Nguyen ���1

Trieu Nguyen

Professor Mehrdad Khosravi

Math 1C

De Anza College

May 29, 2015

Honors Project

Fourier Series

I. Background

On early 19th century, the French mathematician and physicist, Jean-Baptiste Joseph

Fourier (1768-1830), who stated, “Mathematics compares the most diverse phenomena and

discovers the secret analogies that unite them”, came up with a problem of how to describe the

changing of temperature T(x, t) of a thin wire length π on the condition that the two ends held the

same temperature of 0 over the time.

0 π x

T (x = 0, t) = T (x = π, t) = 0. The initial temperature of every single point on the wire

could be expressed by an function of x called f (x):

T (x, t = 0) = f (x)

He then proved that this function could be expanded in a series of sine functions:

with

By doing so, he demonstrated that the change of the heat can be viewed as a linear

combination of sine waves.

f (x) = bn sinnxn=1

∞

∑ bn =2π

f (x)sinnxdx0

π

∫

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At first, Fourier developed this series just to solve the heat equation, it then become

helpful solving many other problems in various fields: electrical engineering, econometrics,

signal and image processing.

II. Definition of Fourier Series:

Now, let us define the Fourier series in general. A periodic change can be described by a

linear combination of both sines and cosines waves and a constant component

While coefficients c0, an, bn are defined as the following integrals:

note that this change has a period of 2π.

In other words, if the change has a period of T, the coefficients becomes:

f (x) = c0 + (an cosnx + bn sinnx)n=1

∞

∑

c0 =12π

f (x)dx−π

π

∫

bn =1π

f (x)sinnxdx−π

π

∫

an =1π

f (x)cosnxdx−π

π

∫

an =2T

f (x)cosnxdx−T

T∫

bn =2T

f (x)sinnxdx−T

T∫

c0 =1T

f (x)dx−T

T∫

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The Fourier series can be explored in an algebraically simpler form when applying Euler

familiar which is:

Hence:

The Fourier series becomes:

while

We can re-write the series above this way:

eiθ = cosθ + isinθ

cosnx = 12(einx + e−inx )

sinnx = 12i(einx − e−inx )

"

#$$

%$$

f (x) = c0 + cneinx + c−ne

−inx"# $%n=1

∞

∑

→ e−iθ = cosθ − isinθ

→cosθ = 1

2(eiθ + e−iθ )

sinθ = 12i(eiθ − e−iθ )

#

$%%

&%%

cn =12π

f (x)e−inx dx−π

π

∫

f (x) = cneinx

n=−∞

t∞

∑

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III. Fourier Series’s Characteristics:

1. Suppose that f is a function as in the definition of Fourier Series, then:

A. The sequence coefficient cn is bounded

B. The coefficients tend to 0 as

Proof:

A. We have

where A is a fixed number which does not depend on n

B. Consider the coefficients of Fourier Series:

Note that f is a periodic function of 2π.

are also periodic functions of 2π

x →∞

cn =12π

f (x)e−inx dx−π

π

∫

cn ≤12π

f (x)−π

π

∫ e−inx dx

cn =12π

f (x) dx−π

π

∫ = A

c0 =12π

f (x)dx−π

π

∫

an =1π

f (x)cosnxdx−π

π

∫

bn =1π

f (x)sinnxdx−π

π

∫

→f (x)cosnxf (x)sinnx

"#$

%$

cn ≤12π

f (x)−π

π

∫ dx

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2. Suppose that f(x) and f’(x) are both continue on period 2π. We say that the Fourier Series

can be differentiated (even if we do not know whether the two series are converge or

not).

Proof:

Consider:

where f(x) is continuos on [-π, π] suggest that f (π) = f (-π)

Therefore the Fourier Series is differentiable

There are a lot more characteristics of Fourier Series which we are not going to cover in

these papers. Since we know the Fourier has many applications in various fields, we will focus

more on this section.

cn ( f ) =12π

f (x)e−inx dx−π

π

∫

→ cn ( f ') =12π

f '(x)e−inx dx−π

π

∫

cn ( f ') =12π

f (x)e−inx"# $% −ππ −

12π

f (x)(−in)e−inx dx−π

π

∫

cn ( f ') =12π

f (π )(−1)n − f (−π )(−1)n"# $%+ incn ( f )

→ cn ( f ') = incn ( f )

→ f '(x) has the series  in cn  einx

n=−∞

∞

∑

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IV. Applications

1. Periodically Forced Oscillation

Consider a mass-spiny system showed we have mass m on the spring constant k damping

c, and force F(t) on mass.

Suppose force F(t) is a periodic function of 2L.

The motion equation of m can be proved as below:

In ideal case, there is no damping, the equation becomes:

This equation has a solution as the following:

where is a steady periodic solution.

In addition, the function F(t) can be described in Fourier Series as below:

So we can assume the steady periodic solution should had the form of:

Plug into the differential equation we will find An, Bn in term of an and bn.

mx ''(t)+ cx '(t)+ kx(t) = F(t)

mx ''+ kx = F(t)

F(t) = c02+ a n cos

nπLt

!

"#

$

%&

n=1

∞

∑ + bn cosnπLt

!

"#

$

%&

x0 (t) =C02+ A n cos

nπLt

!

"#

$

%&

n=1

∞

∑ +Bn cosnπLt

!

"#

$

%&

x0 (t)

x(t) = Acosω0t +Bsinω0t + x0 (t)

ω0 =km; x0 (t)

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2. The periodic variation of gas pressure in a 4-stoke internal combustion engine

P = gas pressure in cylinders

(1) Intake

(2) Compression

(3) Combustion

(4) Expansion

(5) Exhaust

The gas pressure is graphed as:

So, P(t) is a periodic function with period T

with n is any integer number

Therefore, the Fourier Series P(t) will be available if the following conditions are satisfied:

• Function P(t) in one period can be expressed mathematically

→ P(t) = P(t ±T) = P(t ± 2T) = ... = P(t ± nT)

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• P(t) and P’(t) are continuous (or piece-wise continuous) in interval (C, C+T) when C is

zero or any value of t.

Then we can transform P(t) into its Fourier Series

Where c0, an, bn are Fourier coefficients:

an =1T2

P(t)cosC

C+T∫ nπ t

T2

!

"

###

$

%

&&&dt

bn =1T2

P(t)sinC

C+T∫ nπ t

T2

!

"

###

$

%

&&&dt

P(t) = c02+ an cos

nπ tT2

!

"

###

$

%

&&&+ bn sin

nπ tT2

!

"

###

$

%

&&&

'

(

)))

*

+

,,,n=1

∞

∑

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3. Derive a function describing the position x(t) of the sliding block M in one period in a slide

mechanism as illustrated below.

If the crank rotates at a constant velocity of

Position of block M can be described as:

Fourier Series of x (t):

where:

Solve these integral we will find the function of position of block M at time t

an =ωπ

f (t) cos nπ tL

!

"#

$

%&

0

2πω∫  dt

x(t) = R− Rcos ωt( )         t ∈ 0, 2πω

#

$%&

'(

x(t) = R 1− cos ωt( )"# $%        period =2πω

ω

x(t) = c02+ an cos

nπ tπω

!

"

###

$

%

&&&+ bn sin

nπ tπω

!

"

###

$

%

&&&

'

(

)))

*

+

,,,n=1

∞

∑

an =ωπ

f (t) cos nπ tπω

!

"

###

$

%

&&&

0

2πω∫  dt

bn =ωπ

f (t) sin nπ tπω

!

"

###

$

%

&&&

0

2πω∫  dt

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Work Cited

Stewart, James. Calculus: Early Transcendentals. Seventh ed. N.p.: Cengage Learning,

2008. Print.

Walker, James S. "Fourier Series." Springer Reference (2011): n. pag. University of

Wisconsin–Eau Claire. Web. 29 May 2015.

Weisstein, Eric W. "Fourier Series." From MathWorld--A Wolfram Web Resource.