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Math Help - Parametric Representation for a Contour Integral

  1. #1
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    Parametric Representation for a Contour Integral

    Hi, I am having a lot of trouble understanding the concept of "parametric representation" in general. I have the following quiestion,

    For the functions f and contours C, use parametric representations for C or legs of C to evaluate the contour integral of f(z)dz.

    f(z) is defined by the equations:
    {
    1 when y < 0
    4y when y >0
    }

    and C is the arc from z = -1 - i to z = 1 + i along the curve y = x^3.


    I have the solution for this question, but I would much rather understand the intuition into these types of problems. From my vague understanding, (-1 -i) to (1 + i) is where we would like to integrate from and we would like to use y = x^3 to paramertize f(z).

    Am I on the right track? Any help would be very much appreciated.

    Thanks
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  2. #2
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    Opalg's Avatar
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    A parametric representation of a contour C means a continuous function t\mapsto s(t) from an interval on the real line to C, such that as t goes along the interval, s(t) traces out the contour C.

    For example, if C is the unit circle, you can parametrise it by the map s(t) = e^{it} for t in the interval [0,2\pi]. As the parameter t goes from 0 to 2π, s(t) goes right round the contour C.

    If C is the contour given by y = x^3 (where z=x+iy) then you can take x to be the parameter. The parametrisation is then the map s(x) = x + ix^3, defined on the interval [–1,1]. As x goes from –1 to 1, s(x) goes along the curve from –1–i to 1+i.

    If you want to integrate a function f(z) along the contour C, then you substitute z=s(x), dz=s'(x)dx, and the integral becomes an integral of a (complex-valued) function of a real variable, which you can integrate using any of the techniques of ordinary (real-variable) integration. So
    \int_Cf(z)\,dz = \int_{-1}^1f(x+ix^3)\,dx = \int_{-1}^0f(x+ix^3)\,dx + \int_{0}^1f(x+ix^3)\,dx (we have to split up the interval of integration because the function f is defined differently depending on whether y (=x^3) is positive or negative)

    .. .. . . . . . . . . . . . . . . . . . . . = \int_{-1}^01\,dx + \int_{0}^14x^3\,dx

    .. .. . . . . . . . . . . . . . . . . . . . = 1+ \left[x^4\right]_0^1 = 2.
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