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Questions about poles and residues (complex analysis) watch

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    Without going into the precise definition in terms of the Laurent expansion, what exactly is a pole? Is it just where division by zero occurs? I mean, clearly if we have a fraction with (z-a) on the bottom then a will be a pole. But what if we have a fraction with a 1/z term or, say (1 + 1/z)^2 on the top? Is that going to give a pole at 0?

    How can I tell the order of a pole when it's not completely clear? Say we have a pole at a. Do I just take my function f(z), and take the function gn(z) = (z-a)^n * f(z) for each natural number n, and find the least natural number n such that gn(a) is not zero?

    What is the easiest/most straightforward way to find residues? Should I try to get at the -1 term of the Laurent expansion using Taylor/power series expansions, or will it usually be easier to just use the derivative formula
    http://en.wikipedia.org/wiki/Residue...er_order_poles
    ?

    Thanks for any help.
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    (Original post by gangsta316)
    Without going into the precise definition in terms of the Laurent expansion, what exactly is a pole? Is it just where division by zero occurs?
    More or less. Formally, if you have a holomorphic function f : U \to \mathbb{C} with U containing an open annulus around z, we say f has a pole of order n at z if \lim_{\zeta \to z} (\zeta - z)^n f(\zeta) exists and is non-zero.

    I mean, clearly if we have a fraction with (z-a) on the bottom then a will be a pole. But what if we have a fraction with a 1/z term or, say (1 + 1/z)^2 on the top? Is that going to give a pole at 0?
    \displaystyle \left( 1 + \frac{1}{z} \right)^2 = \frac{(z + 1)^2}{z^2}, so this has a double pole at 0.

    How can I tell the order of a pole when it's not completely clear? Say we have a pole at a. Do I just take my function f(z), and take the function gn(z) = (z-a)^n * f(z) for each natural number n, and find the least natural number n such that gn(a) is not zero?
    Yes.

    What is the easiest/most straightforward way to find residues? Should I try to get at the -1 term of the Laurent expansion using Taylor/power series expansions, or will it usually be easier to just use the derivative formula
    http://en.wikipedia.org/wiki/Residue...er_order_poles
    ?
    Do a few practice questions and see which is easier for you. If you know in advance that you have a simple pole you can take a shortcut and try to directly evaluate \lim_{\zeta \to z} (\zeta - z) f(\zeta), for example.
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    (Original post by Zhen Lin)
    More or less. Formally, if you have a holomorphic function f : U \to \mathbb{C} with U containing an open annulus around z, we say f has a pole of order n at z if \lim_{\zeta \to z} (\zeta - z)^n f(\zeta) exists and is non-zero.



    \displaystyle \left( 1 + \frac{1}{z} \right)^2 = \frac{(z + 1)^2}{z^2}, so this has a double pole at 0.



    Yes.



    Do a few practice questions and see which is easier for you. If you know in advance that you have a simple pole you can take a shortcut and try to directly evaluate \lim_{\zeta \to z} (\zeta - z) f(\zeta), for example.
    Thank you.

    If I want to evaluate a real integral with infinity in one (or both) of its limits, I know that I would use the usual semi-circular contour. In these cases, is the integral round the semi-circular arc always going to go to 0? In all of the examples I've seen, this seems to be the case.
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    (Original post by gangsta316)
    Thank you.

    If I want to evaluate a real integral with infinity in one (or both) of its limits, I know that I would use the usual semi-circular contour. In these cases, is the integral round the semi-circular arc always going to go to 0? In all of the examples I've seen, this seems to be the case.
    By no means! Sometimes you will see an integral with something like sinh x in the denominator, in which case, not only would you not have convergence on a large semicircle, but you would be adding an infinite series of residues as the radius of the semicircle increased!

    Look out for a complex analysis book with plenty of examples and you should see some instances of how to use a rectangular contour for example.
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    (Original post by davros)
    By no means! Sometimes you will see an integral with something like sinh x in the denominator, in which case, not only would you not have convergence on a large semicircle, but you would be adding an infinite series of residues as the radius of the semicircle increased!

    Look out for a complex analysis book with plenty of examples and you should see some instances of how to use a rectangular contour for example.
    I see. I just studied Jordan's lemma which guarantees that the semi-circle integral will vanish given certain conditions (the main one being that |f(z)| -> 0 as |z| -> infinity and the convergence is uniform). So how do I decide whether I should use the usual semi-circle or the rectangle contour? It seems that if Jordan's lemma applies, then I should go for the semi-circle and that otherwise I should use the rectangle (and the integrand will usually be something involving exponentials).

    How do I deal with the indented semi-circle contour, where we go round the origin in a semi-circle to avoid a point of non-holomorphy? Usually the main part of the integral we need to evaluate is this one: the integral round the semi-circle in the upper half plane, with center at the origin and radius r (r is small). I know that we can let r->0 before we integrate if we need to. But what happens if I get something/r in the integrand (say exponential/r)? Does this mean that I have definitely done something wrong? This kind of integrand would appear to blow up if we let r->0.
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    (Original post by gangsta316)
    It seems that if Jordan's lemma applies, then I should go for the semi-circle and that otherwise I should use the rectangle (and the integrand will usually be something involving exponentials).

    How do I deal with the indented semi-circle contour, where we go round the origin in a semi-circle to avoid a point of non-holomorphy? But what happens if I get something/r in the integrand (say exponential/r)? Does this mean that I have definitely done something wrong? This kind of integrand would appear to blow up if we let r->0.
    Yes, go with your instinct on this! Most of the basic complex analysis texts will probably just show examples where this automatically applies, but the more comprehensive ones, e.g. the Schaum guide with loads of worked examples, will give you examples where a rectangle (or sector) is appropriate.

    For the small semicircle, remember that you can parameterize it as z = re^(it) where t varies from 0 to pi (or pi to 0 if going clockwise) so dz = (ire^(it))dt and the r's will cancel if you have a 1/r in the integrand. This is typically why you get a finite contribution from the indentation, and if you think about it you can probably see that the integral round a semicircle is half the integral round a circle enclosing the pole - and you get the latter by a simple application of the residue theorem.
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    I'm having trouble understanding the keyhole contour.

    http://www2.imperial.ac.uk/~bin06/M2...pm3l30(11).pdf

    Check near the end of the document. It says "Near gamma 3: argz = 2pi.....". Why is it important that we parametrized it as z = x* e^(2*pi*i)? It clearly made a difference somewhere but why didn't we just replace that e^(2*pi*i) by 1? How can we have an upper branch where the argument is 0 and a lower branch where the argument is 2*pi? I mean, aren't they the same? We could perhaps think of them as slightly different but won't we still get the same things coming out when we plug in 2*pi instead of 0? So I don't understand why this method works, when we would expect the upper integral and lower integral (above and below the axis) to cancel each other out and thus not give us anything useful.
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    (Original post by gangsta316)
    I'm having trouble understanding the keyhole contour.

    http://www2.imperial.ac.uk/~bin06/M2...pm3l30(11).pdf

    Check near the end of the document. It says "Near gamma 3: argz = 2pi.....". Why is it important that we parametrized it as z = x* e^(2*pi*i)? It clearly made a difference somewhere but why didn't we just replace that e^(2*pi*i) by 1? How can we have an upper branch where the argument is 0 and a lower branch where the argument is 2*pi? I mean, aren't they the same? We could perhaps think of them as slightly different but won't we still get the same things coming out when we plug in 2*pi instead of 0? So I don't understand why this method works, when we would expect the upper integral and lower integral (above and below the axis) to cancel each other out and thus not give us anything useful.
    They don't cancel as on the lower side of the branch cut you have  e^{2ai\pi} time the 1 where  0 < a < 1 so you will get

     (1- e^{2ai\pi})I = sum of residues

    The LHS is not zero as a is not an integer.
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    (Original post by thebadgeroverlord)
    They don't cancel as on the lower side of the branch cut you have  e^{2ai\pi} time the 1 where  0 < a < 1 so you will get

     (1- e^{2ai\pi})I = sum of residues

    The LHS is not zero as a is not an integer.
    But if we had parametrized it as z = x instead of z = x*exp(2*pi*i), we would get -I wouldn't we? But why? x = x*exp(2*pi*i) !
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    (Original post by gangsta316)
    But if we had parametrized it as z = x instead of z = x*exp(2*pi*i), we would get -I wouldn't we? But why? x = x*exp(2*pi*i) !
    Because you have cut the plane along the positive real axis,  0\leq argz < 2\pi , you take  \gamma_3 = \displaystyle\lim_{\delta\to 0}x - \delta i.

    As you have cut the plane along the real axis the arguement can't jump back to zero
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    (Original post by thebadgeroverlord)
    Because you have cut the plane along the positive real axis,  0\leq argz < 2\pi , you take  \gamma_3 = \displaystyle\lim_{\delta\to 0}x - \delta i.

    As you have cut the plane along the real axis the arguement can't jump back to zero
    So, rather than having z = x*exp(2*pi*i), are we really saying z = x*exp(c*i) where 0 < c < 2*pi, and then at the end of everything, taking the limit as c -> 2*pi? I just got confused because I thought that, wherever one sees exp(2*pi*i), this can just be replaced by 1.
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    (Original post by gangsta316)
    I thought that, wherever one sees exp(2*pi*i), this can just be replaced by 1.
    No, you can't always do this.
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    (Original post by DFranklin)
    No, you can't always do this.
    Why not? Aren't they equal?

    I've another question. What is the residue of f(z) = 1/z^2 * pi * cot(pi*z) at 0? I tried using the derivative rule by letting g(z) = z * pi *cot(pi*z) [since 0 is a triple pole] and differentiating twice. But then when you plug in 0 we get division by zero.
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    (Original post by gangsta316)
    Why not? Aren't they equal?
    Not when you've got a fractional power in the mix, no.

    I've another question. What is the residue of f(z) = 1/z^2 * pi * cot(pi*z) at 0? I tried using the derivative rule by letting g(z) = z * pi *cot(pi*z) [since 0 is a triple pole] and differentiating twice. But then when you plug in 0 we get division by zero.
    You can't "plug in 0", you need to take the limit as z->0.
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    (Original post by DFranklin)
    Not when you've got a fractional power in the mix, no.

    You can't "plug in 0", you need to take the limit as z->0.
    But why aren't they equal? Isn't it just exp(2*pi*i) = cos(2*pi) + i* sin(2*pi) = 1?

    I see. That does seem to be the proper form of the derivative rule.
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    (Original post by gangsta316)
    But why aren't they equal? Isn't it just exp(2*pi*i) = cos(2*pi) + i* sin(2*pi) = 1?
    No. The reason why is complicated and I'm not going to go over it. You should have been taught it in lectures (particularly if they've discussed things like branch cuts).
 
 
 
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