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Math Help - Does a Laurent series expansion exist?...

  1. #1
    MHF Contributor chisigma's Avatar
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    Does a Laurent series expansion exist?...

    Prove [or even disprove...] that the principal branch of the function cannot be expanded in Laurent series around the point , i.e. a coefficient set such that for a definite region of the complex plane which is not limited to only one point is...

    (1)

    ... doesn't exist.

    Kind regards

    \chi \sigma
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  2. #2
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    How about this: If the logarithm had an (holomorphic) extension in an annulus around zero then, by a continuity argument we could conclude, since logarithm is a primitive of \frac{1}{z} in the slit plane, that \int_C \frac{dz}{z} =0 which is a contradiction.
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    MHF Contributor chisigma's Avatar
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    An useful preliminary is that a complex variable z can be written as...

    \displaystyle z= x + i\ y = r\ e^{i\ \theta} (1)

    ... where...

    r= \sqrt{x^{2} + y^{2}}

    \displaystyle \theta = \tan^{-1} \frac{y}{x} + 2\ k\ \pi\\ , k \in \mathbb{Z}

    ... so that is...

    \displaystyle \ln z= u(x,y) + i\ v(x,y) = r(x,y) + i\ \theta(x,y) (2)

    Now we compute the partial derivatives of u(*,*) and v(*,*) obtaining...

    \displaystyle \frac{\partial{u}}{\partial{x}} = \frac{x}{x^{2}+ y^{2}}

    \displaystyle \frac{\partial{u}}{\partial{y}} = \frac{y}{x^{2}+ y^{2}}

    \displaystyle \frac{\partial{v}}{\partial{x}} = - \frac{y}{x^{2}+ y^{2}}

    \displaystyle \frac{\partial{v}}{\partial{y}} = \frac{x}{x^{2}+ y^{2}} (3)

    ... and we observe that the Cauchy-Riemann conditions...

    \displaystyle \frac{\partial{u}}{\partial{x}} = \frac{\partial{v}}{\partial{y}}

    \displaystyle \frac{\partial{u}}{\partial{y}}= - \frac{\partial{v}}{\partial{x}} (4)

    ... are satisfied for any z \in \mathbb{C} with the only exception of z=0, where none of the (3) exists. So z=0 is a singularity of the function f(z)= \ln z and that means that no 'holomorphic extensions' of it including z=0 may exist . The 'core' of the question is why \ln z has no Laurent series expansion around z=0 like, for example, \displaystyle \frac{1}{z}...

    Kind regards

    \chi \sigma
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    The reason is that the region of convergence of a Laurent series is a (possibly degenerate) annulus. If this region contained more than one point, it would actually contain a whole circle around the origin, which would imply that it is possible to define \mbox{arg }z continuously on a circle around the origin, which is obviously impossible.
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  5. #5
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Bruno J. View Post
    The reason is that the region of convergence of a Laurent series is a (possibly degenerate) annulus. If this region contained more than one point, it would actually contain a whole circle around the origin, which would imply that it is possible to define \mbox{arg }z continuously on a circle around the origin, which is obviously impossible.
    If z= x + i\ y then [as in my previous post...] is...

    \displaystyle \theta= \mbox{arg}\ z = \tan^{-1} \frac{y}{x} + 2\ k\ \pi\\ ,\ \ k\in \mathbb{Z} (1)

    ... and...

    \displaystyle \frac{\partial{\theta}}{\partial{x}}= - \frac{y}{x^{2}+y^{2}}

    \displaystyle \frac{\partial{\theta}}{\partial{y}}= \frac{x}{x^{2}+y^{2}} (2)

    ... so that both the partial derivatives of \mbox{arg}\ z exist everywhere with the only exception of z=0. That means that on any circle around z=0 of radius r>0 the function \mbox{arg}\ z exists and is continous...

    Kind regards

    \chi \sigma
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    MHF Contributor chisigma's Avatar
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    May be it is useful to consider the case of the complex function \displaystyle f(z)=\frac{1}{z}. Setting z= x + i\ y is...

    \displaystyle f(z)= u(x,y)+i\ v(x,y) = \frac{x}{x^{2}+y^{2}} - i\ \frac{y}{x^{2}+y^{2}} (1)

    Is easy to verify from (1) that the Cauchy-Riemann conditions...

    \displaystyle \frac{\partial{u}}{\partial{x}} = \frac{\partial{v}}{\partial{y}}

    \displaystyle \frac{\partial{u}}{\partial{y}} = - \frac{\partial{v}}{\partial{x}} (2)

    ... are satisfied everywhere with the only exception of z=0, exactly as in the case of \ln z. So both the functions have a singularity in z=0 but f(z)=\frac{1}{z} has the Laurent expansion around z=0 [and is the function itself...], f(z)=\ln z don't have it...

    Kind regards

    \chi \sigma
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  7. #7
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by chisigma View Post
    If z= x + i\ y then [as in my previous post...] is...

    \displaystyle \theta= \mbox{arg}\ z = \tan^{-1} \frac{y}{x} + 2\ k\ \pi\\ ,\ \ k\in \mathbb{Z} (1)

    ... and...

    \displaystyle \frac{\partial{\theta}}{\partial{x}}= - \frac{y}{x^{2}+y^{2}}

    \displaystyle \frac{\partial{\theta}}{\partial{y}}= \frac{x}{x^{2}+y^{2}} (2)

    ... so that both the partial derivatives of \mbox{arg}\ z exist everywhere with the only exception of z=0. That means that on any circle around z=0 of radius r>0 the function \mbox{arg}\ z exists and is continous...

    Kind regards

    \chi \sigma
    Sure, the derivatives of \mbox{arg} exist because any two branches only differ by an additive constant. You can define it locally in any small enough neighborhood of a point not the origin. However, it is impossible to define \mbox{arg} continuously on a circle around the origin.
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  8. #8
    MHF Contributor chisigma's Avatar
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    We all [I suppose...] agree on the fact that a 'proof' must have a sort of 'rigorous background', so that I explain now my 'proof' and You all will decide if it is 'rigorous' or not...

    It is well known that the function \ln z is defined unless a constant 2\ k\ i\ \pi and it is not a limitation to 'indagate' on the 'principal branch', i. e. supposing k=0. At this scope let's consider the complex function...

    \displaystyle \varphi(z)= z\ \ln z (1)

    ... in z=1, where it is analytic, so that it can be represented in Taylor series...

    \displaystyle \varphi(z)= \sum_{n=0}^{\infty} \frac{\varphi^{(n)} (1)}{n!}\ (z-1)^{n} (2)

    The computation of the derivatives of (1) in z=1 can be performed considering that \varphi(*) is the solution of the DE...

    \displaystyle \varphi^{'} = 1 + \frac{\varphi}{z}\ \ ,\ \ \varphi(1)=0 (3)

    From (3) we derive in sequence...

    \displaystyle \varphi(1)=0

    \displaystyle \varphi^{(1)}= 1 + \frac{\varphi}{z} \implies \varphi^{(1)} (1)= 1

    \displaystyle \varphi^{(2)}= - \frac{\varphi}{z^{2}} + \frac{\varphi^{(1)}}{z} \implies \varphi^{(2)} (1)= 1

    \displaystyle \varphi^{(3)}= 2\ \frac{\varphi}{z^{3}} -2\ \frac{\varphi^{(1)}}{z^{2}} + \frac{\varphi^{(2)}}{z} \implies \varphi^{(3)} (1)= -1

    \displaystyle \varphi^{(4)}= -6\ \frac{\varphi}{z^{4}} +4\ \frac{\varphi^{(1)}}{z^{3}} - \frac{\varphi^{(2)}}{z^{2}} + \frac{\varphi^{(3)}}{z^{3}}  \implies \varphi^{(4)} (1)= 2

    ... and in general for n \ge 2...

    \displaystyle \varphi^{(n)} (1)= (-1)^{n}\ (n-2)! (4)

    ... so that is...

    \displaystyle \varphi (z)= z\ \ln z = (z-1) + \sum_{n=2}^{\infty} (-1)^{n} \frac{(z-1)^{n}}{n\ (n-1)} (5)


    The (5) is a very interesting formula and more than one 'surprise' is hidden behind it. One of these is visible when we devide both terms of (5) by z obtaining...

    \displaystyle \ln z = \frac{z-1}{z} + \sum_{n=2}^{\infty} \frac{(-1)^{n}}{z}\  \frac{(z-1)^{n}}{n\ (n-1)} (6)

    It is remarkable the fact that neither (5) nor (6) are reported in any of the 'sacred books' I consulted... I wonder why!... anyway (6) offers the possibility to verify if it exists on not a couple of sequences a_{n} and b_{n} such that is...

    \displaystyle \ln z = \sum_{n=0}^{\infty} a_{n}\ z^{n} + \sum_{n=1}^{\infty} b_{n}\ z^{-n} (7)

    From (6) it is immediate to verify that for n >1 is b_{n}=0. For n=1 we derive from (6) that is...

    \displaystyle b_{1}= -1 + \sum_{n=2}^{\infty} \frac{1}{n\ (n-1)} =0

    ... so that all the b_{n} vanish... a little unusual for a function that has a singularity in z=0!... Now a look to the a_{n}... for n=0 we have...

    \displaystyle a_{0}= 1 - \sum_{n=2}^{\infty} \frac{1}{n-1} = -\infty

    ... and that is sufficient to demonstrate that a series like (6) doesn't exist...

    Kind regards

    \chi \sigma
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  9. #9
    MHF Contributor Bruno J.'s Avatar
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    I don't know... to go from (6) to (7), are you just rearranging the series? Because you can only rearrange a power series expansion at a point inside the region of convergence, and the point z=0 surely isn't inside the region of convergence of (6).

    There is nothing wrong with the proof I gave. Such a Laurent series, if it existed, would converge and would be continous on an annulus centered around 0 (do you agree with this?). But this is impossible, because the logarithm has a branch cut from the north pole to the south pole, so the annulus of convergence would intersect the branch cut somewhere (agree with this?). And this would contradict the continuity of the power series expansion inside its region of convergence (agree with this?). It's the same reason why \sqrt z doesn't have a Laurent series expansion about the origin, or, in general, why any function having a branch cut from 0 to \infty does not have a Laurent series expansion around the origin.
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  10. #10
    MHF Contributor chisigma's Avatar
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    In most complex analysis textbooks it is written that the function \ln s is 'non regular' along the negative real axis... according to my [very modest...] opinion that is false!...

    To demostrate that it is sufficient consider the well known Taylor series expansion...

    \displaystyle \ln (1+z)= -\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\ z^{n}\ \ ,\ \ |z|<1 (1)

    Operating tha change of variables z=s-1 and taking into account that \ln s= \ln s + 2\ k\ \pi\ i the (1) becomes...

    \displaystyle \ln s = 2\ k\ \pi\ i - \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\ (s-1)^{n}\ , \ |s-1|<1 (2)

    Now if we set in (2) -s instead of s and consider that is \ln (-s)= \ln s + \pi\ i we obtain...

    \displaystyle \ln s = (2\ k +1)\ \pi\ i - \sum_{n=1}^{\infty} \frac{(s+1)^{n}}{n}\ , \ |s+1|<1 (3)

    Observing (2) and (3) we derive that the first is the Taylor expansion of \ln s in s=1 and the second the Taylor expansion of \ln s in s=-1 so that we can conclude that the function \ln s is analytic both in s=1 and s=-1...

    Kind regards

    \chi \sigma
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  11. #11
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    Quote Originally Posted by chisigma View Post
    In most complex analysis textbooks it is written that the function \ln s is 'non regular' along the negative real axis... according to my [very modest...] opinion that is false!...

    To demostrate that it is sufficient consider the well known Taylor series expansion...

    \displaystyle \ln (1+z)= -\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\ z^{n}\ \ ,\ \ |z|<1 (1)

    Operating tha change of variables z=s-1 and taking into account that \ln s= \ln s + 2\ k\ \pi\ i the (1) becomes...

    \displaystyle \ln s = 2\ k\ \pi\ i - \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\ (s-1)^{n}\ , \ |s-1|<1 (2)

    Now if we set in (2) -s instead of s and consider that is \ln (-s)= \ln s + \pi\ i we obtain...

    \displaystyle \ln s = (2\ k +1)\ \pi\ i - \sum_{n=1}^{\infty} \frac{(s+1)^{n}}{n}\ , \ |s+1|<1 (3)

    Observing (2) and (3) we derive that the first is the Taylor expansion of \ln s in s=1 and the second the Taylor expansion of \ln s in s=-1 so that we can conclude that the function \ln s is analytic both in s=1 and s=-1...

    Kind regards

    \chi \sigma
    Sure, you can make the logarithm analytic in a region containing 1 and -1, the point is you can never construct one defined on a whole circle around the origin (The point of restricting to the principal branch is that it's the easiest domain in which the function can be defined and indeed once defined this way it cannot be extended, which is what I believe authors mean when they say it isn't "regular"). Also it would be helpful if you could adress Bruno J.'s and my responses, do you think they're flawed? why? As it is, the way you're responding makes me think this is the case but it feels as if you're talking to yourself.
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  12. #12
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by chisigma View Post
    In most complex analysis textbooks it is written that the function \ln s is 'non regular' along the negative real axis... according to my [very modest...] opinion that is false!...

    To demostrate that it is sufficient consider the well known Taylor series expansion...

    \displaystyle \ln (1+z)= -\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\ z^{n}\ \ ,\ \ |z|<1 (1)

    Operating tha change of variables z=s-1 and taking into account that \ln s= \ln s + 2\ k\ \pi\ i the (1) becomes...

    \displaystyle \ln s = 2\ k\ \pi\ i - \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\ (s-1)^{n}\ , \ |s-1|<1 (2)

    Now if we set in (2) -s instead of s and consider that is \ln (-s)= \ln s + \pi\ i we obtain...

    \displaystyle \ln s = (2\ k +1)\ \pi\ i - \sum_{n=1}^{\infty} \frac{(s+1)^{n}}{n}\ , \ |s+1|<1 (3)

    Observing (2) and (3) we derive that the first is the Taylor expansion of \ln s in s=1 and the second the Taylor expansion of \ln s in s=-1 so that we can conclude that the function \ln s is analytic both in s=1 and s=-1...

    Kind regards

    \chi \sigma
    See what Jose wrote!

    What is meant by "the logarithm is non-regular along the negative real axis" is that a given branch of the logarithm, if you have chosen the branch cut to run along the negative real axis, is not analytic there. Of course, you can put the branch cut anywhere.
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