- #1
oddiseas
- 73
- 0
Homework Statement
Laurent expansion for 2/z+4 -1/z+2
I can derive the laurent expansion, but i would like a better understanding, so if anyone could tell me if my understanding is right or wrong,
Homework Equations
I know that the region we want is an annulus.
But i am trying to figure this out, so i can have a picture in my head, which seems to work for me. Is this logic correct:
1) we have a circle cenered at -2, and if we keep the magnitude of z greater than 2 than we ensure that -1/z+2 is entirely analytic, and outside of this circle and we don't have any problems.(But really an open disk is analytic but not simply connected, so why can't we just use |z|<2.)
2)We have another circle of radius 4, but we place the singularity on the boundary of the circle, thus if we keep the magnitude of z<4 we ensure that we don't run into the singularity.
QUESTION: why when deriving the laurent series in books if we have:
|z|>2 for the Laurent series to be defined they write 1/z+2=1/z(1+2/z).
It seems to me that they do this because z>2, which meens it won't include z=o and therefore we don't have a problem with 2/z at z=0, thus simplifyibg the calculation.Is this the actual reason?
But if this is so for 2/z+4 we have |z|<4 and thus we may run into the singularity at zero
so we can't express it this way.and we use 1/2(z/2+1). Is this a correct analysis? and if not then why do we do it?
In addition why can't i keep the equation as it originally appeared as 2/z+4 -1/z+2 and derive the expression using the binomial theorm.Why do we change it? is the only reason to simplify the calculation?