Fourier Series

Question 1

What is the equation of a Fourier series?

Answer 1

f(x)=a_0/2+∑a_n cos⁡(nπx/L)+b_n sin⁡(nπx/L)
For a range -L≤x<L and the sum from n=1 to inifinity

Question 2

For which functions can a Fourier series be defined?

Answer 2

For any function which the integral of f(x)^2 between -π and π is less than infinity, a Fourier series can be defined for that function

Question 3

What is point-wise convergence?

Answer 3

If f(x) and f’(x) are continuous as [-π,π) except possibly a finite number of points, the Fourier series converges at every point f ̃(x)=1/2[f(x^+ )+f(x^- )]
Where f(x+)=lim f(x+ε) and f(x-)=lim f(x-ε) where ε is scalar >0

Question 4

What is uniform convergence?

Answer 4

If f(x) and f’(x) exist and are continuous at every value in [-π,π) and f(π)=f(-π), then the Fourier series converges uniformly to f(x)

Question 5

What are the orthogonality relations?

Answer 5

1)∫_sin⁡(nx)cos⁡(mx)dx=0 for n and m are integers
2) ∫sin⁡(nx)sin⁡(mx)dx=0 if n≠m but equals π if n=m
3)∫cos⁡(nx)cos⁡(mx)dx =0 if n≠m but equals π if n=m
All integrated from -π to π

Question 6

What are the trig identities?

Answer 6

a) Cos(a)Cos(b)=[cos(a+b)-cos(a-b)]/2
b) Sin(a)Sin(b)=[cos(a-b)-cos(a+b)]/2
c) Sin(a)Cos(b)=[sin(a+b)+sin(a-b)]/2

Question 7

What is a_0/2 for a range -π to π?

Answer 7

a_0/2=1/2π ∫f(x)dx
Integrated from -π to π

Question 8

What is a_n for a range -π to π?

Answer 8

a_n=1/π ∫f(x)cos(nx)dx
Integrated from -π to π

Question 9

What is b_n for a range -π to π?

Answer 9

b_n=1/π ∫f(x)sin(nx)dx
Integrated from -π to π

Question 10

What is a symmetric function and what does it mean?

Answer 10

f(x)=f(-x)
b_n=0

Question 11

What is an anti-symmetric function and what does it mean?

Answer 11

f(x)=-f(-x)
a_n=0
a_0=0

Question 12

What is Parseval’s theorem?

Answer 12

1/π ∫(f(x)^2 dx=(a_0^2)/2+∑a_n^2+b_n^2
With f(x)^2 integrated from -π to π
And the sum from n=1 to infinity

Question 13

What is a_0/2 for a range -L to L?

Answer 13

a_0/2=1/2L ∫f(x)dx
Integrated from -L to L

Question 14

What is a_n for a range -L to L?

Answer 14

a_n=1/L ∫f(x)cos⁡(nπx/L)dx
Integrated from -L to L

Question 15

What is b_n for a range -L to L?

Answer 15

b_n=1/L ∫f(x)sin⁡(nπx/L)dx
Integrated from -L to L

Question 16

What is a_0/2 for a cosine series range 0 to L?

Answer 16

a_0/2=1/L ∫f(x)dx
Integrated from 0 to L

Question 17

What is a_n for a cosine series range 0 to L?

Answer 17

a_n=2/L ∫f(x)cos⁡(n’πx/L)dx
Integrated from 0 to L

Question 18

What is b_n for a sine series range 0 to L?

Answer 18

b_n=2/L ∫f(x)sin⁡(n’πx/L)dx
Integrated from 0 to L

Question 19

How to solve a Fourier series?

Answer 19

1. Sketch f(x)
2. Periodic extensions
   3.Convergence
   4.Symmetries

Question 20

Fourier series proofs 1

Answer 20

For the interval -π to π for a_0 integrate from -π to π
For a_n and b_n times by cos(mx)/sin(mx) and integrate from -π to π

Question 21

Fourier series proofs 2

Answer 21

For the interval -L to L for a_0 integrate from -L to L
For a_n and b_n times by cos(n’πx/L)/sin(n’πx/L) and integrate from -π to π

Question 22

Fourier series proofs 3

Answer 22

For the interval 0 to L for a_0 integrate from 0 to L
For a_n and b_n times by cos(n’πx/L)/sin(n’πx/L) and integrate from 0 to L