Complex Integration: Evaluating Line Integrals on a Circle

[LostEngKid]

Evaluate the Line Integral (assume counterclockwise orientation)

\oint _{|z| = 2 } z^n \bar{z}^m dz for all m, n \in Z

I have no freaken clue about how to even attempt this...

 

[nicksauce]

By intuition would be to write it as the integral of

(z\bar{z})^{m}z^{n-m}

and evaluate for different cases of n-m.

 

[nrqed]

Evaluate the Line Integral (assume counterclockwise orientation)

\oint _{|z| = 2 } z^n \bar{z}^m dz for all m, n \in Z

I have no freaken clue about how to even attempt this...

Just write z in polar representation z = e^{i r \theta}. Then it's easy.

 

[Dick]

Or just write z=2*exp(i*theta), zbar=2*exp(-i*theta), dz=2*i*exp(i*theta)*d(theta) and integrate theta from 0 to 2pi. Same thing really. If you are following nicksauce's suggestion be sure and replace z*zbar by 4. zbar isn't analytic. Don't try and do the whole thing as a complex integral.

 

[Dick]

Just write z in polar representation z = e^{i r \theta}. Then it's easy.

Uh, z=r*exp(i*theta), right?

 

[nrqed]

Uh, z=r*exp(i*theta), right?

Of course! Sorry for the typo!

 

[Dick]

Of course! Sorry for the typo!

S'alright. Just didn't want to confuse LostEngKid.

 

[nrqed]

S'alright. Just didn't want to confuse LostEngKid.

I know. That's why I apologized. I know that if it was just for you, it would not matter much because it is obvious to you that it's a typo. But I am glad you pointed it out for the OP and others reading this thread!

Regards

 

[LostEngKid]

Thanks for the help guys i really appreciate it, i think ill definitely need this site to pass maths this semester, god i hope i don't have another maths subject next year

Just on a side note, I am doing electrical engineering and I am wondering when complex analysis would be used in a practical sense, i mean at the moment it seems like maths for the sake of maths and no1 has given me an example of an application for it. What is is used for?

 

[Dick]

Thanks for the help guys i really appreciate it, i think ill definitely need this site to pass maths this semester, god i hope i don't have another maths subject next year

Just on a side note, I am doing electrical engineering and I am wondering when complex analysis would be used in a practical sense, i mean at the moment it seems like maths for the sake of maths and no1 has given me an example of an application for it. What is is used for?

I thought electrical engineering was a hotbed of complex numbers, so much so that they use 'j' instead of 'i' so it won't be confused with 'i' for current. Aside from their general uses in differential equations and contour integration, voltage/current and capacitance/inductance are handy to represent as components of complex numbers.