Chap. 10. Fourier Series, Integrals, and TransformsChap. 10. Fourier Series, Integrals, and Transforms

- Fourier series: series of cosine and sine terms - for general periodic functions (even discontinuous periodic func.) - for ODE, PDE problems (more universal than Taylor series)

10.1. Periodic Functions. Trigonometric Series10.1. Periodic Functions. Trigonometric Series- Periodic function: f(x+p) = f(x) for all x; period=p- Examples) Periodic instabilities in rheological processesPeriodic instabilities in rheological processes

Spinning processTubular film blowing process

Flow direction

film thickness

at die

at take-up

- f(x + np) = f(x) for all x (n: integer)

- h(x) = af(x) + bg(x) (f & g with period p; a & b: constants) h(x) with period p.

Trigonometric SeriesTrigonometric Series- Trigonometric series of function f(x) with period p=2: (ak,bk: constant coefficients)

10.2. Fourier Series10.2. Fourier Series- Representation of periodic function f(x) in terms of cosine and sine functions

Euler Formulas for the Fourier Coefficients

0 1 1 2 2f (x) a a cosx b sin x a cos2x b sin 2x

0 n nn 1

f (x) a (a cosnx b sin nx)

These series converge, its sum will be a function of period 2 Fourier SeriesFourier Series

0 n nn 1

f (x) a (a cosnx b sin nx)

(Periodic function of period 2)

(1) Determination of the coefficient term a0

(2) Determination of the coefficients an of the cosine terms

0 n nn 1

0 n n 0n 1

f (x)dx a (a cosnx b sin nx) dx

a dx a cosnx dx b sin nx dx 2 a

01a f (x)dx2

0 n nn 1

0 nn 1

nn 1

f (x)cosmx dx a (a cosnx b sin nx) cosmxdx

1 1a cosmxdx a cos(n m)xdx cos(n m)xdx2 2

1 1a sin(n m)xdx sin(n m)xdx2 2

m1a f (x)cosmx dx (m 1,2,3,...)

(Except n=m)(Except n=m)

(3) Determination of the coefficients bn of the cosine terms

Summary of These Calculations: Fourier Coefficients, Fourier SeriesFourier Coefficients:

Fourier Series:

,...)2,1n(dxnxsin)x(f1b

,...)2,1n(dxnxcos)x(f1a

dx)x(f21a

n

n

0

0 n nn 1

f (x) a a cosnx b sin nx

0 n nn 1

f (x)sinmx dx a a cosnx b sin nx sinmx dx

,...)3,2,1m(dxmxsin)x(f1bm

Ex.1Ex.1) Rectangular Wave f(x) = -k (-<x<0) & k (0<x<); f(x+2)=f(x)

(See Figure. 238 for partial sums of Fourier series )

x5sin51x3sin

31xsink4)x(f

,...5k4b,0b,

3k4b,0b,k4b

)ncos1(nk2dxnxsin)k(dxnxsin)k(1dxnxsin)x(f1b

0dxnxcos)k(dxnxcos)k(1dxnxcos)x(f1a

0dx)x(f21a

54321

0

0

n

0

0

n

0

k

- x

f(x)

-k

Orthogonality of the Trigonometric SystemOrthogonality of the Trigonometric System- Trigonometric system is orthogonal on the interval - x

Convergence and Sum of Fourier SeriesConvergence and Sum of Fourier SeriesTheorem 1Theorem 1: A periodic function f(x) (period 2, - x ) ~ piecewise continuous ~ with a left-hand derivative and right-hand derivative at each point Fourier series of f(x) is convergent. Its sum is f(x).

)nmincluding(0dxnxsinmxcos

)nm(0dxnxsinmxsin),nm(0dxnxcosmxcos

)M)x(''f(dxnxcos)x(''fn1

nnxcos)x('f

dxnxsin)x('fn1

nnxsin)x(fdxnxcos)x(f1a

22

n

2n222n nM2b,

nM2Mdx

n1dxnxcos)x(''f

n1a

22220 3

131

21

2111M2a~)x(fConvergent !

At discontinuous point, Fourier series converge to the average, (f(x+)+f(x-))/2)

10.3. Functions of Any Period p=2L10.3. Functions of Any Period p=2L- Transition from period p=2 to period p=2L- Function f(x) with period p=2L: (Trigonometric SeriesTrigonometric Series)

)v(g)x(f

LxLLvxvLnv

,...)2,1n(dxLxnsin)x(f

L1b

,...)2,1n(dxLxncos)x(f

L1a,dx)x(f

L21a

L

Ln

L

Ln

L

L0

0 n nn 1

n nf (x) a a cos x b sin xL L

,...)2,1n(dvnvsin)v(f1b

,...)2,1n(dvnvcos)v(g1a,dv)v(g21a

n

n0

0 n nn 1

g(v) a a cosnv b sin nv

Ex.1Ex.1) Periodic square wave f(x) = 0 (-2 < x < -1); k (-1 < x < 1); 0 (1 < x < 2) p=2L=4, L=2

Ex. 2Ex. 2) Half-wave rectifier u(t) = 0 (-L < t < 0); Esint (0 < t < L) p = 2L = 2/

k

-1 1 x

f(x)

0dx2xnsink

21b

,...)11,7,3n(nk2a,...),9,5,1n(

nk2a

2nsin

nk2dx

2xncosk

21a,

2kdxk

41dx)x(f

L21a

1

1n

nn

1

1n

1

1

L

L0

x25cos

51x

23cos

31x

2cosk2

2k)x(f

k

-/ 0

u(t)

/

10.4. Even and Odd Functions. Half-Range Expansions10.4. Even and Odd Functions. Half-Range Expansions- If a function is even or odd more compact form of Fourier series

Even and Odd FunctionsEven and Odd Functions Even function y=g(x): g(-x) = g(x) for all x (symmetric w.r.t. y-axis) Odd function y=g(x): g(-x) = -g(x) for all x

Three Key FactsThree Key Facts(1) For even function, g(x),

(2) For odd function, h(x),

(3) Production of an even and an odd function odd function let q(x) = g(x)h(x), then q(-x) = g(-x)h(-x) = -g(x)h(x) = -q(x)

x

g(x)

Even function

x

g(x)

Odd function

L

0

L

Ldx)x(g2dx)x(g

0dx)x(hL

L

In the Fourier series, f(x) even f(x)sin(nx/L) odd, then bn=0

f(x) odd f(x)cos(nx/L) odd, then a0 & an=0

Theorem 1Theorem 1: Fourier cosine series, Fourier sine seriesFourier cosine series, Fourier sine series(1) Fourier cosine series for even function with period 2L

(2) Fourier sine series for odd function with period 2L

L

0n

L

00 ,...)2,1n(dxLxncos)x(f

L2a,dx)x(f

L1a

1n

n0 Lxncosaa)x(f

0n00 ,...)2,1n(dxnxcos)x(f2a,dx)x(f1a

1n

n0 nxcosaa)x(f

For even function with period 2

L

0n ,...)2,1n(dxLxnsin)x(f

L2b

1n

n Lxnsinb)x(f

0n ,...)2,1n(dxnxsin)x(f2b

1n

n nxsinb)x(f

For odd function with period 2

Theorem 2Theorem 2: Sum of functionsSum of functions- Fourier coefficients of a sum of f1 + f2 sums of the corresponding Fourier coefficients of f1 and f2.- Fourier coefficients of cf c times the corresponding coefficients of f.

Ex. 1Ex. 1) Rectangular pulse

Ex. 2Ex. 2) Sawtooth wave: f(x) = x + (- < x < ) and f(x + ) = f(x)

for f2 = :

for odd f1 = x

x5sin

51x3sin

31xsink4)x(f + k+ k

previous result by Ex.1 in 10.2

2k

- x

f(x)

- x

f(x))f,xf(fff 2121

ncos2bn

x3sin

31x2sin

21xsin2)x(f

Half-Range ExpansionsHalf-Range Expansions

Ex. 1Ex. 1) “Triangle” and its-half-range expansions

x

f(x)

L

f(x)

L x

even periodic extension feven periodic extension f11

(period: 2L)

L

f(x)

x

odd periodic extension fodd periodic extension f22

(period: 2L)

x

f(x)

L/2 L0

kLx

2L)xL(

Lk2;

2Lx0x

Lk2)x(f

(a) Even periodic extension: use

1n

n0 Lxncosaa)x(f

(b) Odd periodic extension: use

1n

n Lxnsinb)x(f

10.7. Approximation by Trigonometric Polynomials10.7. Approximation by Trigonometric Polynomials- Fourier series ~ applied to approximation theoryapproximation theory

- Trigonometric polynomial of degree N:

- Total square error:

- Determination of the coefficients of F(x) for minimum E

N

1nnn0 nxsinbnxcosaa)x(f

N

1nnn0 nxsinBnxcosAA)x(F (Minimize the error by usage of the F(x) !)

dx)Ff(E 2

dxFfFdx2dxfE 22

2N

21

2N

21

20

2 BBAAA2dxF

0dx)mx)(sinnx(cos;nxdxsinnxdxcos 22

NN11NN1100 bBbBaAaAaA2fFdx

,dxnxcos)x(f1a,dx)x(f

21a n0

N

1n

2n

2n

20

N

1nnnnn00

2 )BA(A2)bBaA(aA22dxfE

nnnn

N

1n

0n

0n

20

2* bB,aAwhen)ba(a2dxfE

N

1n

2nn

2nn

200

* )bB()aA(aA2EE (E - E* 0)(E - E* 0)

Theorem 1Theorem 1: Minimum square error- Total square error, E, is minimum iff coefficients of F are the Fourier coefficients of f. - Minimum value is E*

- From E*, better approximation as N increases

Bessel inequalityBessel inequality:

Parseval’s equality:

Ex. 1Ex. 1) Square error for the sawtooth wave

dxf1)ba(a2 2

1n

0n

0n

20 for any f(x)

dxf1)ba(a2 2

1n

0n

0n

20