arcsinh(x) = log(1+sqrt(1+x^2)), I seem to remember. [Proof: exercise]
John Cremona
2008/10/31 William Stein <[EMAIL PROTECTED]>:
>
> On Fri, Oct 31, 2008 at 12:57 PM, Jim Clark
> <[EMAIL PROTECTED]> wrote:
>>
>> Hello sage gurus,
>>
>> Using sage to check a manually calculated integral :
>>
>> sage: var('r,h')
>> (r, h)
>> sage: integrate(r/sqrt(r^2 - sqrt(2)*h*r + h^2), r, 0, sqrt(2)
>> *h).factor()
>> sqrt(2)*arcsinh(1)*h
>>
>> My manual result (using an old table of integrals) was sqrt(2)*ln(sqrt
>> (2)+1)*h
>>
>> So, wondering whether arcsinh(1) = ln(sqrt(2)+1), I asked:
>>
>> sage: bool(arcsinh(1) == ln(1+sqrt(2)))
>> False
>>
>> but then,
>>
>> sage: arcsinh(1).n()
>> 0.881373587019543
>> sage: ln(1+sqrt(2)).n()
>> 0.881373587019543
>>
>> They look equal to my eyes...
>>
>> Also,
>> sage: bool(arcsinh(1).n() == ln(1+sqrt(2)).n())
>> False
>>
>> What am I missing here?
>>
>> Thanks,
>> Jim Clark
>
> If expr is a symbolic expression in Sage, then
>
> bool(expr)
>
> evaluates to True only if expr can be proved to be True.
> Otherwise it always evaluates to False.
>
> The actual code that decides this is currently in Maxima.
>
> -- William
>
> >
>
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