Evaluate ∫|z|=4tan⁡zdz tan⁡z=sin⁡zcos⁡z there is a a simple pole at z=π2 Res(f,π2)=limz→π2(z−π2)(sin⁡zcos⁡z) Res(f,3π2)=limz→3π2(z−3π2)(sin⁡zcos⁡z) How to...

Jaylene Franco

Jaylene Franco

Answered

2022-01-29

Evaluate |z|=4tanzdz
tanz=sinzcosz there is a a simple pole at z=π2
Res(f,π2)=limzπ2(zπ2)(sinzcosz)
Res(f,3π2)=limz3π2(z3π2)(sinzcosz)
How to continue ?

Answer & Explanation

ma90t66690

ma90t66690

Expert

2022-01-30Added 7 answers

You have
Res(tanz,±π2)=limz±π2(zπ2)sinzcosz
=limz±π2sinzcos(z)cos(±π2)zπ2
=sin(±π2)cos(±π2)
=1
Besides, ±π2 are the only poles in the disc centered at 0 with radius 4. Therefore
|z|=4tanz,dz=2πi(Res(tanz,π2)+Res(tanz,π2))
=4πi

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