Why,in many problems,in order to get the Laurent series of a function at its singular point,Taylor series is used?For example to find Laurent expansion of zcos(1/z) about z=0,Taylor expansion of cos(1/z) about z=0 is used whereas it is its singular point.

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Answer 1 · 2018-07-18T21:50:22

The Lorentz series may be more easily defined as the Taylor series at infinity. This is identical to the expansion of 1/z at zero.

That's why we use a Taylor series: we consider #cos(1/z)# around its singularity:
#cos(1/z) = 1 - 1/(2z^2) + 1/(24z^4) - ... #

So the original is
#zcos(1/z) = z - 1/(2z) + 1/(24z^3) - ... #

This represents the nature near the pole.