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: 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.
[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
Dn(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 Dn 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
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
[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)