MAT334-2018F > Final Exam

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Victor Ivrii:
Find all singular points, classify them, and find residues at these points of
$$
f(z)= \tan (z) + z\cot^2(z);
$$
infinity included.

hz12:
f(z) = $\frac{\mathrm{sin}\mathrm{}(z)}{\mathrm{cos}\mathrm{}(z)}+z\frac{\mathrm{cos}\mathrm{}\wedge 2(z)}{\mathrm{sin}\mathrm{}\wedge 2(z)}$
      =$ \frac{{{\mathrm{sin}}^{\mathrm{3}}\left(z\right) +\ }{\mathrm{zcos}}^{\mathrm{3}}\left(z\right)\ \ \ \ =g}{{\mathrm{cos} \left(z\right)\ }{sin}^2\left(z\right)\ \ \ \ =h}$
     

Cos(z)sin$\mathrm{\wedge}$2(z) = 0

cos(z) = 0 or sin$\mathrm{\wedge}$2(z) = 0

so z =k$\pi $ or $z=\frac{\pi }{2}+k\pi $
 
1, when z = k$\pi $

g = ${sin}^3(z)+z{cos}^3\left(z\right)\neq 0$                            h= ${\mathrm{cos} \left(z\right)\ }{sin}^2\left(z\right)=0$
                                                                     $h^`=-{sin}^3\left(z\right)+2{cos}^2\left(z\right){\mathrm{sin} \left(z\right)\ }=0$                    ${\ h}^{''}\neq 0$


So pole of order = 2

 2, when z =$\frac{\pi }{2}+k\pi $               

g = ${sin}^3(z)+z{cos}^3\left(z\right)\neq 0$                            h= ${\mathrm{cos} \left(z\right)\ }{sin}^2\left(z\right)=0$
                                                                    $h^`=-{sin}^3\left(z\right)+2{cos}^2\left(z\right){\mathrm{sin} \left(z\right)\ }\ \neq 0$
                               
So pole of order = 1

Ziqi Zhang:
I think when z=0, it should have pole of order 1. Because at z=0, the numerator is not 0 when taking derivative once and denominator is not 0 when taking derivative twice.

Ziqi Zhang:
Residue at z=0: 0+cos2(0)=1

Residue at z=kπ, but k≠0: 0-2(kπ)cos(kπ)sin(kπ)+cos2(kπ)=(-1)n

Residue at z=0.5π+kπ: -1+0=-1

Zixuan Miao:
for infinity case, it is a non-isolated singularity

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