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Math Help - Complex analysis - contour integrals and branch cuts

  1. #1
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    Complex analysis - contour integrals and branch cuts

    Hi.


    I'm having some difficulties understanding when I'm supposed to take branch cuts into consideration when facing contour integrals. So far in my textbook, the exercises will always point out whether we are working with branches or not. But does that mean that I only have to consider branch cuts when the exercise specifically points it out?


    To illustrate, here is one example given in the book:



    The function f(z) = 1/(z^2), which is continuous everywhere except at the origin, has an antiderivative F(z) = -1/z in the domain |z| > 0, consisting of the entire plane with the origin deleted. Consequently,

    INTEGRAL (1/(z^2))dz = 0

    when C is the positively oriented circle:

    z = 2e^(iθ)

    (- pi < θ < 2 pi)


    And here the example ends.




    Later in the same section I run into this example:



    Let us use an antiderivative to evaluate the integral

    INTEGRAL (z^(1/2))dz

    where the integrand is the branch

    f(z) = z^(1/2) = exp((1/2)log(z)) = r^(1/2)*e^(iθ/2)

    (r > 0, 0 < θ < 2 pi )

    of the square root function and where C is any contour from z = -3 to z = 3 that, except for its end points, lies above the x-axis.

    The example then goes on to explain that the branc of z^(1/2) is not defined at θ = 0, and that thus we have to establish a new branch before we can proceed to solve the integral.


    I don't have any problems performing this latter process. However, my question is, how do we know that we have to take branch cuts into consideration in example 2 but not in example 1? Is it just because it is specifically pointed out in the example that we now work with an initial branch? Does this mean that if there is no such initial branch specified in an exercise, I can just solve away without taking it into consideration? (As long as the integral itself does not involve a Log of course).


    I would greatly appreciate it if anyone can explain this to me!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by krje1980 View Post
    I don't have any problems performing this latter process. However, my question is, how do we know that we have to take branch cuts into consideration in example 2 but not in example 1? Is it just because it is specifically pointed out in the example that we now work with an initial branch? Does this mean that if there is no such initial branch specified in an exercise, I can just solve away without taking it into consideration? (As long as the integral itself does not involve a Log of course).

    We define \int_{\gamma}\omega for the differential form \omega=f(z)dz if f is a continuous map in an open set D that contains the range of \gamma \in C^1[a,b] .


    At first, the problem should specify clearly what is the integrand funcion. In our case, the expression f(z)=\sqrt{z} does not define a map, only its branches. If the problem does not specify which one, then choose a branch:

    \subset \mathbb{C}\rightarrow \mathbb{C}, \quad (\textrm{rg}(\gamma)\subset D)" alt="f\subset \mathbb{C}\rightarrow \mathbb{C}, \quad (\textrm{rg}(\gamma)\subset D)" />

    in which f is continuous (in this case holomorphic).


    Fernando Revilla
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  3. #3
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    Thanks a lot!

    Since I'm quite new to complex analysis, I must admit that I am not that familiar with all the rigorous proofs yet. However, I think I understand your explanation quite well. It is because the second problem actually defines only a branch and not a map that we have to take this into consideration, right? And since, in the first problem, the problem is not defined only to one single branch, we do not have to consider it, right?

    Also, like I said, I have no problems identifying a branch which will make the function continuous (holomorphic) once I know that this is required.

    I'm sorry if it takes me a little while to digest it .
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