Modern Math CAtanh

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CAtanh

Returns the inverse hyperbolic tangent of a complex number, or of any real number.

Syntax

CAtanh( byref z as Complex ) as Complex
CAtanh( byval x as float64 ) as Complex

Notes

• Reached through the inverse trigonometric functions, so it inherits the Hull–Fairgrieve–Tang conditioning rather than carrying its own copy. atanh z = -i·atan( iz ).
• Cut along the real axis beyond ±1, and that segment cannot be reached through CAtan: forming 1 + iz adds +0.0 to -0.0, IEEE gives +0.0, and the bit that decides the side of the cut is gone before the logarithm sees it. It is computed directly — atanh( x ± 0i ) = atanh( 1/x ) ± iπ/2.
• The imaginary axis is answered directly too: atanh( ±0 + yi ) = ±0 + i·atan( y ).
• The float64 overload is AfxNova's CArcTanHReal.
• CAtanh( ±1 ) is infinite, which is the true value.

Examples

#import "std/numeric"
using numeric

print CToString( CAtanh( CSet( 0.5, 0.25 ) ) )  '' 0.5003700000525311+0.3143981432077165i
print CToString( CAtanh( 2.0 ) )                '' 0.5493061443340549+1.5707963267948966i   -- outside the real function's domain
print CToString( CAtanh( CSet( 2, -0.0 ) ) )    '' 0.5493061443340549-1.5707963267948966i   -- the other side of the cut
print CToString( CAtanh( 0.5 ) )                '' 0.5493061443340549+0i   -- inside it, so simply real

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Library: Math Library
Category: Math Library
Documented under: Members