Saluton!

* Rudolf Polzer; 2003-08-25, 12:41 UTC:
> Until now tan and tanh don't have much in common, mathematically.
> But this makes them nearly the same functions:
> 
>   irb(main):003:0> require 'complex'
>   => true
>   irb(main):004:0> Math.atanh(17)
>   => Complex(0.0588915178281917, 1.5707963267949)
>   irb(main):005:0> Math.atanh(17 * Complex::I)
>   => Complex(0.0, 1.51204050407917)
> 
> That makes it obvious that it would be good to use the same prefix
> for the inverse functions.

You mean that

 sin(Complex::I * x) = Complex::I * sinh(x)
sinh(Complex::I * x) = Complex::I *  sin(x)
 cos(Complex::I * x) =              cosh(x)
cosh(Complex::I * x) =               cos(x)

explains why to use the symbols asin, asinh, acos, and acosh? Well,
that's quite obvious if one expresses trigonometric and hyperbolic
functions in terms of complex exponential functions:

    sinh(x) = (exp(x) - exp(-x)) / 2

and

    sin(x) = (exp(ix) - exp(-ix)) / 2i

If you apply sinh on ix the result obviously is i times that of
sin(x).

Gis,

Josef 'Jupp' Schugt
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