xf in the interval bxa .

o USE OF FOURIER SERIES IN ODEsExample:Non-homogeneous linear ODE with constant coefficients

- The inhomogeneous term is expanded into a Fourier series.- The solution of the ODE is the sum of the complementary solution and the particular solution

obtained by the method of undetermined coefficient.