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Math Help - Strange complex integral/path question

  1. #1
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    Strange complex integral/path question

    Hey guys,

    I've been trying to work out what this textbook question means! Well the first part that is, is it supposed to be say we must find those constants before proceeding? I've tried adding them all together just alebratically but its ugly and also integrating each fraction over the path but I feel like these constant need to vanish. It's an odd number so there are no solutions provided. I thought I would be able to use the same logic as the previous question, but its completely different! I'm really struggling with this one, enough to have made an account on here and ask for help!
    Could anyone point me in the right direct?



    Thanks alot,
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Strange complex integral/path question

    Quote Originally Posted by Angela11 View Post
    I interpret that you have to express the integral in terms of the given constants. Taking into account that \pi<\sqrt{10}<2\pi, we have

    (i)\;\int_{|z|=\pi}\frac{dz}{z^3(z^2+10)}=P\int_{|  z|=\pi}\frac{dz}{z}+Q\int_{|z|=\pi}\frac{dz}{z^2}+  R\int_{|z|=\pi}\frac{dz}{z^3}=\ldots

    (ii)\;\int_{|z|=2\pi}\frac{dz}{z^3(z^2+10)}=P\int_  {|z|=2\pi}\frac{dz}{z}+\ldots +T\int_{|z|=2\pi}\frac{dz}{z+\sqrt{10}i}=\ldots

    Now, apply the Cauchy integral formula.
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    Re: Strange complex integral/path question

    Thanks,

    I did something similar to that though I let z = pie^(it), so dz/dt = i pi e^(it) and for the P integral i got 2ipiP, and so on, then I got stuck at the S and T parts, I wasn't sure if f(z) was just = S or T.
    If f(z)=S, say, would f(-isqrt10) just = S, how come the S and T parts cancelled in part i? I tried rationalising and adding the S and T parts together but it didn't look helpful
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Strange complex integral/path question

    Quote Originally Posted by Angela11 View Post
    and for the P integral i got 2ipiP
    Right, but before continuing: have you covered the Cauchy integral formula?.
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    Re: Strange complex integral/path question

    Briefly,

    I know that int over c is f(z)/(z-z0) = 2ipif(z0) if z0 is enclosed within C, But we have more been using that approach above, kind of parametrizing first.

    We have used Cauchys integral formula as far as find the integral of 2z/(z-3) for example,

    So i thought for S the z0 would be isqrt(10) so the integral would be -2pisqrt(10) or S2ipi depending on if f(z)=S and if f(isqrt(10)) just = S,

    Just a little confused at this point
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Strange complex integral/path question

    Well, using the Cauchy integral formula or parametrizing, you'll easily obtain:

    \int_{|z|=\pi}\frac{dz}{z^3(z^2+10)}=2P\pi i

    \int_{|z|=2\pi}\frac{dz}{z^3(z^2+10)}=2P\pi i+2S\pi i\color{red}+\color{black}2T\pi i
    Last edited by FernandoRevilla; October 16th 2011 at 03:42 AM.
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  7. #7
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    Re: Strange complex integral/path question

    I understand that the Q and R parts are zero, but i'm confused with the S and T parts,

    for ii)

    Is this the correct logic for doing the S and T parts, letting the Cauchy integral formula be int f(z)/(z-a) = 2ipif(a)

    S:
    a = isqrt(10), f(z) = S, f(a) = S so the integral is 2Sipi

    T:

    a=-isqurt(10) f(z)=T, f(-sqrt(10))=-f(sqrt(10))=-T so -2Tipi


    and for part i, are the S and T integrals zero as pi<sqrt(10) so a=isqrt(10) is outside |z|=pi
    Last edited by Angela11; October 16th 2011 at 02:42 AM.
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Re: Strange complex integral/path question

    Quote Originally Posted by Angela11 View Post
    for ii) Is this the correct logic for doing the S and T parts, letting the Cauchy integral formula be int f(z)/(z-a) = 2ipif(a)
    Yes.

    S: a = isqrt(10), f(z) = S, f(a) = S so the integral is 2Sipi
    Right.

    T: a=-isqurt(10) f(z)=T, f(-sqrt(10))=-f(sqrt(10))=-T so -2Tipi
    No, it is f(a)=T (I had a typo in my previous post)


    and for part i, are the S and T integrals zero as pi<sqrt(10) so a=isqrt(10) is outside |z|=pi
    Right.
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    Re: Strange complex integral/path question

    Many thanks Fernando!
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