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Math Help - Complex Variables

  1. #1
    Member ronaldo_07's Avatar
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    Complex Variables



    Please help solve this question i do not understand it.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ronaldo_07 View Post


    Please help solve this question i do not understand it.
    a) The equivalent parametric equation for the line from z=0 to z=1 is z\!\left(t\right)=t;~0\leqslant t\leqslant 1. Thus, z^{\prime}\!\left(t\right)=1.

    Therefore, \int_{\gamma}\left(z-i\right)^2\,dz=\int_0^1 t^2-2it-1\,dt. Can you continue from here?

    b) The equation of the line segment is z\!\left(t\right)=1+it;~0\leqslant t\leqslant1. Thus, z^{\prime}\!\left(t\right)=i.

    Therefore, \int_{\gamma}\left(z-i\right)^2\,dz=i\int_0^1\left(1+it-i\right)^2\,dt. Can you continue from here?

    c) The equation of the circle in parametric form is z\!\left(t\right)=i+e^{it};~-\frac{\pi}{2}\leqslant t\leqslant0. Thus, z^{\prime}\!\left(t\right)=ie^{it}.

    Therefore, \int_{\gamma}\left(z-i\right)^2\,dz=i\int_{-\frac{\pi}{2}}^0e^{3it}\,dt. Can you continue from here?


    Does this make sense?
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