For each of the functions f(z) find the Laurent Series expansion on for the given isolated singularity (specify R). Then classify as an essential singularity, a pole (specify the order), or a removable singularity. Then find
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suppose is an isolated singularity of f, then
is removable the Laurent expansion is a Taylor series - there are no negative powers of in the expansion.
is a pole of order n there are exactly n terms in the Laurent expansion with negative powers of
is an essential singularity there is infinite number of terms in the Laurent expansion with negative powers of
There are two singularities for this ...
The solution shows how to expand a complex function to a Laurent series, identify the type of singularity and find the residue.
Series and singularities of complex functions
1. Find the series representations for the function
f(z) = z/[(z-2)(z^2-1)]
in powers of z that is valid when
1) |z| < 1
2) 1 < |z| < 2
3) |z| > 2
2. Fine all singularities of the function f in C, and determine whether it is a pole (find its order), a removable singularity, or an essential singularity.
1) f(z) = ze^(3/z)
2) f(z) = (cos z)/[(z - (pi/2))(sin z)]
3) f(z) = (cos z)/[(z^2)(z - pi)^3]