Fourier series & Transform

Example: $\sum_{r=1}^{\infty}r^{-4}$

1. for $f(x)=x^2$, in $-2\leq x<2$

2. Then,

3. Parseval’s theorem:

Fourier transform

• f(x) satisfied dirichlet Condition
• f(x) must be periodic, with fundamental period $L<\infty$

Q: Can we find F.S, for $f(x): \mathbb{R}\rightarrow\mathbb{R}$ with no periodic? A: Yes! F.T. which is one of the integral transform

Arrangement:

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

Suppose that $f(x)$ is periodic, which fundamental period is $T$,

by formula,

(we define $\omega , \Delta \omega$)

Q: What happens to the above formula when $T\rightarrow +\infty$?

AKA$\Delta\omega\rightarrow 0, \int_{-\frac{T}{2}}^{\frac{T}{2}}\rightarrow\int_{-\infty}^{+\infty}$

Let $g ( w _ { r } ) = \int _ { - \frac { I } { 2 } } ^ { \frac { T } { 2 } } f ( u ) e ^ { - i \omega _ { r } u } d u$

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

• Def:
• F.T. of $f(t)$: $\tilde{f} ( w ) = \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { + \infty } f ( t ) \exp ( - i \omega t ) d t$
• I.F.T.: $f ( t ) = \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } \tilde { f } ( \omega ) \exp ( i \omega t ) d w$

Find the F.T. of f(t)

A:

 

 

 

The Dirac $\delta$ function

• Def: The Dirac $\delta$ 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: $\int_{-\infty}^{+\infty}\delta(x)dx=1$.

• Property:

 

Last Updated on 3 years by Yichen Liu