Complex Integrals: Find Values of (1+3z+5z^2)/z^n

[bballife1508]

Find all possible values of the integral (1+3z+5z^2)/z^n about the circle of radius 1 centered at 0. n is any integer.

Please help.

 

[hgfalling]

Well, what do you know about integrals of this type? It's generally required that you show us some work or something.

Suppose you let n=0; that's a pretty simple thing to do. Then what is the integral?

Try some different values of n. Maybe you'll see what's going on.

 

[bballife1508]

when n=0 it becomes just the integral of 1+3z+5z^2 which is z+3/2z^2+5/3z^3, correct?

 

[hgfalling]

When we talk about complex integrals, we are generally talking about something that is loosely related to the concept of line integrals in $\mathbb{R}^2$.

So we are going to travel along a curve (for example, the circle of radius 1 in the complex plane), and integrate the value of the function as we go. We can do this by parametrizing the curve as a function of t, and making a real integral:

$$\intop_{z_0(t)}^{z_1(t)} f(z(t))z'(t) dt$$

But while this is valid and sometimes useful, it basically misses the point of complex analysis, which is basically all the miracles that come out of treating complex integrals differently from real ones.

Now, back to your problem...is it from a class? What have you covered so far? Do you know about Cauchy's Theorem? Residues? etc?

 

[bballife1508]

We have talked about cauchy and everything, I am preparing for my final and need help with this practice problem asap.

 

[hgfalling]

OK, to do this problem you need two facts:

$$\intop_{|z|=1} z^n dz = 0, n\in\mathbb Z, n \neq -1$$
$$\intop_{|z|=1} z^{-1} dz = 2\pi i$$

So, for example, if in the original problem, $n=0$, then

$$\intop_{|z|=1} 1+3z+5z^2 dz = 0$$

Can you take it from here?

 

[bballife1508]

for n=0 should it be 5z^3 not 5z^2? and just to make sure,

when n=1 it would be the integral of 1/z+3+5z^2 which would be 2pi(i)

and n=2 integral of 1/z^2+3/z+5z which would be 6pi(i)

and n=3 again would be 0

n=4 would be 10pi(i)

so the possilbe values are 0,2pi(i), 6pi(i) and 10pi(i)


is this correct?

 

[bballife1508]

sorry, i mistyped that original question, it should have been the integral of (1+3z+5z^3)/z^n

 

[hgfalling]

Yes, that's right.

 

[bballife1508]

thank you, if you can please help with this one as well it would be greatly appreciated

https://www.physicsforums.com/showthread.php?t=456776

 

[HallsofIvy]

I hate to say this, but you are preparing for a final in this course and you seem to know nothing about it? Surely, if you are expected to be able to do problems like this, you should have learned a while ago that the integral around a closed path of an analytic function is always 0. For n= 0 you should not even be thinking about an anti-derivative, you should recognize immediately that the integral is 0.

What you have is
$$\int_{|z|= 1} \frac{1+3z+5z^2}{z^n} dz= \int_{|z|= 1}z^{-n}dz+ 3\int_{|z|= 1}z^{1- n}dz+ 5\int_{|z|= 1} z^{2- n} dz$$,

And a good parameterization for the circle is [itex]z= e^{i\theta}[/tex].

I can tell you now, without doing any calculation, that this integral is equal to 0 for all except 3 values of n.