- #1
jjr
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Homework Statement
Describe all the singularities of the function ##g(z)=\frac{z}{1-\cos{z}}## inside ##C## and calculate the integral
## \int_C \frac{z}{1-\cos{z}}dz, ##
where ##C=\{z:|z|=1\}## and positively oriented.
Homework Equations
[/B]
Residue theorem: Let C be a simple closed contour, described in the positive sense. If a function ##f## is analytic inside and on C except for a finite number of singular points ##z_k (k=1,2,...,n)## inside C, then
##\int_C f(z)dz=2\pi i\sum_{k=1}^{n}Res_{z=z_k}f(z)##
and the series
## \cos{z}=(1-\frac{z^2}{2!}+\frac{z^4}{4!}-...) ##
Laurent series representation of a function
##f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n+\frac{b_1}{z-z_0}+\frac{b_2}{(z-z_0)^2}+...##
The Attempt at a Solution
The function has singularities whenever ##\cos{z}=1 ##, i.e. when ##z=2\pi n,\hspace{3mm} n=0,\pm 1,\pm 2##. Within the boundary this only happens once, when ## n=0##.
The integral itself is easily calculated by the residue theorem, but I have to find the residues first.
I use the series expansion and get
##g(z)=\frac{z}{1-\cos{z}}=\frac{z}{1-(1-\frac{z^2}{2!}+\frac{z^4}{4!}-...)}=\frac{1}{\frac{z}{2!}-\frac{z^3}{4!}+...}##
I am a bit stuck at this point. To be able to find the residues and to classify the singularity I need to know how many of the ##b_n## terms of the Laurent series representation vanish. But I'm not sure how I can get it on the form ##...\frac{b_1}{z-z_0}+\frac{b_2}{(z-z_0)^2}+...##
Any suggestions would be very helpfulJ