So I have the set $a=\{x+iy|y=x^3-3x^2+4x-1\}$ that connects $1+i$ and $2+3i$.
How do I parametrize a complex path of this?
Eventually I want to find $\int_a(12z^2-4iz)dz$ and it seemed to me the best way was to parametrize a path, but I wouldn't know how. Can somebody please help me?
Ok i will try to solve this as a 2nd order integration by curve.
$\int_a (12x^2-12y^2+2ixy -4ix +4y)(dx+idy)$
$$ =\int_a (12x^2-12y^2+4y)dx+(-2xy+4x)dy + i\int_a(12x^2-12y^2+4y)dy+(2xy-4x)dx $$
These two integrals depend on the path chosen since $\frac{\partial Q}{\partial x}\ne \frac{\partial P}{\partial y}$
So i will take the path that you wanted
$\gamma(t)=(t,t^3−3t^2+4t−1)$ for $t\in[1,2]$
That gives us the integral by substituting $x$ for $t$ , $y$ for $t^3−3t^2+4t−1$ ,
$dx$ for $1dt$,
$dy$ for $(3t^2-6t+4)dt$
And the borders are $ [1,2] \ni t$
There is some work to be done but the integral itself is easy.