Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

1. 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 process
Tubular film blowing process
Flow
direction
film thickness
at die
at take-up

2. - 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= + + + + +L
0 n n
n 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 n
n 1
f (x) a (a cosnx b sin nx)
∞
=
= + +∑ (Periodic function of period 2)

3. (1) Determination of the coefficient term a0
(2) Determination of the coefficients an of the cosine terms
0 n n
n 1
0 n n 0
n 1
f (x)dx a (a cosnx b sin nx) dx
a dx a cosnx dx b sin nx dx 2 a
π π
π π
π π π
π π π
π
∞
− −
=
∞
− − −
=
 
= + + 
 
 = + + = ÷
 
∑∫ ∫
∑∫ ∫ ∫
0
1
a f (x)dx
2
π
ππ −
= ∫
0 n n
n 1
0 n
n 1
n
n 1
f (x)cosmx dx a (a cosnx b sin nx) cosmxdx
1 1
a cosmxdx a cos(n m)xdx cos(n m)xdx
2 2
1 1
a sin(n m)xdx sin(n m)xdx
2 2
π π
π π
π π π
π π π
π π
π π
∞
− −
=
∞
− − −
=
∞
− −
=
 
= + + 
 
  
= + + + −  ÷
  
  
+ + + −  ÷
  
∑∫ ∫
∑∫ ∫ ∫
∑ ∫ ∫
m
1
a f(x)cosmx dx (m 1,2,3,...)
π
ππ −
= =∫
(Except n=m)(Except n=m)

4. (3) Determination of the coefficients bn of the cosine terms
Summary of These Calculations: Fourier Coefficients, Fourier Series
Fourier Coefficients:
Fourier Series:
,...)2,1n(dxnxsin)x(f
1
b
,...)2,1n(dxnxcos)x(f
1
a
dx)x(f
2
1
a
n
n
0
=
π
=
=
π
=
π
=
∫
∫
∫
π
π−
π
π−
π
π−
( )0 n n
n 1
f (x) a a cosnx b sin nx
∞
=
= + +∑
( )0 n n
n 1
f (x)sin mx dx a a cosnx b sin nx sin mx dx
π π
π π
∞
− −
=
 
= + + 
 
∑∫ ∫
,...)3,2,1m(dxmxsin)x(f
1
bm =
π
= ∫
π
π−

5. 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 )






+++
π
=⇒






π
==
π
==
π
=
π−
π
=


 +−
π
=
π
=
=


 +−
π
=
π
=
=
π
=
∫∫∫
∫∫∫
∫
π
π−
π
π−
π
π−
π
π−
π
π−
x5sin
5
1
x3sin
3
1
xsin
k4
)x(f
,...
5
k4
b,0b,
3
k4
b,0b,
k4
b
)ncos1(
n
k2
dxnxsin)k(dxnxsin)k(
1
dxnxsin)x(f
1
b
0dxnxcos)k(dxnxcos)k(
1
dxnxcos)x(f
1
a
0dx)x(f
2
1
a
54321
0
0
n
0
0
n
0
k
-  x
f(x)
-k

6. 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 Series
Theorem 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
==
≠=≠=
∫
∫∫
π
π−
π
π−
π
π−
)M)x(''f(dxnxcos)x(''f
n
1
n
nxcos)x('f
dxnxsin)x('f
n
1
n
nxsin)x(f
dxnxcos)x(f
1
a
22
n
<
π
−
π
=
π
−
π
=
π
=
∫
∫∫
π
π−
π
π−
π
π−
π
π−
π
π−
2n222n
n
M2
b,
n
M2
Mdx
n
1
dxnxcos)x(''f
n
1
a ==
π
<
π
= ∫∫
π
π−
π
π−






+++++++ 22220
3
1
3
1
2
1
2
1
11M2a~)x(fConvergent !
At discontinuous point, Fourier series
converge to the average, (f(x+)+f(x-))/2)

7. 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
LxL
Lv
xv
L
n
v
→
≤≤−
π
=⇔π≤≤π−
π
=
,...)2,1n(dx
L
xn
sin)x(f
L
1
b
,...)2,1n(dx
L
xn
cos)x(f
L
1
a,dx)x(f
L2
1
a
L
L
n
L
L
n
L
L
0
=




 π
=
=




 π
==
∫
∫∫
−
−−
0 n n
n 1
n n
f (x) a a cos x b sin x
L L
π π∞
=
    
= + + ÷ ÷  ÷
    
∑
,...)2,1n(dvnvsin)v(f
1
b
,...)2,1n(dvnvcos)v(g
1
a,dv)v(g
2
1
a
n
n0
=
π
=
=
π
=
π
=
∫
∫∫
π
π−
π
π−
π
π−
( )0 n n
n 1
g(v) a a cosnv b sin nv
∞
=
= + +∑

8. 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); Esint (0 < t < L)
p = 2L = 2/
k
-1 1 x
f(x)
0dx
2
xn
sink
2
1
b
,...)11,7,3n(
n
k2
a,...),9,5,1n(
n
k2
a
2
n
sin
n
k2
dx
2
xn
cosk
2
1
a,
2
k
dxk
4
1
dx)x(f
L2
1
a
1
1
n
nn
1
1
n
1
1
L
L
0
=




 π
=
=
π
−==
π
=⇒
π
π
=




 π
====
∫
∫∫∫
−
−−−






−
π
+
π
−
π
π
+= x
2
5
cos
5
1
x
2
3
cos
3
1
x
2
cos
k2
2
k
)x(f
k
-/ 0
u(t)
/

9. 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
L
dx)x(g2dx)x(g
0dx)x(h
L
L
=∫−

10. In the Fourier series,
f(x) even  f(x)sin(nx/L) odd, then bn=0
f(x) odd  f(x)cos(nx/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






=




 π
== ∫∫
L
0
n
L
0
0 ,...)2,1n(dx
L
xn
cos)x(f
L
2
a,dx)x(f
L
1
a∑
∞
=





 π
+=
1n
n0
L
xn
cosaa)x(f
( ) 





=
π
=
π
= ∫∫
ππ
0
n
0
0 ,...)2,1n(dxnxcos)x(f
2
a,dx)x(f
1
a( )∑
∞
=
+=
1n
n0 nxcosaa)x(f
For even function with period 2






=




 π
= ∫
L
0
n ,...)2,1n(dx
L
xn
sin)x(f
L
2
b∑
∞
=





 π
=
1n
n
L
xn
sinb)x(f
( ) 





=
π
= ∫
π
0
n ,...)2,1n(dxnxsin)x(f
2
b( )∑
∞
=
=
1n
n nxsinb)x(f
For odd function with period 2

11. 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 






+++
π
=⇒ x5sin
5
1
x3sin
3
1
xsin
k4
)x(f + k+ k
previous result by Ex.1 in 10.2
2k
-  x
f(x)
-  x
f(x)
)f,xf(fff 2121 π==+=
π
π
−= ncos
2
bn






−+−+π=⇒ x3sin
3
1
x2sin
2
1
xsin2)x(f

12. 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
k
Lx
2
L
)xL(
L
k2
;
2
L
x0x
L
k2
)x(f <<−<<=
(a) Even periodic extension: use ∑
∞
=





 π
+=
1n
n0
L
xn
cosaa)x(f
(b) Odd periodic extension: use ∑
∞
=





 π
=
1n
n
L
xn
sinb)x(f

13. 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
1n
nn0 nxsinbnxcosaa)x(f
( )∑=
++=
N
1n
nn0 nxsinBnxcosAA)x(F (Minimize the error by usage of the F(x) !)
∫
π
π−
−= dx)Ff(E 2
∫∫∫
π
π−
π
π−
π
π−
+−= dxFfFdx2dxfE 22
( )2
N
2
1
2
N
2
1
2
0
2
BBAAA2dxF ++++++π=∫
π
π−






 =π== ∫∫∫
π
π−
π
π−
π
π−
0dx)mx)(sinnx(cos;nxdxsinnxdxcos 22
( )NN11NN1100 bBbBaAaAaA2fFdx ++++++π=∫
π
π−







π
=
π
= ∫∫
π
π−
π
π−
,dxnxcos)x(f
1
a,dx)x(f
2
1
a n0

14. 





++π+





++π−=⇒ ∑∑∫ ==
π
π−
N
1n
2
n
2
n
2
0
N
1n
nnnn00
2
)BA(A2)bBaA(aA22dxfE
nnnn
N
1n
0
n
0
n
2
0
2*
bB,aAwhen)ba(a2dxfE ==





++π−= ∑∫ =
π
π−
( ) ( )





−+−+−π=− ∑=
N
1n
2
nn
2
nn
2
00
*
)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
∫∑
π
π−
∞
= π
≤++ dxf
1
)ba(a2 2
1n
0
n
0
n
2
0 for any f(x)
∫∑
π
π−
∞
= π
=++ dxf
1
)ba(a2 2
1n
0
n
0
n
2
0