# Partial Differential Equations/Fourier-analytic methods

### 1. Fourier Series of Even and Odd Functions

[edit | edit source]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.

[edit | edit source]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

[edit | edit source]**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)

[edit | edit source]

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

**5. Remainder of fourier series**

[edit | edit source]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

[edit | edit source]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**

[edit | edit source].

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**

[edit | edit source].

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**

[edit | edit source]**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**

[edit | edit source]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**

[edit | edit source]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)