# Parametric Representation for a Contour Integral

• Nov 19th 2007, 04:58 PM
scothoward
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
• Nov 20th 2007, 03:19 AM
Opalg
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)

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

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