All Laurent series expansion around 1.

[roy240]

Homework Statement

Question is= Find all Laurent series expansion of f(z)=z^4/(3+z^2) around 1. I will be very very thankful if someone can help me to do this question.

Homework Equations

The Attempt at a Solution

can I assume (z-1=u) here and change the function in terms of $u$. then i will have

f(u+1)=(u+1)^4/(u+2)^2

It should have singularity at $u=-2$. now I m big confuse about in what regions I should compute Laurent series.

I have trouble of thinking that how many laurent series I will have and what will be the conditions on $z$.

Thanks !

 

[lanedance]

first, subbing in for z = u+1
g(u) = f(u+1)=(u+1)^4/(3+(u+1)^2) =(u+1)^4/(u^2+ 2u +4)

 

[lanedance]

there will be a laurent series for each disk about z-1 = u = 0 defined by the poles of the function

 

[lanedance]

maybe have a look at these, in particular the 2nd
http://math.furman.edu/~dcs/courses/math39/lectures/lecture-38.pdf

 

[Gib Z]

You've done the substitution to center the Laurent expansion around u=0, now draw a diagram, plotting the poles of the new function and draw the annuli these poles split the plane into. For each annuli, decide what the rule is for that region (eg 8< |u| < 23 or something).

The next part is a bit harder: Try to manipulate the expression you have ( $$\frac{ (u+1)^4}{(u+1)^2+3}$$ ) into the product of two terms, 1 a polynomial, and another that you can interpret as the sum of a geometric series. Then you can expand the geometric series into a sum, and multiply the polynomial into it to get the Laurent series.