What is the residue of cosec^{2} z (\text{at}\ z=0)?

dedica66em

dedica66em

Answered question

2022-01-16

What is the residue of cosec2z(at z=0)?

Answer & Explanation

Esta Hurtado

Esta Hurtado

Beginner2022-01-17Added 39 answers

Find the series of
The series for
sinz at z=0
is sinz=zz33!+z55!
=zz36z5120.
Square this. Multiplying the first three terms of each factor is enough. Then
sin2z
=(zz36+z5120)(zz36+z5120)
=z2z43+2z645.
Now divide the first three terms of this into 1 using polynomial long division to get as much of the Laurent series at z=0 needed to determine the residue:
cosec2z=1z2+13+z215+
=1z2+0z+13+.
The residue is the coefficient of z1 and that is 0.

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