- #1
moo5003
- 207
- 0
Homework Statement
f(z) = sin(z)/[cos(z^3)-1]
Show that z=0 is a non-essential singularity with an ORD(f:0)=-5 and determine the co-eff. a_-1
Homework Equations
The Attempt at a Solution
So, after expanding the power series z^5f(z) I showed that h(0) = 2, ie removable and has order -5. My question comes when I'm finding the co-eff. I can't seem to split the function into explicit h(1/z) + g(z) form so its somewhat hard to find the a_-1 coeff. Any tips would be appreciated.
My power expansion thus far: (Sigma is always from n=0 to infinity for this)
f(z) = Sigma[(-1)^n/(2n+1)! * z^(2n+1)] / Sigma[(-1)^(n+1)/(2n+2)! * z^(6n+6) ]
How do I get a_-1 from this ?!
Alright, I think I may have gotten it:
Just written out the series goes something like:
[z-z^3/3!+z^5/5!-z^7/7!...] / [-z^6/2!+z^12/4!...]
The pattern I saw for the co-eff's of 1/z is as follows, tell me if you see the same thing :P.
-Sigma[(3n+5)!/(2n+2)!]
Last edited: