How can I find the order of the pole z=π2 for f(z)=1(2log⁡(z))(1−sin⁡(z))? I know the...

Mabel Breault

Mabel Breault

Answered

2022-01-14

How can I find the order of the pole z=π2 for f(z)=1(2log(z))(1sin(z))? I know the answer should be 2, but I cant

Answer & Explanation

Durst37

Durst37

Expert

2022-01-15Added 37 answers

First, note that 2log(z) does not vanish at z=π2, so we can ignore it; it does not contribute to the pole. So consider simply 11sin(z). We want to write things in terms of w=zπ2,
so let sin(z)=sin(w+π2)=cos(w), where at the end I have used a simple trig formula. Then we can expand cos(w)=1w22+w424, so
11cos(w)=1w22w424+=1w2(12w224+)
Thus, we can see that the order of the pole is 2.
kaluitagf

kaluitagf

Expert

2022-01-16Added 38 answers

you have the function h(z)=log(z)(1sinz)
then h(π2)=0,
h(z)=1sinzzlogzcosz then h(π2)=0
h(z)=zcosz(1sinz)z2coszz+logzsinz thenh(π2)0 hence π2 is a zero of order two for h, therefore is a pole for 1h(z) of order two.
alenahelenash

alenahelenash

Expert

2022-01-24Added 366 answers

Write f(z)=1/logz2(1sinz)=g(z)h(z)Observe that g(z) is analytic and non- zero at z=π2 hence the order of zero of h(z) there determines the order of pole for rational function f(z).h(π2)=h(π2)=0 and h(π2)0 suggests h(z) has zero of order 2 at z=π2 so f(z) has pole of order 2 there.

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