Re: circumference of an ellipse

• To: mathgroup at smc.vnet.net
• Subject: [mg19390] Re: circumference of an ellipse
• From: "Allan Hayes" <hay at haystack.demon.co.uk>
• Date: Mon, 23 Aug 1999 13:57:08 -0400
• References: <7p017c\$778@smc.vnet.net> <7p301g\$anl@smc.vnet.net> <7pl5l0\$cch@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Stephen,
Yes, and I should have used

quartercircumference[a_, b_] :=
Integrate[Sqrt[D[a Cos[t], t]^2 + D[b Sin[t], t]^2], {t, 0, Pi/2},
Assumptions -> {a > 0, b > 0}]

quartercircumference[a, b]

If[Arg[b^2/a^2] != Pi,
-((b*MeijerG[{{1/2, 3/2}, {}}, {{0, 1}, {}}, a^2/b^2])/(2*Pi)),
Integrate[Sqrt[b^2*Cos[t]^2 + a^2*Sin[t]^2], {t, 0, Pi/2}]]

But now we see that Mathematica does not deduce that if a and be are positive then
b^2/a^2 cannot be negative.
We clearly need

%[[2]]

-((b*MeijerG[{{1/2, 3/2}, {}}, {{0, 1}, {}}, a^2/b^2])/(2*Pi))

Allan
---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565

Stephen P Luttrell <luttrell at signal.dra.hmg.gb> wrote in message
news:7pl5l0\$cch at smc.vnet.net...
> Allan Hayes <hay at haystack.demon.co.uk> wrote in message
> news:7p301g\$anl at smc.vnet.net...
> > Marcel,
> >
> > circumference[a_, b_] :=
> >   Integrate[Sqrt[D[a Cos[t], t]^2 + D[b Sin[t], t]^2], {t, 0, 2Pi}]
> >
> >...
>
> (Preamble: I have \$Version = "4.0 for Microsoft Windows (April 21, 1999)")
>
> I agree with this parametric solution, but it exposes a bug in Mathematica
> when you evaluate the following symbolic expression:
>
> circumference[a, b]
>
> This gives zero!
>
> Furthermore, if you define
>
> halfcircumference[a_, b_] :=
>   Integrate[Sqrt[D[a Cos[t], t]^2 + D[b Sin[t], t]^2], {t, 0, Pi}]
>
> and then evaluate halfcircumference[a, b], you get "Infinite expression
1/0
> encountered".
>
>
> Steve Luttrell
> Signal Processing and Imagery Department
> Malvern, United Kingdom, WR14 3PS
>
> +44 (0)1684 894046 (tel)
> +44 (0)1684 894384 (fax)
> luttrell at signal.dera.gov.uk (email)
>
>
>

```

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