Homework Help: Is this line integral correct?

1. Jan 15, 2012

Jamin2112

1. The problem statement, all variables and given/known data

C1(0) dz / (z * sin2(z))

2. Relevant equations

Residue Theorem material

3. The attempt at a solution

z * sin2(z)
= z * (1/2 - cos(2z)/2)
= z * [1/2 - (1/2)∑(-1)n(2z)n/(2n)! ]
= z3 + ...

---> z * sin2(z) has a zero of order 3 at z = 0
---> 1/(z * sin2(z)) has a pole of order 3 at z = 0

The circle C1(0) encloses z = 0 (its only singularity), so we have

2πi * Res[f, 0] = 2πi * 1/(3-1)! * limz-->0 d2/dz2 (z-0)3 1/(z * sin2(z)),

which I don't want to calculate if there's an easier way to do the problem. Is there an easier way? I'm too cool for calculating double derivatives.

2. Jan 15, 2012

Dick

You could expand it in a Laurent series around z=0 and look for the coefficient of the 1/z term if you are so dead set against finding derivatives.

3. Jan 15, 2012

Jamin2112

Ah, I see! So perform long division using the series from the trig identity I invoked?

4. Jan 15, 2012

Dick

Factor out all of the z's like you did so you should be left with something like z^3*(1-g(z)). Then use the expansion 1/(1-g(z))=1+g(z)+g(z)^2+... Remember?

Last edited: Jan 15, 2012
5. Jan 16, 2012

Jamin2112

I got 1/3. That seem legit?

6. Jan 16, 2012

Dick

That's what I got. Not sure it makes it legit.

7. Jan 16, 2012

Jamin2112

Awesome, brah. Now let's look at ∫C f(z), where f(z) = dz/(z-1)2(z2+4) and C = C4(0). The integrand clearly has 3 singularities, all of which lie inside C. They are z = 1, which has order 1; z = 2i, which has order 2; and z = -2i, which has order 2. The residue is therefore

2πi * (Res[f, 1] + Res[f, 2i], Res[f, -2i]).​

But hang on a sec, brah. For Res[f, 1] I got limz-->1 (z-1) * 1/(z-1)2(z2+4) = ∞. Wat do?

8. Jan 16, 2012

Dick

The pole at z=1 has order 2. The other two have order 1. Why do you think otherwise?

9. Jan 16, 2012

Jamin2112

Let me redeem myself. To find the order of the pole z=1, I need to write (z-1)2(z2+4) in the form ∑an(z-1)n. Is this supposed to be incredibly obvious? I can't figure it out.

10. Jan 16, 2012

Dick

No. You are supposed to write f(z)=g(z)/(z-1)^n where is g(z) is analytic in a neighborhood of z=1 and g(1) is not zero. What are n and g(z)? Yes, it's supposed to be pretty obvious.

11. Jan 16, 2012

Jamin2112

A punctured neighborhood of z=1?

12. Jan 16, 2012

Dick

Ok. Punctured. It might have a removable singularity at z=1.

13. Jan 16, 2012

Jamin2112

You must be referring to g(z) = 1/(z2+4), n = 2.

14. Jan 16, 2012

Dick

Sure. Pole of order 2.

15. Jan 16, 2012

Jamin2112

Ah, I understand the reasoning now.

Since g(z) = 1/(z2+4) is analytic in a neighborhood about z=1, it will equal some power series centered at z=1:

g(z) = ∑an(z-1)n. ​

Thus
g(z)/(z-1)2
= (z-1)-2 (a0 + a1(z-1) + a2(z-2)2 + ...)
= a0/(z-1)2 + ....

Thus f(z) = g(z)/(z-1)2 has a finite number of terms of the form a-k/(z-1)k (with k>0), the first of them being being k = 2. Is that basically right?

16. Jan 16, 2012

Dick

Sure. The point is near z=1 f(z) behaves like a0/(z-1)^2 plus small corrections.

17. Jan 16, 2012

Jamin2112

Sweet deal. (Maybe I should actually buy the textbook and stop sleeping in class.)

I have another question, though. (See this as an opportunity to become an even higher contributor to the homework help sub-forum.)

Am I correct that z-2csc(z) has a pole of order 3 at z=0?

For z2sin(z)
= z2(z - z3/3! + z5/5! - z7/7! + z9/9! + ...)
= z3 - z5/3! + ...
undeniably has a zero of order 3 at z=0.

And in that case,
Res[z-2csc(z)] = 1/(2-1)! limz-->0 ((z3 * z-2csc(z))'' = limz-->0 (z*sin2(z) - 2cos(z)sin(z) + 2z*cos2(z))/sin3(z) ????????????

18. Jan 16, 2012

Dick

Sure again. Now you can either use l'Hopital on the trig stuff to find the limit, or you can use the Laurent series trick like in the first problem.

Last edited: Jan 16, 2012
19. Jan 16, 2012

Jamin2112

Sweet deal. Again, how do I solve z6+1=0?

I tried this:

z6+1=0 <---> (z3+i)(z3-i)=0

z3+i
= (x+iy)3+i = 0
<---> (x2+2xyi - y2)(x+iy) + i = (x3 - 3xy2) + i * (3x2y - y3 + 1) = 0
<---> x2 = 3y2; 3x2y - y3 = -1
<---> y = (1/8)-1/3 = -1/2

So two of our zeroes are √3/2 -1/2 * i and -√3/2 - 1/2 * i

Ditto for (z3-i)=0? amidoinitrite?

20. Jan 16, 2012

Dick

Now can't you think of a simpler way to do it than that? You are trying to find the six sixth roots of -1. Try using polar form r*exp(i*theta).