Finding Maclaurin & Laurent Series for f(z)

In summary, the conversation focused on finding the Maclaurin and Laurent series for the function f(z) = (z + 2)/(z - 2) on specific domains. The attempt at a solution involved using geometric series to manipulate the function and simplify it into a more manageable form.
  • #1
bondi97
1
0

Homework Statement



f(z) = (z + 2)/(z - 2)

a) Find the Maclaurin Series for f on the doman |z| < 2.

b) Find the Laurent Series for f centered at z0 = 0 on domain 2 < |Z| < inf.


Homework Equations




The Attempt at a Solution



I'm having a hard time figuring out how (z + 2)/(z - 2) = 1 + (4/(z-2)) = 1 - (2/(1 - (z/2)).

I tried referring to a geometric series, but I don't think I have the right approach.
 
Last edited:
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  • #2
hmmmm... been a while but how about
[tex] \frac{z+2}{z-2} = (\frac{1/z}{1/z})\frac{z+2}{z-2} = \frac{1+2/z}{1-2/z} = (1+\frac{2}{z})\frac{1}{1-2/z} [/tex]
 
  • #3
or for your specific question working back
[tex] 1 + \frac{4}{z-2} = \frac{z-2}{z-2} + \frac{4}{z-2} = \frac{z-2+4}{z-2} = \frac{z+2}{z-2} [/tex]
is that your question?
 

FAQ: Finding Maclaurin & Laurent Series for f(z)

1. What is the difference between Maclaurin and Laurent series?

The Maclaurin series is a special case of the Laurent series, where the center of expansion is at the origin. Laurent series, on the other hand, can have a center of expansion at any point within the radius of convergence.

2. How do you find the Maclaurin series for a given function?

To find the Maclaurin series for a function f(z), you can use the Taylor series formula and plug in the value of z=0. This will give you a polynomial expression that represents the Maclaurin series for f(z).

3. What is the purpose of finding a Laurent series for a function?

Finding the Laurent series for a function helps us to better understand the behavior of the function, particularly near singularities or poles. It also allows for easier calculation of complex integrals involving the function.

4. How does one determine the radius of convergence for a Laurent series?

The radius of convergence for a Laurent series is determined by the distance from the center of expansion to the nearest singularity or pole of the function. This means that the series will converge for all values of z within this radius.

5. Can any function be represented by a Laurent series?

No, not every function can be represented by a Laurent series. The function must be analytic (meaning it can be represented by a power series) in the region of interest, and it must have at least one singularity or pole within that region.

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