Integral Transforms

Fourier series & Transform


  1. for , in
  1. Then,
  1. Parseval’s theorem:

Fourier transform

  • f(x) satisfied dirichlet Condition
  • f(x) must be periodic, with fundamental period

Q: Can we find F.S, for with no periodic? A: Yes! F.T. which is one of the integral transform


  1. introduction of F.T.
  2. Laplace transform (other integral transform

Suppose that is periodic, which fundamental period is ,

by formula,

(we define )

Q: What happens to the above formula when ?



The above formula is from the Fourier inversion theorem 逆傅立叶变换

  • Def:
  • F.T. of :
  • I.F.T.:

Find the F.T. of f(t)





The Dirac function

  • Def: The Dirac function can be loosely thought of as a function on the real line which is zero everywhere except at the origin, which is infinite.

    and is also constrained to satisfy the identity: .

  • Property:


Last Updated on 2 years by Yichen Liu