Can I Use Antiderivatives to Evaluate this Complex Integral?

[Macykc2]

Homework Statement

I need to evaluate the following integral using the antiderivative:
$$\int log^2(z) \, dz$$
I don't know how to make a subscript for the integral sign, there should be a "c" on the bottom part. C is any contour from $π$ to $i$, not crossing the non-positive x-axis.

Homework Equations

Given above

The Attempt at a Solution

The only thing I can think of is to do a substitution, such as u=logz, like in the real case but I haven't officially learned if that's possible so I don't know if I can do it, nor if I even have to. And it specifically says to use the antiderivative so I can't parameterize.

 

[mfb]

Did you try integration by parts?

 

[sunnnystrong]

Integration by parts...
u = (log(x))^2
dv = 1dx

 

[mfb]

Integration by parts...
u = (log(x))^2
dv = 1dx

If I understand that post correctly, that will make it worse. There is a better choice of the two parts.

 

[sunnnystrong]

If I understand that post correctly, that will make it worse. There is a better choice of the two parts.

well, i don't want to post the solution but if you use u = log^2(x) than it will reduce the power on the log by 1 and leave you with an easier problem to integrate :)

 

[FactChecker]

well, i don't want to post the solution but if you use u = log^2(x) than it will reduce the power on the log by 1 and leave you with an easier problem to integrate :)

It works out fine with u=ln(z) and v'=ln(z), but you are right that your way is easier.

 

[mfb]

Oh wait, for post 4 I was imagining logs in the denominator for some reason.
Ignore post 4, both approaches work and the one from sunnnystrong is easier.

 

[Ray Vickson]

Homework Statement

I need to evaluate the following integral using the antiderivative:
$$\int log^2(z) \, dz$$
I don't know how to make a subscript for the integral sign, there should be a "c" on the bottom part. C is any contour from $π$ to $i$, not crossing the non-positive x-axis.

Homework Equations

Given above

The Attempt at a Solution

The only thing I can think of is to do a substitution, such as u=logz, like in the real case but I haven't officially learned if that's possible so I don't know if I can do it, nor if I even have to. And it specifically says to use the antiderivative so I can't parameterize.

You just put a '_C' next to your int instruction, to get $\int_C \log^2 (z) \, dz$. Right-click on the formula and ask for a display of math as tex commands, to see how it is done.

As for using antiderivatives: see, eg.,
https://en.wikipedia.org/wiki/Antiderivative_(complex_analysis).