Contour Integral Routines

• To: mathgroup at smc.vnet.net
• Subject: [mg18641] Contour Integral Routines
• From: p_mclean at postoffice.utas.edu.au (Patrick McLean)
• Date: Thu, 15 Jul 1999 01:45:38 -0400
• Organization: Maths Dept, University of Tasmania
• Sender: owner-wri-mathgroup at wolfram.com

```Here are some lil' routines I knocked up to do some contour integration. I
think that they're pretty dodgy but a first approximation nonetheless!!

Things I'd like to do:
Fix dodgy bits
Include other contours
Put little arrows on the contours
Conformal Maps??

If there are any suggestions or similar approaches out there I'd be interested.

\!\(\( (*\
Version\ 4\ users\ can\ omit\ this\ *) << Calculus`\n\(X[a_, b_]\)[t_] :=
\(UnitStep[t - a] -
UnitStep[t - b]\n
\n (*\ Integral\ of\ f\ over\ path\ made\ up\ of\ gs\ *) \n
i[f_, gs_] :=
\(Sum[NIntegrate[
f[\(gs[\([i]\)]\)[t]] \(\(Derivative[1]\)[gs[\([i]\)]]\)[t], {
t, 0, 1}], {i, 1, Length[gs]}]\n
\n (*\ Plot\ of\ path\ made\ up\ of\ gs\ *) \nplot[gs_] :=
\(ParametricPlot[\n\t\t
Evaluate[
Table[{Re[\(gs[\([i]\)]\)[t]], Im[\(gs[\([i]\)]\)[t]]}, {i,
1, Length[gs]}]]\n\t\t\t, {t, 0, 1}, PlotRange -> All]\n
\n (*\ Reverses\ orientation\ of\ path\ segment\ *) \nm[g_] :=
\(\(g[1 - #]\ &\)\n
\n (*\ Circles\ And\ Some\ Ellipses\ *)
\(c[zc_, r_, t0_, t1_]\)[t_] :=
\(zc + r\ Exp[2  Pi\ I\ \((t0 + \((t1 - t0)\)\ t)\)]\n
\(el[t0_, t1_, p_]\)[t_] :=
\(Cos[2\ Pi\ \((t0 + \((t1 - t0)\) t)\) - I\ Log[p]]\n
\n (*\ Lines\ *) \n\(l[a_, b_]\)[t_] :=
\(a\ \((1 - t)\)\  +
t\ \ b\n\(l[z0_, DirectedInfinity[z_]]\)[t_] :=
\(z0 + z\ Tan[Pi\ t/2]\n
\(l[z0_, DirectedInfinity[z_]]\)[1] :=
\(DirectedInfinity[z]\nl[DirectedInfinity[z_], z0_] :=
m[l[z0, DirectedInfinity[z]]]\)\)\)\)\)\)\)\)\); \n
\n (*\ Now\ try\ plotting\ this\ path\ *) \np = 6; r = 1/p; t1 = 1/8;
t2 = 1 - t1;
t4 = \(-\(ArcSin[\(Sqrt[2]\ p\ r\)\/\(1 - p\^2\)]\/\(2\ \[Pi]\)\)\);
t3 = 1 - t4; \n
plot[{\n\t\tm[c[1, r, t1, t2]], \n\t\t
l[\(m[c[1, r, t1, t2]]\)[1], \(el[t4, t3, p]\)[0]], \t\t\n\t\t
el[t4, t3, p], \n\t\t
l[\(el[t4, t3, p]\)[1], \(m[c[1, r, t1, t2]]\)[0]]\n\t}]\n
\n (*\ or\ doing\ this\ integral\ *) \ni[Cos, {c[0, 2, 0, 1/4]\t}]\)\)

--
Patrick McLean

No news is good news...

```

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