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Some questions on contour integals watch

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    I've been trying to read a bit about contour integrals, and was wanting to check a few things.

    For example, if this is the question:

    "Evaluate \displaystyle \int_{\gamma}{\frac{1}{z}}\,dz where \gamma is the unit circle in the complex plane."

    I start by parametrising the unit circle by z=e^{i\theta} for 0 \leq \theta \leq 2\pi, so the integral becomes

    \displaystyle \int_0^{2\pi}{\frac{ie^{i\theta}  }{e^{i\theta}}}\,d\theta, and so I'd say the required answer is 2i\pi

    Is this the right idea or am I completely off? Do other contour integrals behave like this, i.e. is this the method you would use?

    Also, I'd like to know if there is any visualisation behind these, much like a normal integral is the area under a line or the limit of a sum. What do these integrals actually mean? Thanks for your patience with this list of questions
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    (Original post by coffeym)
    I've been trying to read a bit about contour integrals, and was wanting to check a few things.

    For example, if this is the question:

    "Evaluate \displaystyle \int_{\gamma}{\frac{1}{z}}\,dz where \gamma is the unit circle in the complex plane."

    I start by parametrising the unit circle by z=e^{i\theta} for 0 \leq \theta \leq 2\pi, so the integral becomes

    \displaystyle \int_0^{2\pi}{\frac{ie^{i\theta}  }{e^{i\theta}}}\,d\theta, and so I'd say the required answer is 2i\pi

    Is this the right idea or am I completely off? Do other contour integrals behave like this, i.e. is this the method you would use?
    This is fine, and you can evaluate other integrals like this. But it's not the method you would usually use. There are lots of important results (i.e. tricks) to do with contour integration that make life easier.

    For example, if f is analytic, then \int_\gamma f(z)dz = 0 for any contour. More generally, the http://en.wikipedia.org/wiki/Residue_theorem lets you evaluate such integrals even where f is not analytic.

    Also, I'd like to know if there is any visualisation behind these, much like a normal integral is the area under a line or the limit of a sum. What do these integrals actually mean?
    I'm not sure there's a visualisation beyond what you've already found: the limit of the sum \sum f(z) \delta z. But one of the important uses of contour integration is that you can consider it a representation of a normal integral (see the wiki example, for instance), and this lets you solve a normal integral that would otherwise be insoluble.

    Hope that's some help.
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    Thanks very much, that link was good

    (Although I think I'm getting a bit ahead of myself...maybe I should leave complex analysis for a couple of years hehe)
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    I'd reccomend Needham's 'Visual Complex Analysis' for a superbly readable (though pretty heavy) introduction to this stuff. As the title suggests, there's a pretty big focus on the visual interpretation of what's going on - it's probably just what you're looking for.
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    (Original post by HPSH)
    I'd reccomend Needham's 'Visual Complex Analysis' for a superbly readable (though pretty heavy) introduction to this stuff. As the title suggests, there's a pretty big focus on the visual interpretation of what's going on - it's probably just what you're looking for.
    Thanks that looks good.

    Another question is: when can we use

     \oint

    instead of

     \int? I think it's something to do with when the contour is closed. Is this correct?
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    (Original post by coffeym)
    Thanks that looks good.

    Another question is: when can we use

     \oint

    instead of

     \int? I think it's something to do with when the contour is closed. Is this correct?
    The looped integral signifies that the contour is closed and positively oriented - mathematicians tend not to use it, I've only ever seen it in physics/engineering books.
 
 
 

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