MSE 102, Fall 2018 Problem Set #4
Prof. Daryl C. Chrzan
(1) A function f(x) can be expressed in terms of a Fourier transform.

with

[30 points] Plotting this function poses a few challenges. First, the function is an infinite 
sum. We cannot plot all the terms of the function. However, the terms for large |?| are less 
important that those for small |?|, such that the function converges (almost everywhere) if 
the sum is computed from ? = -? to ? = ?, for some ?. Plot the finite sums for N=3, 20,
and 100 for ? = -5 to ? = 5. Describe in words what you see happening as ? increases.
[20 points] What function did I use to calculate the values for ?'? Write down the integral 
that gives the value of ?' quoted above.

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(2) [50 points] The structure below represents atoms of three types arranged to form a 2D 
crystal. Give the plane group that describes the symmetry of this crystal, and sketch the 
unit cell with all of its symmetry operations on the crystal so that the symmetry elements 
of the group are positioned properly in the crystal.