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Thread: integrating in the complex plane

  1. #1
    Sep 2009

    integrating in the complex plane

    hi here is the question, by integrating explicitly in the complex plane, evaluate

     \int_C (z-1)^6 dz

    where C is the half-circle centered at z = 1 with radius 1, starting at z_0 = 2 and ending at z_1=0, and traveresed in the upper half-plane. Show that the fundamental theorem of calculus holds for this example.

    here's my working out:

    parametrise:  z(t) = 1 + e^{it}
     dz = ie^{it}, 0 <=t<= \pi

     \int^\pi_0 (e^{it} - 1 + 1)^6 i e^{it} dt

    let  x = e^{it} and  dx = ie^{it} dt

     \frac{x^7}{7} ]^\pi_0
     = -2/7

    my fundamental part doesnt equal, coz when i did it i got -128/7
    so can someone please help me where i got it wrong?

    coz when i FIRST did it i parametrise this way:
     z(t) = e^{it}

     \int^\pi_0 (e^{it} - 1 )^6 i e^{it} dt
    and i got the same answer as the fundamental part, but since the question says centered at z = 1 dont you have to add the 1 to z(t)?
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  2. #2
    Apr 2010
    Adelaide, Australia
    You got the first part right:

    A parametrisation of C is C\left(t\right)=1+e^{it}.

    Now, for f\left(z\right)=\left(z-1\right)^6 the integral is

    \int_C f=\int_C f\left(z\right) =\int_a^b  f\left(C\left(t\right)\right)C^\prime\left(t\right  )~dt =\int_0^\pi\left(1+e^{it}-1\right)^6\left(ie^{it}\right)~dt =\int_0^\pi ie^{7it} =\left[\frac{1}{7}e^{7it}\right]_0^\pi=-\frac{2}{7}.

    You didn't show your working for the FTC part, but I believe you got that part wrong.

    For f=\left(z-1\right)^6, the anti-derivative F=\frac{1}{7}\left(z-1\right)^7.

    The fundamental theorem of calculus states that

    \int_C f=F\left(C\left(b\right)\right)-F\left(C\left(a\right)\right) =F\left(C\left(\pi\right)\right)-F\left(C\left(0\right)\right) =F\left(0\right)-F\left(2\right) =-\frac{1}{7}-\frac{1}{7} =-\frac{2}{7}.

    As expected, the result derived from the fundamental theorem and the result from explicit integration are the same.

    Maths help
    Last edited by lovek323; Apr 30th 2010 at 09:36 PM.
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