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MathGroup Archive 2004

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Re: ArcCos[x] with x > 1

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49322] Re: ArcCos[x] with x > 1
  • From: ab_def at prontomail.com (Maxim)
  • Date: Wed, 14 Jul 2004 07:29:27 -0400 (EDT)
  • References: <cclev9$kb3$1@smc.vnet.net> <cco3io$4ig$1@smc.vnet.net> <cctaff$c11$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"Carl K. Woll" <carlw at u.washington.edu> wrote in message news:<cctaff$c11$1 at smc.vnet.net>...
> 
> Unfortunately, I think the use of Integrate for definite integrals may
> always have these problems with branches. After all, even if it were
> possible to rotate the branch cuts of functions on the fly, it will not
> always be possible to orient all of the branch cuts of the functions that
> arise in the indefinite integral so that they don't cross the path of
> integration.

Rotating the branch cuts can be a fruitful idea for handling integrals
of rational functions. Here's a quick implementation:

In[1]:=
ClearAll[int]
Options[int] = {Assumptions :> $Assumptions};
int::idiv = "Integral of `1` does not converge on `2`.";
int[$fz_, {$z_?(Variables[#] === {#} &), z1_, z2_}, Sopt___?OptionQ]
:=
Module[
    {main, assum, ans},
    main[] := Module[
        {lambdaQ, dummyint, fz = $fz,
         pz, qz, modz, adz, cond, Lz0, La, z},
        lambdaQ = Element[#, Reals] && 0 <= # <= 1 &;
        If[z1 == z2, Return[0]];
        fz = Together[fz /. $z -> z];
        {pz, qz} = {Numerator[fz], Denominator[fz]};
        If[! PolynomialQ[pz, z] || ! PolynomialQ[qz, z],
          Return[$Failed] ];
        Lz0 = Union@Array[Root[Function @@ {z, qz}, #] &,
          Exponent[qz, z]];
        La = Residue[pz/qz, {z, #}] & /@ Lz0;
        modz = Together[fz - Total[La/(z - Lz0)]];
        adz = Total[La*Log[(z - Lz0)/(z2 - z1)]];
        cond = Refine[
            z1 != z2 &&
              And @@ (!lambdaQ[(# - z1)/(z2 - z1)] &) /@ Lz0];
        If[!cond,
          Message[int::idiv, $fz, {z1, z2}];
          Return[$Failed] ];
        If @@ {cond,
              Integrate[modz, {z, z1, z2},
                  GenerateConditions -> False] +
                (adz /. z -> z2) - (adz /. z -> z1),
                dummyint[$fz, {$z, z1, z2},
                  Assumptions -> !cond && $Assumptions]} /.
          dummyint -> int
        ];
    assum = Assumptions /. Flatten@{Sopt, Options[int]};
    Block[
      {$Assumptions = assum},
      ans /; (ans = main[]) =!= $Failed
      ]
    ]

An example:

In[5]:=
Assuming[Element[{a, b}, Reals] && a != b,
  int[1/(z^2 - 1), {z, -2 + I*a, -2 + I*b}] // Simplify]

Out[5]=
(Log[(I + a)/(-a + b)] - Log[(3*I + a)/(-a + b)] -
  Log[(I + b)/(-a + b)] + Log[(3*I + b)/(-a + b)])/2

This approach has several advantages: it is fast; it returns
relatively simple conditions (compare with Mathematica's output for
Integrate[1/(z^2-1),{z,-2+I*a,-2+I*b}], which has a size of 250kb);
and most importantly, it gives the complete answer, that is, there is
no need to reevaluate the resulting If expression: either the If
condition is true or the integral doesn't converge.

I wonder whether this method can be extended to handle fractional
powers; it certainly would be hard to generalize it to antiderivatives
containing nested radicals or logarithms, because the branch cuts
aren't necessarily straight lines then.

Maxim Rytin
m.r at inbox.ru


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