4.6.3.2 Transcendental functions

In general, the transcendental functions \log, \sin^{-1} (arcsine), \cos^{-1} (arccosine), and \tan^{-1} are multiply defined. The value of \log z is defined to be the one whose imaginary part lies in the range from -\pi (inclusive if -0.0 is distinguished, exclusive otherwise) to \pi (inclusive). \log 0 is undefined.

The value of \log z for non–real z is defined in terms of log on real numbers as:

\log z = \log |z| + (\angle z) i

where \angle z is the angle of z = a * e^{(i * b)} specified as:

\angle z = b + 2 \pi n

with -\pi <= \angle z <= \pi and \angle z = b + 2 \pi n for some integer n.

With the one–argument version of \log defined this way, the values of the two–argument–version of \log z, \sin^{-1} z, \cos^{-1} z, \tan^{-1} z, and the two–argument version of \tan^{-1} are according to the following formulae:

\log_b z = (\log z)/(\log b)
\sin^(-1) z = -i \log (i z + \sqrt(1 - z^2))
\cos^(-1) z = \pi/2 - sin^(-1) z
\tan^(-1) z = (\log (1 + i z) - \log (1 - i z)) / (2 i)
\tan^(-1) (x, y) = \angle(x + yi)

The range of \tan^{-1} (x, y) is as in the following table; the asterisk (‘*’) indicates that the entry applies to implementations that distinguish minus zero.