gives the arc tangent of the complex number .
gives the arc tangent of , taking into account which quadrant the point is in.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- All results are given in radians.
- For real , the results are always in the range to .
- For certain special arguments, ArcTan automatically evaluates to exact values.
- ArcTan can be evaluated to arbitrary numerical precision.
- ArcTan automatically threads over lists.
- ArcTan[z] has branch cut discontinuities in the complex plane running from to and to .
- If or is complex, then ArcTan[x,y] gives . When , ArcTan[x,y] gives the number such that and .
- ArcTan can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcTan is the inverse tangent function. For a real number x, ArcTan[x] represents the radian angle measure such that . The two-argument form ArcTan[x,y] represents the arc tangent of y/x, taking into account the quadrant in which the point lies. It therefore gives the angular position (expressed in radians) of the point measured from the positive axis. ArcTan is consequently useful when converting from Cartesian to polar coordinate systems and for finding the phase in phasor notation .
- ArcTan automatically threads over lists. For certain special exact arguments, ArcTan automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcTan may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcTan include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcTan is defined for complex argument via . ArcTan[z] has branch cut discontinuities in the complex plane.
- Related mathematical functions include Arg, Tan, ArcCot, ArcTanh, and Gudermannian.
Examplesopen allclose all
Basic Examples (7)
Numerical Evaluation (6)
Evaluate ArcTan efficiently at high precision:
ArcTan threads elementwise over lists and matrices:
Specific Values (6)
Function Properties (12)
ArcTan is defined for all real values:
ArcTan achieves all real values from the interval :
ArcTan is an odd function:
ArcTan has the mirror property :
ArcTan is injective:
ArcTan is not surjective:
ArcTan is neither non-negative nor non-positive:
ArcTan is neither convex nor concave:
Series Expansions (4)
Integral Transforms (3)
Function Identities and Simplifications (3)
Find integrals of rational functions in terms of ArcTan:
Branch cuts of ArcTan run along the imaginary axis:
The cumulative distribution function (CDF) of the standard distribution of the hyperbolic secant is given in terms of ArcTan:
This is a scaled and shifted version of the Gudermannian function:
Properties & Relations (4)
Possible Issues (1)
Because ArcTan is a multivalued function,
Wolfram Research (1988), ArcTan, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTan.html (updated 2021).
Wolfram Language. 1988. "ArcTan." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcTan.html.
Wolfram Language. (1988). ArcTan. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcTan.html