Confused about computing Laurent series

[quasar_4]

I am very confused about how to actually compute a Laurent series. Given an analytic function, we can write down its poles. Then, if I understand correctly, we have to write a Laurent series for each pole. What I'm confused about is the actual mechanics of writing one down. I know that for f(z) with pole at f(z0) that we can write

f(z) = (a_p)/(z-z0)^p + ...+ a_1/(z-z0) + a0 + a1(z-z0) + ...

what I don't understand is how to get the a^{n} coefficients. I know we have the formula a_{n} = (1/2*\pi*i) * \oint\frac{f(z)}{(z-z0)^{n+1}} dz, but all the examples I have just pop out the series (no one is doing any integrals). I must be missing something obvious!

If I can put it into the form f(z) = 1/(1+z) then I can use a geometric series to write this out...

but what if it's something like 1/z?

 

[HallsofIvy]

?? 1/z is already of that form! 1/z = z^-1 so a^-1= 1 and all the other aⁿs are 0.

 

[quasar_4]

oh no, then I am really confused about what's going on. In general, how is a Taylor series used in a Laurent series? Maybe that will help me get started. In the meantime, I guess I'm going to go read through that chapter again... :(

 

[HallsofIvy]

A function is "analytic" at a point if and only if its Taylor's series at that point exists and converges to the function in some neighborhood of that point. A function, f(z), has a "pole of order n at z₀" if and only if (z- z₀)ⁿf(z) is analytic at z₀ but no lower power of z is. Write zⁿf(z) as a Taylor's series and divide each term by zⁿ to get the Laurent series for f(z).

For example, if f(z)= z^-1, then zf(z)= 1 is analytic (and that IS its "Taylor's series: 1+ 0z+ 0z²+ ...) so f(z) has a pole of order 1 at 0 and the Laurent series for f(z) is 1/z+ 0+ 0z+...

 

[quasar_4]

oh, I see. Then when I am starting a problem of this type, it should perhaps be necessary tp start by checking analycity of the function. Once I've found the poles, I should be able to use a Taylor series exp. to write the Laurent series, then find the residue...

I think it is beginning to make sense, at least computationally. It might take a few days of computing these things before the theory part all sinks in.