Describe the issue:

According to Python array API standard, arcsin(z) is defined as

-1j * ln(z*1j + sqrt(1 - z**2))

As demonstrated below, the evaluation of numpy.arcsin and using the definition above lead to different results on the real line when abs(z.real) > 1.

Reproduce the code example:

import numpy
z = complex(2, 0)
print(numpy.arcsin(z))                            # -> (1.5707963267948966+1.3169578969248166j)
print(-1j*numpy.log(z*1j + numpy.sqrt(1 - z*z)))  # -> (1.5707963267948966-1.3169578969248166j)

Error message:

No response

Python and NumPy Versions:

2.0.0rc1
3.11.8 | packaged by conda-forge | (main, Feb 16 2024, 20:53:32) [GCC 12.3.0]

Runtime Environment:

No response

Context for the issue:

See also mpmath/mpmath#786 that contains some data points on how other software evaluate arcsin on its branch cuts.