Fourier Series

Question 1

What defines an even/odd function? Which are the equations sin(x)and cos(x)?

Answer 1

Even - g(+x) = +g(-x)
Odd - g(+x) = -g(-x)
sin(x) is odd, whereas cos(x) is even.

Question 2

What is the Fourier Series equation for a periodic function f(x)?

Answer 2

f(x) = a0/2 * sum between n=1 and infinity of (an cos(nx) + bn sin(nx))

Question 3

What is the equation for a0?

Answer 3

a0 = 1/pi * int between -pi and pi of f(x) dx

Question 4

What is the equation for an?

Answer 4

an = 1/pi * int between -pi and pi of cos(nx) * f(x) dx

Question 5

What is the equation for bn?

Answer 5

bn = 1/pi * int between -pi and pi of sin(nx) * f(x) dx

Question 6

What can you do for an even function?

Answer 6

Remove the bn*sin(nx) term from the series and change the integrals for a0 and an to an integral between 0 and pi. Also the multiplier changes to 2/pi.

Question 7

What can you do for an odd function?

Answer 7

Remove the an*cos(nx) term from the series and change the integral for bn to an integral between 0 and pi. Also the multiplier changes to 2/pi.

Question 8

How do you derive the a0 term?

Answer 8

Integrate f(x) from -pi to pi.

Question 9

How do you derive the an term?

Answer 9

Integrate f(x) * cos(mx) from -pi to pi.

Question 10

How do you derive the bn term?

Answer 10

Integrate f(x) * sin(mx) from -pi to pi.

Question 11

How do you change a Fourier series integral for a even/odd function?

Answer 11

Make the substitution s = -x in the integral between 0 and pi of g(x).

Question 12

How can you split an integral up?

Answer 12

If, for example, the integral is between -pi and pi, you can split it into an integral between -pi and 0, and 0 and pi.

Question 13

How can you express an odd number algebraically?

Answer 13

odd number = 2k - 1

Question 14

What does it mean if a function is ‘piecewise smooth’?

Answer 14

The function is continuous except for a finite number of “jump” discontinuities.

Question 15

What does the Fourier convergence theorem state for continuous values?

Answer 15

That if f(x) and df(x)/dx are piecewise smooth then the sum of the Fourier Series for f(x) is convergent to f(x) for values of x where f(x) is continuous.

Question 16

What does the Fourier convergence theorem state for discontinuous values?

Answer 16

That for a value x=a where f(x=a) is discontinuous, the sum of the Fourier series for f(x=a) is the average of the left and right hand limits (1/2*(f(a+) + f(a-))

Question 17

How do you write the Fourier series with periodicity 2L?

Answer 17

f(x) = a0/2 * sum between n=1 and infinity of (an cos((npix)/L) + bn sin((npix)/L))

Question 18

What is the equation for a0, an and bn?

Answer 18

Same as other one but 1/L in front instead and (npix)/L in the functions. Also the integral limits are -L and +L.

Question 19

How can you prove this 2L periodicity equation?

Answer 19

Start with the normal expression and substitute in z = (pi*x)/L. Thus a change in x equal to 2L gives a change of 2pi with is no change, so can write cos(nx) and sin(nx) as cos(nz) and sin(nz). You can then integrate: 1/pi * integral between -pi and pi of F(z), by changing back to x to get the new terms.

Question 20

How do you derive Parseval’s Theorem?

Answer 20

Start with the 2L Fourier series expression, multiply both sides by f(x) and integrate from -L to +L. Then, remembering the expression for a0, an and bn, substitute these in and simplify.

Question 21

What is a periodic extension?

Answer 21

When a function is not periodic and only defined over a finite interval, you can add either an odd or even periodic extension.

Question 22

How do you apply an even/odd periodic extension?

Answer 22

Even: f(-x) = f(x). Fourier series will only have cosine terms
Odd: f(-x) = -f(x). Fourier series will only have sine terms.

Question 23

What does a question mean by: find the sine fourier series or the cosine fourier series?

Answer 23

Sine = odd series
Cosine = even series
(periodic extensions)