Statistical Mechanics - Physics at Oregon State University

5.1. FERMIONS IN A BOX. 97
fα(z) =
∞
n=1
z n
n α (−1)n+1 , |z| < 1 (5.34)
This defines the function for a disc around the origin. The values for arbitrary
values of z are obtained by analytical continuation. For example:
∞
f0(z) = z n (−1) n+1 ∞
= 1 − z n (−1) n = 1 − 1 z
=
1 + z 1 + z
n=1
n=0
and this defines a function everywhere, except at the pole z = −1.
Why is this important?
(5.35)
One might argue that now we have computers available and we do not need
all this analytical detail anymore. Since one use of power series is to be able
to evaluate functions over a reasonable domain of arguments, we can simply let
computers take over and do the integrations for all values. This is not really
correct, however. The analysis above tells us where we have poles and other
singularities. These are places where computer calculations will fail! Relying on
computers also assumes that there are no errors in the computer calculations,
and that is not always true. Therefore, we need to be able to check our computer
calculations against the analytical solutions we can obtain via power series and
other means. The analytical solutions can tell us that near a limiting point
or singularity an energy will behave like |T − T0| −0.3 , for example. This is
very difficult to extract precisely from a computer calculation. But, of course,
computational physics also has its place, because it can extend the analytical
solutions to domains of the parameters where we cannot get analytical solutions.
Simple example.
As a simple example, study the differential equation
¨x + x = 0 (5.36)
and try a power series of the form x(t) = ∞
0 cnt n . This gives:
∞
cnn(n − 1)t n−2 ∞
+ cnt n = 0 (5.37)
0
or after combining terms with equal powers of t, and defining c−2 = c−1 = 0 we
get:
∞
0
0
(cn+2(n + 2)(n + 1) + cn)t n = 0 (5.38)