Phy 515 class notes for Spring, 1999

Jan 20 Feb 1 Mar 1 Apr 5 Phy 515 page
Jan 25 Feb 8 Mar 8 Apr 12
Feb 15 Mar 22 Apr 19
Feb 22 Mar 29 Apr 26

Phy 515, Week 9

22 Mar How did Euler originally sum 1 / n2? A very clever trick involving the roots of polynomials.
More on Riemann surfaces. Try to examine the square root of z2 - a2. For very large z this looks like z itself, with no branch points, but near z = a or z = - a, clearly something happens. Start to analyze it near z = a and you see that the function is almost equal to

( 2 a )1/ 2 ( z - a )1/ 2
so that there is a branch at z = a. To analyze it in detail, start at a value of z just above a; then this is the square root of a positive number, and so you can take it to be positive. Follow it counterclockwise around z = a, and the value of the function becomes proportional to +i and then after a full rotation to -1. Keep going, and you have a two-sheeted Riemann surface.
Now what happens near z = -a? It starts to get confusing. The value of the function just to the left of -a -- is it positive or negative? You have to make a choice. To see what is going on, go back to the plain old square root of z function itself and ask how you reallyknow how to find the values of the functions.

24 Mar Complete the examination of the function ( z2 - a2 )1/ 2, and show various ways to draw the Riemann surface for it. Emphasize that you start with a choice for the value of the function at one point and move around from there. The surface can be thought of in a less geometric way as a pair of objects: one is a complex number (z), and the other is the path by which you got to z from your starting point. Actually, you don't need to know the details of the path, only which of several classes it belongs to. [Did it encircle +a or did it not?] [Did it encircle -a or did it not?]

Do in detail the analysis of the contour integral around the origin of the function

e( x / 2 )( t - 1/ t ) 1
t
This has an essential singularity at the origin, and it's necessary to find the coefficient of 1 / t in the Laurent series expansion there. The result, after some straight-forward calculation, is an infinite series,
¥
å
 0 
 ( - 1 )m ( x / 2 )2 m
( m! )2
.
This is a Bessel function, J0( x ).

26 Mar In relation to one of the homework problems, how doyou evaluate a residue? Must you use that formula involving derivatives? (No!) Essentially, you have to know the precise meaning of the word and then you're better off working the answer out for yourself in the individual case. When you hit a complicated problem and don't quite know what's happening, make up a simpler problem that captures some of the essential features and solve that first. Then a residue at a fifth order pole won't seem as formidable. Try

1 / ( z - i )2 or z / ( z - i )2
and understand these before going on to the hard cases.

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