Modern Math CAtan

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CAtan

Returns the inverse tangent of a complex number.

Syntax

CAtan( byref z as Complex ) as Complex

Notes

• The logarithmic form, (i/2)·( Log( 1 - iz ) - Log( 1 + iz ) ), which is well conditioned — the cancellation that ruins the naive arcsine does not arise for the arctangent.
• The cuts are on the imaginary axis beyond ±i, and the log form cannot see which side of one it is on: forming 1 + iz adds +0.0 to -0.0, and IEEE resolves that to +0.0, destroying the sign bit that decides. So that segment is answered directly.
• The whole imaginary axis is, in fact, and not only for the sign: the log form cancels for a small argument and gave atan( 1e-8·i ) as 9.999999994736e-9 for an answer that is 1e-8.
• CAtan( 0 ) is exactly zero.

Examples

#import "std/numeric"
using numeric

print CToString( CAtan( CSet( 0.5, 0.25 ) ) )  '' 0.4842544903299662+0.2005866181312343i
print CToString( CAtan( CSet( 0, 2 ) ) )       '' 1.5707963267948966+0.5493061443340549i   -- on the cut, above it
print CToString( CAtan( CSet( -0.0, 2 ) ) )    '' -1.5707963267948966+0.5493061443340549i   -- below it
print CImag( CAtan( CSet( 0, 1e-8 ) ) )        '' 1e-008   -- exactly 1e-8, where the log form cancels

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