Fourier Integral Theorem:

If f(x) is a given function defined in (-l, l) and satisfies dirichlet’s conditions then

F(x) = 1/π   ₀∫^∞_-∞∫^∞ f(t) cosλ (t-x) dt dλ

This is known as Fourier integral Theorem or Fourier integral formula.

Point of discontinuity

{F(x+0) – F(x-0)}/2 = ₀∫^∞_-∞∫^∞ f(t) cosλ (t-x) dt dλ

Fourier Sine integral:

F(x) = 2/π   ₀∫^∞ sin λx ₀∫^∞ f(t) sin λt dt dλ

 Fourier cosine integral:

F(x) = 2/π   ₀∫^∞cos λx ₀∫^∞ f(t) cos λt dt dλ

Complex form of Fourier integral:

F(x) = 1/2π   _-∞∫^∞e^-iλx _-∞∫^∞f(t) e^iλt dt dλ

Fourier Transforms

Complex Fourier Transforms and its inversion formula

F[ f(x) ] = 1/√2π _-∞∫^∞f(t) e^ist dt is called the complex Fourier transform of f(x).

f(x) =  1/√2π _-∞∫^∞ F[ f(t) ] e^-isx ds is called the inversion formula for the complex Fourier transform of F[ f(t) ]

Fourier Sine Transform:

F_s[ f(x) ] = √(2/π) ₀∫^∞f(t) sin st dt   is called the Fourier sine transform of the function f(x).

f(x)=  √(2/π) ₀∫^∞ F_s[ f(x) ]  sin sx ds    is called the inversion formula for the Fourier Sine Transform.

Fourier Cosine Transform:

F_c[ f(x) ] = √(2/π) ₀∫^∞f(t) cos st dt   is called the Fourier cosine transform of the function f(x).

f(x)=  √(2/π) ₀∫^∞ F_c[ f(x) ]  cos sx ds    is called the inversion formula for the Fourier Cosine Transform.

Convolution of two functions:

If f(x) and g(x) are any two functions defined in (-∞, ∞) then the convolution of these two functions is defined by

1/√2π _-∞∫^∞f(t) g(x-t) dt and is denoted by

 f * g = 1/√2π _-∞∫^∞f(t) g(x-t) dt

Convolution Theorem for Fourier Transforms:

F[(f * g) x ] = F(s) . G(s)

1/√2π _-∞∫^∞ (f * g) x e^isx  dx = {1/√2π _-∞∫^∞f(x) e^isx  dx} {1/√2π _-∞∫^∞ g(x) e^isx  dx }

Parseval’s Identity:

If f(x) is a given function defined in (-∞, ∞) then it satisfy the identity

_-∞∫^∞ [ f(x) ]² dx =  _-∞∫^∞ [ f(s) ]²  ds  where F(s) is the fourier transform of f(x).