[seqfan] Re: Beauty

[Benoît Jubin]

There are a few formulas connecting hyperbolic and circular
trigonometric functions.  Yours is easily derived from:
cosh(x)=cos(ix)=(e^x+e^{-x})/2
sinh(x)=-i.sin(ix)=(e^x-e^{-x})/2
Since arcsinh(x)=ln(x+sqrt(1+x^2)), on has by the above
arcsin(x)=-i.ln(ix+sqrt(1-x^2)).

Best,
Benoit


On Wed, Oct 29, 2008 at 6:49 PM, Artur <grafix at csl.pl> wrote:
> Dear Seqfans,
> Formula discovered today by me which merge hyperbolic World with eliptic
> World (the stranges which I know in Mathematics in general):
> Cosh[2 n ArcSinh[Sqrt[2]]]= (-1)^n Cos[2 n ArcSin[Sqrt[3]]]
>
> Best wishes
> Artur
>
>
>
>
>
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