Arg¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

Arg[z] gives the argument (phase angle in (-Pi, Pi]) of numeric z; 0 for nonnegative reals, Pi for negative reals.

Examples¶

No verified examples yet for this function.

Implementation notes¶

builtin_arg returns the phase angle in (-Pi, Pi]. A pure MPFR real folds to exact 0 or Pi by sign. For a Complex[re, im] whose parts are exact (Integer/Rational), it recognises the special directions and returns exact multiples of Pi: 0 for positive reals, Pi for negatives, ±Pi/2 on the imaginary axis, and ±Pi/4, ±3Pi/4 on the diagonals; otherwise it returns the symbolic ArcTan[re, im]. When either component carries MPFR it evaluates mpfr_atan2 at the combined precision; an inexact machine Real falls through to the libm atan2(im, re). Symbolic inputs return NULL.

Attributes: Listable, NumericFunction, Protected.

Implementation status¶

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References¶

Source: src/complex.c
Specification: docs/spec/builtins/arithmetic.md

Notes & additional examples¶

Worked examples¶

In[1]:= Arg[1]
Out[1]= 0

In[2]:= Arg[-1]
Out[2]= Pi

In[3]:= Arg[1 + I]
Out[3]= 1/4 Pi

Powers wrap the phase into the principal branch:

In[1]:= Arg[(1 + I)^10]
Out[1]= 1/2 Pi

In[2]:= Arg[-2 + 2 I]
Out[2]= 3/4 Pi

Stepping the argument of (1 + I)^k traces the unit circle and folds back at Pi:

In[1]:= Table[Arg[(1 + I)^k], {k, 0, 8}]
Out[1]= {0, 1/4 Pi, 1/2 Pi, 3/4 Pi, Pi, -3/4 Pi, -1/2 Pi, -1/4 Pi, 0}

Generic complex numbers evaluate to high precision:

In[1]:= N[Arg[2 + 3 I], 40]
Out[1]= 0.98279372324732906798571061101466601449686

Notes¶

Arg[z] gives the phase angle in the range (-Pi, Pi]: 0 for positive reals, Pi for negative reals.