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Complex integration question

  1. Aug 11, 2008 #1
    Evaluate the Line Integral (assume counterclockwise orientation)

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

    I have no freaken clue about how to even attempt this....
     
  2. jcsd
  3. Aug 11, 2008 #2

    nicksauce

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    By intuition would be to write it as the integral of

    [tex](z\bar{z})^{m}z^{n-m}[/tex]

    and evaluate for different cases of n-m.
     
  4. Aug 11, 2008 #3

    nrqed

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    Just write z in polar representation [tex] z = e^{i r \theta} [/tex]. Then it's easy.
     
  5. Aug 11, 2008 #4

    Dick

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    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.
     
  6. Aug 11, 2008 #5

    Dick

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    Uh, z=r*exp(i*theta), right?
     
  7. Aug 12, 2008 #6

    nrqed

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    Of course! Sorry for the typo!!
     
  8. Aug 12, 2008 #7

    Dick

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    S'alright. Just didn't want to confuse LostEngKid.
     
  9. Aug 12, 2008 #8

    nrqed

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    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
     
  10. Aug 12, 2008 #9
    Thanks for the help guys i really appreciate it, i think ill definately need this site to pass maths this semester, god i hope i dont have another maths subject next year

    Just on a side note, im doing electrical engineering and im 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?
     
  11. Aug 12, 2008 #10

    Dick

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    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.
     
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