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Math Help - Integrating complex logs

  1. #1
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    Integrating complex logs

    Hi,

    I'm having trouble getting my head around integrating a function such as:

    f(z)=log(z+2),

    where z is a complex number.

    Any ideas on where to start?

    Thanks

    Tony
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  2. #2
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    Quote Originally Posted by Tony2710 View Post
    Hi,

    I'm having trouble getting my head around integrating a function such as:

    f(z)=log(z+2),

    where z is a complex number.

    Any ideas on where to start?

    Thanks

    Tony
    How about (z+2)\log(z+2) - (z+2) + \alpha where \alpha\in \mathbb{C} ?
    (Treat this as a regular calculus problem).
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  3. #3
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    Quote Originally Posted by Tony2710 View Post
    Hi,

    I'm having trouble getting my head around integrating a function such as:

    f(z)=log(z+2),

    where z is a complex number.

    Any ideas on where to start?

    Thanks

    Tony
    Yea, plot it. But first plot Log(z) then draw some contours over the real and imaginary surfaces and interpret:

    \mathop\int\limits_{C} Log(z)dz

    and:

    \oint Log(z)dz

    I personally think until you understand the geometry of a multifunction (it's real and imaginary Riemann surfaces), contour integrations over these surfaces will always be difficult to understand. Take time to draw them and understand them, then the integrals become a breeze.
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  4. #4
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    I want to add something to what I said. The function \log (z+5) has a primitive g on \mathbb{C} - (-\infty,-5]. On this "slit plane" (or cut-plane) g'(z) = \log (z+5). However, on (-\infty,-5] the primitive is not even continous (because the logarithm jumps its value as it comes around). Therefore, if you are trying to compute the line integral from A to B and the curve that you are integration does not cross over (-\infty,-5] then the integral is simply g(B) - g(A). However, if the curve does cross over then you need to be careful and cannot use this fundamental theorem for line integrals.
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