### About

# Fourier coefficients

The Fourier series of a periodic function *f(x*) with period *x =
2π* is of the form

*f(x) = a _{0 }/2
+ Σ (a_{n}cos(nx) + b_{n}sin(nx)); n=1,2,3....∞*

To calculate the coefficients of the series, one starts with the following assumed identities:

*∫f(x)cos(mx)dx = ∫cos(mx) Σ(a _{0}/2
+ Σ a_{n}cos(nx) + b_{n}sin(nx))dx*

*∫f(x)sin(mx)dx = ∫sin(mx) Σ(a _{0}/2
+ Σ a_{n}cos(nx) + b_{n}sin(nx))dx*

where one integrates over one base period *(m = 1)*.

Suppressing constants, the following types of integral are to be
evaluated, summed over index *n*:

With *m = 1,2,3...∞* and *n = 1,2,3...*∞: order of the
harmonic (fundamental *m, n = *1)

*cos (mx) *

*sin (mx) *

*cos (mx) * (a*cos (nx) + b*sin (nx)) *

*sin (mx) * (a*cos (nx) + b*sin(nx)) *

All integrals are zero except of those few where the indices are
identical: *m = n* __and__ the function types are the same (sine
or cosine). Therefore every sum for a specific index *n* has only
one member and the coefficients can easily be derived from the reduced
equations as:

*a _{0} = 2/T∫f(t) dt *

*a _{n}= 2/T∫cos(nx)
f(t) dt *

*b _{n }= *

*2/T∫sin(nx) f(t) dt*

This simulation demonstrates the different types of functions and their integral.

# Operation of the simulation

A **ComboBox** holds a list of all function combinations described on
the *Fourier Coefficients *page. When one is selected it is
displayed in red. The antiderivative is calculated for the fundamental
period and drawn as a blue curve. Its end value at **x = 2pi** is the
definite integral over one fundamental period, which is needed for the
calculation of the coefficients. The integration process is slowed down
to visualize more clearly the consequence of changes in parameters or
indices.

Some of the selectable functions contain parameters *a* and* b*
which can be changed continuously by **sliders a/ b . **Two
other

**sliders**select the critical indices

*m/n**m*and

*n*as real numbers between 1 and 10.

Parameters and indices are maintained when functions are changed.
Integration is started automatically at any change as long as the
selection **Integral** remains active.

By means of sliders *a* and *b* scaling of the ordinate can be
adjusted to the specific function. They also allow phase shifting of
functions.

**E1:** Choose *cosx* in the **comboBox**. It will be
calculated and displayed in red. Activate the **Integral*** *check
box. The integration process will begin with the initial value of the
function at *x = 0* and will progress in red to the end of the
fundamental period *x = 2 π. *Reflect why the end value and
hence the definite integral over the interval *[0, 2pi]* is zero
for integer *n*.

**E2:** Change index *n *with the slider and watch the integral
curve. Reflect again why the definite integral is always zero for
integer *n*.

**E3:** Choose *sinx *and verify the experiments for it.

**E4: **Choose* asinx* + *bcosnx *and assure yourself by
varying *a, b, n* that the superposition is always a simple, phase
shifted periodical, whose definite integral is zero for integer* n*.

**E5: **Choose* cosx * sinx *and assure yourself that the
definite integral is always zero for integer *n*.

**E6:** Choose the remaining "mixed" functions and assure yourself
that the definite integral is __non zero only__ when both terms are
of the same type and have identical indices.

**E7: **Integrate some of the functions analytically and verify the
experimental findings.

**E8: **Conclude in general which characteristics of the
functions *sine *and *cosine* are the base of your results.

This file was created by Dieter Roess November 2008

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

### Translations

Code | Language | Translator | Run | |
---|---|---|---|---|

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

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### Sample Learning Goals

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### For Teachers

This Fourier Series simulation is essentially identical to the basic version for all intents and purposes, containing the entirety of the basic version within itself.

However, the difference between this "advanced" version and the "basic" version is that more functions have been added for the user to explore.

The additional functions are as follows:

- **Sawtooth Cosine**

- **Sawtooth Sine**

- **Rectangular Cosine**

- **Rectangular Sine**

- **Rectangular Pulse Cosine**

- **Rectangular Pulse Sine**

- **Gaussian Pulse Cosine**

- **Gaussian Pulse Sine**

Research

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### Video

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### Version:

- http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
- http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

### Other Resources

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