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Re: ArcCos[x] with x > 1

"David W. Cantrell" <DWCantrell at> wrote in message news:<cd0730$9g9$1 at>...

> But Paul's case is for version 5, as is what I showed above. So at least
> in the current version, some indefinite integrals are wrong.

Actually this is a quite curious example, because Mathematica 5.0
doesn't always return the same answer for this integral:

indef1 = Integrate[r*Sqrt[4*r^2+1]*ArcCos[1/(2*r)], r]
def1 = Integrate[r*Sqrt[4*r^2+1]*ArcCos[1/(2*r)], {r, 1/2, 1/Sqrt[2]}]
Integrate[Sqrt[4*r^2+1]*ArcCos[1/(2*r)], r]
indef2 = Integrate[r*Sqrt[4*r^2+1]*ArcCos[1/(2*r)], r]
def2 = Integrate[r*Sqrt[4*r^2+1]*ArcCos[1/(2*r)], {r, 1/2, 1/Sqrt[2]}]

(1/24)*Sqrt[1 + 4*r^2]*((-Sqrt[4 - 1/r^2])*r + (2 +
8*r^2)*ArcCos[1/(2*r)]) -
  (r*((-4 + Sqrt[16 - 1/r^4])*(-1 + 16*r^4) + 2*Sqrt[16 -
1/r^4]*Sqrt[-1 + 16*r^4]*
      Log[4*r^2 + Sqrt[-1 + 16*r^4]]))/(24*Sqrt[4 + 1/r^2]*(-1 +
4*r^2)*Sqrt[1 + 4*r^2])

(1/48)*(2 - 4*Sqrt[3] + 3*Sqrt[3]*Pi - 4*Log[2 + Sqrt[3]])

Integrate[Sqrt[4*r^2+1]*ArcCos[1/(2*r)], r]

(1/24)*Sqrt[1 + 4*r^2]*((-Sqrt[4 - 1/r^2])*r + (2 +
8*r^2)*ArcCos[1/(2*r)]) +
  (Sqrt[4 - 1/r^2]*r*Sqrt[-1 + 16*r^4]*(ArcTan[1/Sqrt[-1 + 16*r^4]] -
2*Log[4*r^2 + Sqrt[-1 + 16*r^4]]))/
   (24*(-1 + 4*r^2)*Sqrt[1 + 4*r^2])

(1/144)*((-2 + 9*Sqrt[3])*Pi - 6*(Sqrt[3] + Log[7 + 4*Sqrt[3]]))

This quirk is 100% reproducible on my machine (except for one time
when it crashed the kernel); apparently, evaluating In[3] changes some
internal states/settings and indef2, def2 are not the same as indef1,
def1. Out[4] and Out[5] are correct while Out[1] and Out[2] are not.

Similar things sometimes occur after a computation is interrupted, and
this is probably one of the most annoying and hard to track types of
glitches; here's an example where it is easy to see what is going on:

Sum[D[Pochhammer[1,k]^2/Pochhammer[n,k]*2^-k/k!, n], {k, 0, Infinity}]
(*interrupt the computation with Alt-.*)

{Analytic -> True}

Infinity::indet: Indeterminate expression 0*n*Infinity*Gamma[n]
Infinity::indet: Indeterminate expression 0*Infinity*Gamma[1 + n]
Infinity::indet: Indeterminate expression 0*n*Infinity*Gamma[n]
General::stop: Further output of Infinity::indet will be suppressed
during this calculation.


{Analytic -> False}

If you press Alt-. after you see the warning messages, Mathematica
forgets to restore the previous setting for Analytic.

Maxim Rytin
m.r at

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