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Re: Have I found a bug?

• To: mathgroup at smc.vnet.net
• Subject: [mg84253] Re: Have I found a bug?
• From: Norbert Marxer <marxer at mec.li>
• Date: Sun, 16 Dec 2007 05:35:56 -0500 (EST)
• References: <fjtgdu$7hm$1@smc.vnet.net>

On 14 Dez., 09:55, Louise Hoffman <louise.hoff... at gmail.com> wrote:
> Dear readers,
>
> When I calc. this in Mathematica 5.2
>
> g[x_] := 1 - x^2
> h[x_] := -x
> Exp[-Integrate[ (D[g[x], x] - h[x])/g[x], x ] ]
>
> it returns
> 1/Sqrt[-1+x^2]
>
> where I would expect
>
> 1/Sqrt[1-x^2]
>
> Have I found a bug, or have I made a mistake?
>
> Lots of love,
> Louise

Hello

Whe you perform an indefinite integration the solution is only unique
up to an additive constant. In Mathematica you can add this additive
constant (c[1]) by hand ...

g[x_] := 1 - x^2
h[x_] := -x
Exp[-(Integrate[(D[g[x], x] - h[x])/g[x], x] + c[1])]

... and get a solution depending on this addive constant ...

1/(E^c[1]*Sqrt[-1 + x^2])

For c[1] -> 0 you get the standard solution: 1/Sqrt[-1 + x^2]
For c[1] -> (I*Pi)/2 you get your solution 1/Sqrt[1- x^2] which
corresponds to -(I/Sqrt[-1 + x^2])

... as is shown by the following command:

Exp[-(Integrate[(D[g[x], x] - h[x])/g[x], x] + c[1])] /. c[1] -> 0
Exp[-(Integrate[(D[g[x], x] - h[x])/g[x], x] + c[1])] /. c[1] ->
(I*Pi)/2

Best Regards
Norbert Marxer