Complex line integral

Question

Can someone recommend an online article or introductory tutorial that will show me how to do real and complex line integrals using Mathematica?

Answer 1

Here's a quick function that does a real line integral:

SyntaxInformation[
   lineIntegrate] = {"LocalVariables" -> {"Plot", {3, 3}}, 
   "ArgumentsPattern" -> {_, _, _}};


lineIntegrate[r_?VectorQ, f_Function, {t_, tMin_, tMax_}] := 
 Module[{param, localR}, localR = r /. t -> param;
  Integrate[(f[localR, #] Sqrt[#.#]) &@D[localR, param], {param, tMin,
     tMax}]]

lineIntegrate[{Cos[t], Sin[t]}, 1 &, {t, 0, 2 Pi}]

(* ==> 2 Pi *)

The second argument is a function to be evaluated at points along the curve. This function in turn can take two arguments: the position $\vec{r}$ and the derivative of the position with respect to the parameter, $d\vec{r}/dt$.

One could modify this definition so the function takes only the curve parameter as the argument, but I settled on this so that I can calculate things like work integrals easily. For example, take a closed path in 3D, and show that the work along it under a conservative force $\vec{F} = (0, 0, z)$ vanishes:

ParametricPlot3D[{Cos[t], Sin[t], Sin[2 t]}, {t, 0, 
  2 Pi}, PlotStyle -> Tube[.01]]

work[r_, tangent_] := {0, 0, r[[3]]}.tangent

lineIntegrate[{Cos[t], Sin[t], Sin[2 t]}, 
 work[#1, #2/Sqrt[#2.#2]] &, {t, 0, 2 Pi}]

(* ==> 0 *)