You are here:

Advanced Math/Q1:functions of complex variable


dear sir,
functions of complex variable
Q: ∫1-cos2(Z-3)/(z-3)³ (z-3 is having power of 3)  

it need to be solved by residue theorem

first calculate residue then use theorem 2pi i (sum of residue)

can you provide solution to this I am having exam very soon (i am getting 4pi i) but
the ans given is 8pi i(i is iota)


First of all, around what contour are you integrating? That's a pretty important part of this question! I will assume counterclockwise around some closed loop containing z=3.

I suggest you use the residue theorem, if that is what the problem says, but could also use the Cauchy integral formula, so I'm not sure why you think it "need" to be solved by the residue theorem. Here is the correct/best solution, which uses the Cauchy formula:

∫ f(z)/(z-a)^(n+1) = 2 π i f^(n)(a) / n!

where n! is factorial, f^(n) is the nth derivative of f, and i is the imaginary unit, i = &sqrt;(-1).

So f(z) = 1 - cos²(z), a=3, and n=2.

f''(z) = 2 ( cos²(z-3) - sin²(z-3) )

f''(3) = 2 ( cos²(0) - sin²(0) ) = 2

2 π i f''(3) / 2! = 2 π 2 i / 2! = 2 π i

That is the correct answer.

Likewise, if you compute the residue -- which can be done by expanding the series:

( 1 - cos²(z) ) / (z-3)³ = 1/(z-3) - (z-3)/3 + (2/45) (z-3)³ - ...

the residue being the coefficient of the (z-3)^(-1) term, so residue = 1.

Then 2 π i is still the answer.

Since your answer is 4 π i, and the answer given is 8 π i, there are two possibilities:

1. You and the book are wrong.

2. I am wrong, and one of the other answers is right. This would be true if the curve is not a simple loop. If the winding number of the curve is 2, you are right. If it is 4, the book is right.

Advanced Math

All Answers

Answers by Expert:

Ask Experts


Clyde Oliver


I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks.


I am a PhD educated mathematician working in research at a major university.


Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

BA mathematics & physics, PhD mathematics from a top 20 US school.

Awards and Honors
Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

Past/Present Clients
In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

©2016 All rights reserved.