On Tuesday, October 18, 2016 at 4:27:22 PM UTC-4, digxx wrote:
>
> do u have an example for how to use a contour?
> quadgk(cis,0,1+1*im)=
>
probably integrates over the straight line so how can I integrate over the
> line gamma(t)=t+im*t^2
>
By contour, I just meant straight-line segments. For example:
quadgk(z -> 1/z, 1, 1im, -1, -1im)
integrates 1/z over a closed counter-clockwise diamond-shaped contour
around the origin in the complex plane, returning 2πi by the residue
theorem.
Of course, you can integrate over an arbitrary contour by a change of
variables, but then you have to include the Jacobian factor yourself. e.g.
you can integrate f(z) over z(t) from t=0..1 by doing quadgk(t -> f(z(t)) *
z′(t), 0, 1), where z′(t) is dz/dt. In your example,
z(t) = t + t^2*im
z′(t) = 1 + 2t*im
--SGJ
