Laurent Series and Taylor Series, when to use which? | Complex Analysis #10

How to determine if you need to use a Taylor Series or a Laurent Series when determining power series for a complex function. The only things you need to know is if the point of expansion is an isolated singularity and what the shape of the region looks like.

I made the flowchart in the video to give you a logical method on how to determine the appropriate series (Laurent Series or Taylor Series) since I think this is one part that is not really explained so often. Note that you should not be able to find it in any textbooks since I made it from scratch (let me know otherwise). I have tested the flowchart for the most common cases/examples a student should meet in complex analysis and please let me know if you can find a case that contradicts the method.

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METHOD - FLOWCHART:
1) start by marking the important points (point of expansion z_0 and the isolated singularities) on the z-plane.

2) create as many regions as you can from the image. The rule of
thumb is that a region can't contain an isolated singularity (exception: the point of expansion z_0 can be an isolated singularity).

3) for each region use the flowchart.

3.1) if the point of expansion z_0 is an isolated singularity then you will have to use a Laurent Series (since this is the only power series which still works if the z_0 is an isolated singularity). Otherwise, continue to step 3.2

3.2) If the region is an annulus or the outside of a circle, then you will need to use a Laurent Series. If the region is the inside of a circle, then you will need to use a Taylor Series.

CONCEPTS FROM THE VIDEO
►Taylor Series
A Taylor Series is a specific kind of power series and is used to approximate a analytic function f(z) around some point z_0 in the complex plane with the help of the functions derivative evaluated at this point
f(z_0) + f'(z_0)*(z-z_0) + f''(z_0)*(z-z_0)^2 + f'''(z_0)*(z-z_0)^3 + ...

► Laurent Series
A Laurent Series is a specific kind of power series and is used to approximate an analytic function f(z) around some point z_0 (note that z_0 can be an isolated singularity) in the complex plane

... + (a_-3) *(z-z_0)^-3 + (a_-2) *(z-z_0)^-2 + (a_-1) *(z-z_0)^-1 + a_0 + a_1 *(z-z_0) + a_2 *(z-z_0)^2 + a_3 *(z-z_0)^3 + ...

where the coefficients a_n are determined by contour integration, but 99,9 % of the cases these are determined by geometric series. Note that: "... + (a_-3) *(z-z_0)^-3 + (a_-2) *(z-z_0)^-2" is called the principal part, while "a_0 + a_1 *(z-z_0) + a_2 *(z-z_0)^2 + a_3 *(z-z_0)^3 + ..." is called the analytic part.

► Isolated Singularity
An isolated singularity is a singularity (point there the function is not defined or is not well behaved) that has no other singularities close to it. The most common isolated singularity to stumble on in complex analysis is poles.

► Poles
A pole is a specific kind of singularity, the short and the most intuitive definition is that poles are points z_0 in the complex plane so that f(z_0) = g(z_0)/0, where g(z_0) =\= 0.


TIMESTAMPS
Flowchart - 00:16

Examples: Determine all possible Power Series for the following functions
f(z) = 1/(z-1) around z=0 - 00:43
f(z) = 1/((z-1)(z-2)) around z=1 - 01:53
f(z) = 1/((z-1)(z-2)) around z=0 - 02:36

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