       Re: ArcCos[x] with x > 1

• To: mathgroup at smc.vnet.net
• Subject: [mg49347] Re: ArcCos[x] with x > 1
• From: ab_def at prontomail.com (Maxim)
• Date: Thu, 15 Jul 2004 07:00:12 -0400 (EDT)
• References: <cclev9\$kb3\$1@smc.vnet.net> <cco3io\$4ig\$1@smc.vnet.net> <cctaff\$c11\$1@smc.vnet.net> <cd0730\$9g9\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```"David W. Cantrell" <DWCantrell at sigmaxi.org> wrote in message news:<cd0730\$9g9\$1 at smc.vnet.net>...

> 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:

In:=
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}]
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}]

Out=
(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])

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

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

Out=
(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])

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

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

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

Out=
{Analytic -> True}

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

Out=
\$Aborted

Out=
{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 inbox.ru

```

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