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) dtDn(x) = Dirichlet kernel = Comments 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 . 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) dxb(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 ) dya(y) =   f(t) cos ty dtb(y) =   f(t) sin ty dtf(x) =    dy  f(t) cos (y(x-t)) dt 9. Special Cases of Fourier Integral if f(x) = f(-x) thenf(x) =    cos xy dy  f(t) cos yt dtif f(-x) = -f(x) thenf(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) thenFourier Cosine Transform ( Fourier Cosine Transform (f(x)) ) = f(x)If f(-x) = -f(x) thenFourier Sine Transform (Fourier Sine Transform (f(x)) ) = f(x)