# 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: 5x$^3$- 3x
Sums of even powers of x are even: -x$^6$ + 4x$^4$+ x$^2$-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: x$^2$cos 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
f$^2$(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)