Periodic function and Fourier expansion

Periodic function and Fourier expansion

The theorem is that if a function f is periodic with frequency w (period a), the function can be expanded as Fourier polynomials. i.e.

If f(x+na)=f(x)

Then f(x)=\sum_{k=nK}f(k)e^{-ikx}

where K=\frac{2\pi}{a}

.

The proof actually relies on common sense.

define:

 f(k)=\int f(x)e^{-ikx}dx

One can see that both f(x) and exp(-ikx) are perodic, where average of exp(-ikx) is zero. So if the two periods are not commensurate, f(k) will be zero.

The only possible nozero f(k) occurs when kx=2npi, where k=nK.

pydao0.979 released

Pydao, a new software for data organize and analysis is released as a trial version.

Data organization is based on hierachical data file (HDF) structure.

Data analysis is based on the plugins designed for special usage.

Data visualization is based on matplotlib and mayavi pakage.

Now I have lattice dynamics as a useful plugin.

There are some build in analysis and visualization tools, not as much as the xpy1.xx. But we will make pydao more and more complete in the future.

Right now, only source code is provided. Win32 compiled will come soon.