Partial Differential Equations/Fourier-analytic methods

1. Fourier Series of Even and Odd Functions

A function f(x) is said to be even if f(-x) = f(x).

The function f(x) is said to be odd if f(-x) = -f(x)

Graphically, even functions have symmetry about the y-axis, whereas odd functions have symmetry around the origin.

Examples:

Sums of odd powers of x are odd: 5x³- 3x

Sums of even powers of x are even: -x⁶ + 4x⁴+ x²-3

sin x is odd, and cos x is even

The product of two odd functions is even: x sin x is even

The product of two even functions is even: x²cos x is even

The product of an even function and an odd function is odd: sin x cos x is odd

2. Integrating even functions over symmetric domains.

Let p > 0 be any fixed number. If f(x) is an odd function, then

Intuition: The area beneath the curve on [-p, 0] is the same as the area under the curve on [0, p], but opposite in sign. So, they cancel each other out!

Let p > 0 be any fixed number. If f(x) is an even function, then

Intuition: The area beneath the curve on [-p, 0] is the same as the area under the curve on [0, p], but this time with the same sign. So, you can just find the area under the curve on [0, p] and double it!

3. Periodic functions

Definition:

A function f(x) is said to be periodic if there exists a number

T > 0 such that f(x + T) = f(x) for every x. The smallest such

T is called the period of f(x).

Intuition: periodic functions have repetitive behavior.A periodic function can be defined on a finite interval,

then copied and pasted so that it repeats itself.

4. The fourier series of the function f(x)

a(k) =  f(x) cos kx dxb(k) =  f(x) sin kx dx

5. Remainder of fourier series

Sn(x) = sum of first n+1 terms at x.

remainder(n) = f(x) - Sn(x) =   f(x+t) Dn(t) dt

Sn(x) =   f(x+t) Dn(t) dt

D_n(x) = Dirichlet kernel =

Comments

The Dirichlet kernel is also called the Dirichlet summation kernel. There is also a different normalization in use: the kernels D_n and  are often multiplied by 2. They are then represented also by the series

7. Riemann's Theorem

.

If f(x) is continuous except for a finite # of finite jumps in every finite interval then:

lim_(k->)  f(t) cos kt dt = lim_{(k-> )} f(t) sin kt dt = 0

The fourier series of the function f(x) in an arbitrary interval.

A(0) / 2 + (k=1..) [ A(k) cos (k(Π)x / m) + B(k) (sin k(Π)x / m) ]

a(k) = 1/m  f(x) cos (k(Π)x / m) dx

b(k) = 1/m  f(x) sin (k(Π)x / m) dx

8. Parseval's Theorem

.

Parseval's theorem usually refers to the result that the Fourier transform is unitary, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

If f(x) is continuous; f(-PI) = f(PI) then

 f²(x) dx = a(0)² / 2 + (k=1..) (a(k)² + b(k)²)

Fourier Integral of the function f(x)

f(x) =   ( a(y) cos yx + b(y) sin yx ) dy

a(y) =   f(t) cos ty dt

b(y) =   f(t) sin ty dt

f(x) =    dy  f(t) cos (y(x-t)) dt

9. Special Cases of Fourier Integral

if f(x) = f(-x) then

f(x) =    cos xy dy  f(t) cos yt dt

if f(-x) = -f(x) then

f(x) =    sin xy dy  sin yt dt

10. The Fourier Transforms

Fourier Cosine Transform

g(x) = () f(t) cos xt dt

Fourier Sine Transform

g(x) = () f(t) sin xt dt

11. Identities of the Transforms

If f(-x) = f(x) then

Fourier Cosine Transform ( Fourier Cosine Transform (f(x)) ) = f(x)

If f(-x) = -f(x) then

Fourier Sine Transform (Fourier Sine Transform (f(x)) ) = f(x)