Validating Complex Integration Using Polar Form and the Unit Circle

[futurebird]

I've been asked to find the value of:

$$\int_{-1}^{1}z^{\frac{1}{2}}dz$$

Here is how I did it:

I want to change to polar form. To do that I'll use the fact that:

$$z=e^{i\theta}$$ For a unit circle radius 1.
$$dz=ie^{i\theta}d\theta$$

I replace z and dz:

$$= \int_{-1}^{1}e^{\frac{i\theta}{2}}ie^{i\theta}d\theta$$

$$= \int_{-1}^{1}ie^{\frac{i\theta}{2}}e^{i\theta}d\theta$$

$$= \int_{-1}^{1}ie^{\frac{3i\theta}{2}}d\theta$$

$$= \frac{2}{3}\int_{-1}^{1}\frac{3i}{2}e^{\frac{3i\theta}{2}}d\theta$$

$$= \frac{2}{3}(e^{\frac{3i\theta}{2}})^{-1}_{1}$$

$$= (\frac{2}{3})(e^{\frac{3i}{2}}-e^{\frac{-3i}{2}})$$

Did I do this correctly? How can I check my work?

 

[Dick]

Definitely not right. You changed the integration variable from z to theta, but then you plugged the z limits into theta. That's definitely wrong. You also have to be careful about how you define z^(1/2). You know about branch cuts, right?

 

[futurebird]

Definitely not right. You changed the integration variable from z to theta, but then you plugged the z limits into theta. That's definitely wrong. You also have to be careful about how you define z^(1/2). You know about branch cuts, right?

Okay thanks. I know a *little* about branch cuts. I think that I need one on the -y axis since I want to trace a circut from theta= 0 to pi. (but not go through the origin.)

But, how do I change the -1 and 1? I see why what I did was wrong. I didn't change the interval... But, I can't quite see how to change it.

 

[Hurkyl]

If z = 1 and z = Exp(i theta)...

 

[Dick]

If you know the limits on theta are 0 and pi, those are the limits you should put into your antiderivative, not 1 and -1. Those are the z limits.

 

[futurebird]

If z = 1 and z = Exp(i theta)...

okay

z=1
z = Exp(i theta)

Then theta = 2pi

z=-1
z = Exp(i theta)

then theta = 1 NO!
then theta = pi !

I don't see how this helps. I think the values of a and b should be 0 and pi, but I don't know WHY.

I also don't know how to relate the branch cut -pi/4 <= theta < 3pi/4 to all of this.

 

[futurebird]

If you know the limits on theta are 0 and pi, those are the limits you should put into your antiderivative, not 1 and -1. Those are the z limits.

This makes sense to me, but I don't know if I'll know what to do on the next problem, becuse it might not be that easy to see. Is it because I'm changing 1 and -1 from rectangular to polar? And I can just ignore r because it is a constant?

 

[futurebird]

Okay let me try this again. I feel like I'm going to cry this stuff is so confusing.
grrr.. Okay

$$z=e^{i\theta}$$ For a unit circle radius 1.
$$dz=ie^{i\theta}d\theta$$
1 becomes 0 and -1 becomes $$\pi$$ in terms of $$\theta$$.

I replace z and dz AND I also replace a and b:

$$= \int_{0}^{\pi}e^{\frac{i\theta}{2}}ie^{i\theta}d\theta$$

$$= \int_{0}^{\pi}ie^{\frac{i\theta}{2}}e^{i\theta}d\theta$$

$$= \int_{0}^{\pi}ie^{\frac{3i\theta}{2}}d\theta$$

$$= \frac{2}{3}\int_{0}^{\pi}\frac{3i}{2}e^{\frac{3i\theta}{2}}d\theta$$

$$= \frac{2}{3}(e^{\frac{3i\theta}{2}})^{\pi}_{o}$$

$$= (\frac{2}{3})(e^{\frac{3i\pi}{2}}-e^{0})$$

$$= (\frac{2}{3})(-i-1)$$

$$= (\frac{-2i-2}{3})$$
Did I do this correctly? How can I check my work?

 

[Dick]

It's because you are changing variables, period. If I want to integrate f(x^2) from 1 to 2 and I change variables using u=x^2 and I get an antiderivative of g(u), I evaluate that from u=1 to 4. Not 1 to 2.

 

[Dick]

I don't want to make you cry, but your integration variables are backwards. I would evaluate it from pi to 0, not 0 to pi. I get (2/3)(1+i). The only way to check that I can think of is to evaluate along another path from -1 to 1 that avoids your branch cut. Wherever it is.

 

[futurebird]

It's because you are changing variables, period. If I want to integrate f(x^2) from 1 to 2 and I change variables using u=x^2 and I get an antiderivative of g(u), I evaluate that from u=1 to 4. Not 1 to 2.

That makes sense 1^2 = 1 and 2^2=4 and you have u=x^2.

I'm having a hard time seeing what u is in my problem. Well, I guess it would be theta since it's (d theta) that I'm working with... so theta = "a function of z"

z = x + iy

theta = arctan y/x

Is that it? arctan 0 ? that makes no sense...

 

[futurebird]

I don't want to make you cry, but your integration variables are backwards. I would evaluate it from pi to 0, not 0 to pi. I get (2/3)(1+i). The only way to check that I can think of is to evaluate along another path from -1 to 1 that avoids your branch cut. Wherever it is.

No I'm not going to cry you've been a BIG help! Okay, I mixed up the -1 and the 1. But when I make that change it should change the sign.

I have 3 more of these to do. Maybe I should go practice some calc 3 change of variable problems before I attempt them. Is that what it's called "change of variables" ? It's been 8 years since I had calc and I'm really rusty.

 

[Dick]

You said, "I want to trace a circut from theta= 0 to pi". When did you decide that made no sense?

 

[Dick]

No I'm not going to cry you've been a BIG help! Okay, I mixed up the -1 and the 1. But when I make that change it should change the sign.

I have 3 more of these to do. Maybe I should go practice some calc 3 change of variable problems before I attempt them. Is that what it's called "change of variables" ? It's been 8 years since I had calc and I'm really rusty.

Practice is never bad. But I think you know the what the problem is. If you decide to use theta as a variable instead of z, you should use theta limits instead of z limits. That's not rocket science. I'd worry more about branch cut issues.

 

[futurebird]

You said, "I want to trace a circut from theta= 0 to pi". When did you decide that made no sense?

It made sense but only becuase I drew a little sketch.

But I found another error that I made and I think I see how to change a and b in general...

It's this:

z=1
z = Exp(i theta)

Then theta = 2pi

z=-1
z = Exp(i theta)

then theta = 1 NO!
then theta = pi !

 

[Dick]

Sketches also help. Now start drawing branch cuts. They are pretty important with functions like z^(1/2).

 

[HallsofIvy]

One problem I see right from the start is that you do not specify any path for the integral. If z is a complex variable then there are an infinite number of paths you could use to go from -1 to 1. Taking $z= e^{i\theta}$ with $\theta$ from $-\pi$ to $\pi$ is choosing the upper unit semi-circle. Seeing only "-1 to 1" my first thought would have been to use the x-axis as path. Are those two integrals the same?

 

[futurebird]

One problem I see right from the start is that you do not specify any path for the integral. If z is a complex variable then there are an infinite number of paths you could use to go from -1 to 1. Taking $z= e^{i\theta}$ with $\theta$ from $-\pi$ to $\pi$ is choosing the upper unit semi-circle. Seeing only "-1 to 1" my first thought would have been to use the x-axis as path. Are those two integrals the same?

I don't think you can go through the origin. It's a branch point. That means you can't take that path or integral. (Right?)

I'm still fuzzy on how to use the branch cut. I know it needs to go from the origin to $$z_{\infty}$$, and if I think of the stereographic projection that means that any line from the origin off in some direction will do. So, I'll make it go down along the -y axis... so it won't hit my line integral in the upper half plane. You can't take the intergral and go through a branch cut.

What impact does this choice have on my answer? Or is the branch cut only important for cases like:

$$\int_{C}^{}\frac{dz}{z}$$ on a closed circuit called "C" around the origin, where if you ignore the branch cut you might think the value is 0 (and this is not correct.)

 

[Dick]

You CAN go through the origin, if you are careful, you just don't want to go through the negative y axis. The values of the function from 0 to 1 are just x^(1/2) and the values from -1 to 0 are i*|x|^(1/2). So you can see that the absolute values you are integrating are the same over both intervals and the absolute value of each integral is 2/3. So the [-1,0] gives i*2/3 and [0,1] gives 2/3. Total (2/3)*(1+i).

 

[futurebird]

You CAN go through the origin, if you are careful, you just don't want to go through the negative y axis. The values of the function from 0 to 1 are just x^(1/2) and the values from -1 to 0 are i*|x|^(1/2). So you can see that the absolute values you are integrating are the same over both intervals and the absolute value of each integral is 2/3. So the [-1,0] gives i*2/3 and [0,1] gives 2/3. Total (2/3)*(1+i).

Wow. I didn't know that. Thank you so much for you help!