- idea: to express complicated functions by using simple functions

for example:

power series :

- Besides power series, we do have other ways, such as
**Fourier series**

Advantages:

- easy to differentiate and integrate
- each term contain only one
**characteristic frequency** - To solve PDE and ODE (forbid some numerical methods)
- can be used to describe non continuous (non differetiable) functions

- A very famous algorithm, FFT Math7270 十大算法

Q: Under what kind of conditions **Dirichlet conditions**

- the function must be periodic:
- must be
**single-valued**and**continuous**except possibly at a finite number of finite discontinuities - must have only a limit number of maximum and minimum within 1 period
- the integrated over 1 period of
must converge

- If Dirichlet conditions are satisfied, the fourier series converge to
at all point where f(x) is continuous. - roughly, why
can be expressed by sine and cosine?

For a general function

,

## fourier series of

where **Fourier coefficients**

L: period of

**MAIN TASK**: to determine the**Fourier coefficients**ofwhen its fourier series exists. **remark**: All the terms of a Fourier series are mutually orthogonal. (Trigonometric Identities)

Q: If Fourier Series of

Why?

when

: when

: Hence,

Let

be **angular frequency**

Example: square-wave function

First of all,

is a odd function so we only need to solve the sine part in F.S.

## Symmetric properties

- Odd functions has no cosine terms (all
) - Even functions has no cosine terms (all
)

## Discontinuous Function

If

However, Fourier Series will overshoot its value although more terms would reduce the overshoot, it never disappear even in the limit. (**Gibbs’ Phenomenon**)

Size of overshoot his proportioned to the magnitude of the discontinuity.

Example: square wave

## Non-periodic function

to expand(延拓为奇函数或偶函数)

Example: Find F.S. for

L=4

For

, by formula So

Remark:

## Integration and differentiation

Fourier Series of a function id obtained by integration and differentiation of another one

Example:

.

By integration,

and we have

F.S. don’t have! to deal with

, by differentiation,

## Complex Fourier Series

So we have

with coefficient

Q: Why we have

- orthogonallity:

- prove

- The relation between
and and :

Why?

Example:

when

:

Hint:when

:

## Parsval’s thm.

Remark:

**P:**

- Consider
and with period ;

Last Updated on 2 years by Yichen Liu