## Saturday, March 6, 2010

### Fourier Series and other amazing feats on WolframAlpha

As occasional visitors of my blog would know, I am a great enthusiast for http://www.wolframalpha.com/. For those unfamiliar with the Wolframalpha website, it is a powerful online tool that allows you perform quite sophisticated algebraic feats with excellent graphical solutions provided with details of the algebra. Fortunately, the commands for the software are quite intuitive and easy to learn. I suggest going to the website and start playing (try "plot x^2sinx", "Integrate xcos(x^2) from x =1 to 3", "Differentiate x^2In(x)" and "Solve x^3-2x^2 + 6x - 10 =0" for starters - there is an example page to help you with syntax and common commands).

For students studying Fourier Series, the site is particularly useful. Some of the exercises I can recommends for students of the Fourier Series:

A. Visualising the periodicity of function with multiple terms

e.g. Compare a "Plot Sin(x/2) + Sin(x) + Sin(3x)" with "Plot Sin(5x) + Sin(x) + Sin(3x)"

B. Integrating terms in evaluating the coefficients of the Fourier Series

e.g. If you are determining the "a1" coefficients for f(x) = 2x + 3 over the period 2pi, "Integrate (2x+3) cos(x) from x = -pi to pi"

C. Carrying out a full Fourier expansion of f(x) over a period of 2pi for n terms

e.g. "FourierTrigSeries 2x+3, x, 6"

D. Checking whether a certain function is odd or even

e.g. "Plot x^2, sinx" to compare an even with an odd function, and "Plot x^2 sinx" to see what happens when you multiply an odd and even function.

E. Performing a half range cosine expansion of a function with a period of 2pi

e.g. "FourierCosSeries 2x+3, x, 6"

F. Performing a half range sine expansion of a function with a period of 2pi

e.g. "FourierSinSeries 2x+3, x, 6"

G. Carry our a Fourier series expansion in complex form

e.g. "FourierSeries 2x+3, x, 6"

In all cases, the software can be used to aid learning and also check the answers you are calculating or deriving, AND IT IS ABSOLUTELY FREE !

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