Series de Fourier

Fourier Series

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1 Foundation of Fourier Series

This section includes questions regarding parts of the Fourier Series that are necessary to understand in order to make the next part of the series much easier. If you are comfortable with the topics listed below, feel free to skip this section and go straight into the ”2. Fourier Series”.

Topics covered in this section: • Graphing Functions

• Integration (Basics) • Advanced Integration • Dirichlet Conditions

1.1 Graphing Functions

Question 1.

In each of the following functions, plot the function on the x-y axis and state the (i) amplitude and (ii) the period.

a) y = sin(x) b) y = 13 cos(2x) + 23

c) y = 3sin(x − π2 ) d) y = 12 cos( 12 x + π2 ) + 3

Question 2.

In each of the following Non-sinusoidal periodic functions, describe the func- tion analytically and state its period.

a)

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c)

d)

e)

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Question 3.

Sketch the graphs of the following functions and label all the relevant values.

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 x 2 0 < x < 2 d)f(x)=2 2<x<4 1 4<x<6

a) f(x)=

f(x + 8) = f(x)

􏰂5 0<x<4 0 4<x<8

b) f(x) = x − x 0 < x < 2 f(x + 2) = f(x)

􏰂

c) f(x)= 5sin(x) 0<x<π 0 π<x<2π

f(x + 2π) = f(x)

1.2 Integration - Basics Question 4.

Complete the following: a)􏰃x4 dx

c) 􏰃 √x dx

e) 􏰃 cos(x) dx

g) 􏰃 √ 1 dx x2 −1

1.3 Advanced Integration Question 5.

f(x + 6) = f(x)

b) 􏰃 e3x dx

d) 􏰃 sin(x) dx

f) 􏰃 4 dx x

h)􏰃 1 dx 1+x2

Complete the following: a) 􏰃 π sin(nx) dx

b) 􏰃 π −π

d) 􏰃 π −π

cos(nx) dx cos2(nx) dx

−π

c)􏰃π sin2(nx)dx

−π Question 6.

Complete the following: a) 􏰃π x cos(nx) dx

b) 􏰃π x sin(nx) dx 00

c) 􏰃 π x2 cos(nx) dx d) 􏰃 π sin(nx) cos(nx) dx 00

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1.4 Dirichlet Conditions

Question 7.

In each of the following cases, state whether or not the function can be represented by a Fourier Series. Each function is defined over the interval -π < x < π and f(x) = f(x+2π).

a) f(x) = x2 c)f(x)=1

x

e) f(x) = tan(x) g) f(x) = cos2(x)

b) f(x) = 2x + 3 d)f(x)= 1

x+4

f) f(x) = y where x2 + y2 = 4

h) y = sec(x)

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2 Fourier Series

2.1 Fourier Series for functions of period 2π

Question 8.

Determine the Fourier series of the functions presented below and plot them on the x-y axis.

a) f(x) =

f(x + 2π) = f(x)

b) f(x) =

f(x + 2π) = f(x)

􏰂2 0<x<π 0 π<x<2π

􏰂x 0<x<π 0 π<x<2π

c) f(x) = 3x + 3 − π < x < π π

f(x + 2π) = f(x)

2.2 Even and odd functions

Question 9.

In each of the following functions, state whether the function is even, odd, or neither.

a)

d) f(x) =

f(x + 2π) = f(x)

􏰂x2 0<x<π 4 π<x<2π

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c)

d)

e)

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Question 10.

Determine

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