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Semiconductor NanostructuresQuantum states and electronic transport$

Thomas Ihn

Print publication date: 2009

Print ISBN-13: 9780199534425

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199534425.001.0001

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(p.521) A Fourier transform and Fourier series

(p.521) A Fourier transform and Fourier series

Source:
Semiconductor Nanostructures
Publisher:
Oxford University Press

A.1 Fourier series of lattice periodic functions

Let v(r) = v(r + R) be a lattice periodic function. Its expansion in a Fourier series is

A Fourier transform and Fourier series
with
A Fourier transform and Fourier series
where the space integration has to be taken over the unit cell (UC) with volume V 0, and K is a vector of the reciprocal lattice.

A.2 Fourier transform

The Fourier transform of a function is given by

A Fourier transform and Fourier series
with the inverse transform
A Fourier transform and Fourier series

A.3 Fourier transform in two dimensions

In two dimensions the Fourier transform of a function is given by

A Fourier transform and Fourier series
with the inverse transform
A Fourier transform and Fourier series
(p.522) If the function U(r) possesses radial symmetry, i.e. it depends only on r = |r|, then
A Fourier transform and Fourier series
and correspondingly for the inverse transform
A Fourier transform and Fourier series

We talk about the Fourier–Bessel expansion, because the J 0(kr) are Bessel functions.

Fourier transform of the Coulomb potentials. The two-dimensional Fourier transform of the Coulomb potential

A Fourier transform and Fourier series
is given by
A Fourier transform and Fourier series