I am stuck at two questions about Laurent expansion.
Below are the problems:
1.
Determine the Laurent expansion of (z^2 + 2z)/(z^3 -1) in the annulus 0 < | z - 1 | < R about z = 1. Find the large R one can use.
2.
Obtain a Laurent expansions of f(z) =1/[ (z-j) (z-2) ] in the region 0 < | z - j | < square root of 5
Any help would be appreicated.
Thank you for your reply, but I have problem in finding the Laurent expansion of (1).
Actually, I tried to transfer (1) into the form of f(s)[1/(1-s)], but I failed. So, I cannot use the Binomial Theorem of (1+z)^-1. I also tried other methods such as appiling the equation of Taylor series directly on to (1) which does not work as well.
It would be kind of you if you could give me some idea on this.
All right!... so that the next step is to find the Taylor expansion of...
(1)
Because f(s) is analytic in s=0 we can write and the can be computed imposing the condition...
(2)
From (2) we derive first...
(3)
... and for n>1 the satisfy the 'recurrence relation'...
(4)
... with the 'initial conditions (3). The 'formal' solution is possible but a little 'uncorfortable' ... some of the derived 'directly' from (4) are...
...
Kind regards
A seen before, the coefficients of the Taylor expansion of are solution of the 'recurrence relation' ...
(1)
The (1) can be solved finding the roots of the characteristic equation...
(2)
... that are so that is...
(3)
... where and can be derived from the 'initial conditions'. It is obvious that the can be easily iteratively computed from (1) but (3) is important in order to extablish the radious of convergence of the Taylor series that is ...
Kind regards