Modern Math CAsin

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CAsin

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

Syntax

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

Notes

• The Hull–Fairgrieve–Tang method, transcribed from .NET's Asin_Internal. The textbook -i·Log( iz + CSqrt( 1 - z² ) ) cancels catastrophically near the branch points at ±1, which is exactly where an inverse sine is most often evaluated. This does not.
• Exactly ±1 on the real axis is answered directly. Asin_Internal divides by zero there — its t term contains 1/( s + ( x - 1 ) ) and s is zero — so .NET's own Asin( 1 ) is NaN rather than π/2.
• The cuts are the real axis beyond ±1, and the sign of a zero imaginary part decides the side. .NET tests imaginary < 0.0 here, and a negative zero is not less than zero, so .NET lands on the wrong branch for half the axis: its Asin( 2 - 0i ) has a positive imaginary part where C99 gives a negative one. This tests the sign bit.
• The float64 overload exists because the real arcsine is undefined outside [-1, 1] while the complex one is not — CAsin( 2.0 ) is a perfectly ordinary complex number. AfxNova spelled it CArcSinReal.
• CSin( CAsin( z ) ) returns z on the principal strip.

Examples

#import "std/numeric"
using numeric

print CToString( CAsin( CSet( 0.5, 0.25 ) ) )  '' 0.5016088532755009+0.2813960562452928i
print CToString( CAsin( CSet( 1, 0 ) ) )       '' 1.5707963267948966+0i   -- PI/2 -- .NET's own Asin gives NaN here
print CToString( CAsin( 2.0 ) )                '' 1.5707963267948966+1.3169578969248166i   -- the double overload
print CToString( CAsin( CSet( 2, -0.0 ) ) )    '' 1.5707963267948966-1.3169578969248166i   -- the other side of the cut

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