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ArcTan

Status: Stable

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

Description

ArcTan[z]
    gives the principal inverse tangent of z, in (-Pi/2, Pi/2).
ArcTan[y, x]
    gives the argument of the complex number x + I y, in (-Pi, Pi]
    (two-argument atan2 form).
ArcTan is Listable.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_arctan (src/trig.c) handles one- and two-argument forms. For ArcTan[z]: (1) odd_fold for oddness; (2) trig_i_fold for ArcTan[I y] -> I ArcTanh[y]; (3) exact inversion via exact_arctan, scanning n in [-d/2, d/2] for d in {1,2,3,4,5,6,10,12} against exact_tan(n,d) and returning n/d * Pi; (4) numeric fallback MPFR mpfr_atan/mpfr_complex_atan, else get_approx + C99 catan. For the two-argument ArcTan[x, y] (argument of x + I y): integer inputs are resolved exactly by quadrant (the axes and the four diagonals ±1 map to 0, ±Pi/2, Pi, ±Pi/4, ±3Pi/4); real inputs use MPFR mpfr_atan2 or C99 atan2. Indeterminate ArcTan[0,0] and unhandled cases return NULL.

Data structures. Expr* trees; the single-arg exact path reuses the forward exact_tan table.

Attributes: Listable, NumericFunction, Protected.

Implementation status

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

References

Notes & additional examples

Worked examples

In[1]:= ArcTan[1]
Out[1]= 1/4 Pi

In[2]:= ArcTan[Sqrt[3]]
Out[2]= 1/3 Pi

In[1]:= ArcTan[-1, 1]
Out[1]= 3/4 Pi

In[2]:= ArcTan[{0, 1}]
Out[2]= {0, 1/4 Pi}

A pure-imaginary argument crosses over to the inverse hyperbolic tangent:

In[1]:= ArcTan[I/2]
Out[1]= I ArcTanh[1/2]

The Taylor series recovers the Gregory series, and the antiderivative is closed-form:

In[1]:= Series[ArcTan[x], {x, 0, 9}]
Out[1]= x - 1/3 x^3 + 1/5 x^5 - 1/7 x^7 + 1/9 x^9 + O[x]^10

In[2]:= Integrate[ArcTan[x], x]
Out[2]= 1/2 (2 x ArcTan[x] - Log[1 + x^2])

Machin's 1706 formula then evaluates Pi to 40 digits:

In[1]:= N[16 ArcTan[1/5] - 4 ArcTan[1/239], 40]
Out[1]= 3.1415926535897932384626433832795028841975

Notes

ArcTan[z] gives the principal inverse tangent, in (-Pi/2, Pi/2). The two-argument form ArcTan[x, y] is the quadrant-aware atan2, giving the argument of x + I y in (-Pi, Pi]. ArcTan is Listable.