ArcCot

ArcCot[z]

gives the arc cotangent of the complex number .

Details

  • 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 , excluding 0.
  • For certain special arguments, ArcCot automatically evaluates to exact values.
  • ArcCot can be evaluated to arbitrary numerical precision.
  • ArcCot automatically threads over lists.
  • ArcCot[z] has a branch cut discontinuity in the complex plane running from to .
  • ArcCot can be used with Interval and CenteredInterval objects. »

Background & Context

  • ArcCot is the inverse cotangent function. For a real number , ArcCot[x] represents the radian angle measure (excluding 0) such that .
  • ArcCot automatically threads over lists. For certain special arguments, ArcCot automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCot may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCot include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
  • ArcCot is defined for complex argument via . ArcCot[z] has a branch cut discontinuity in the complex plane.
  • Related mathematical functions include Cot, ArcTan, and ArcCoth.

Examples

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Basic Examples  (5)

Results are in radians:

Divide by Degree to get results in degrees:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Scope  (46)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate ArcCot efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix ArcCot function using MatrixFunction:

Specific Values  (4)

Values of ArcCot at fixed points:

Values at infinity:

Singular points of ArcCot:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization  (3)

Plot the ArcCot function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties  (12)

ArcCot is defined for all real values:

Complex domain:

ArcCot achieves all real values except 0 from the interval :

Function range for arguments from the complex domain:

ArcCot is an odd function:

ArcCot has the mirror property cot^(-1)(TemplateBox[{x}, Conjugate])=TemplateBox[{{{cot, ^, {(, {-, 1}, )}}, (, x, )}}, Conjugate]:

ArcCot is not an analytic function:

Nor is it meromorphic:

ArcCot is neither non-decreasing nor non-increasing:

ArcCot is injective:

ArcCot is not surjective:

ArcCot is neither non-negative nor non-positive:

ArcCot has both singularity and discontinuity at zero:

ArcCot is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of ArcCot:

Definite integral of ArcCot over an interval centered at the origin is 0:

More integrals:

Series Expansions  (4)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcCot around :

General term in the series expansion of ArcCot:

Find series expansions at branch points and branch cuts:

ArcCot can be applied to a power series:

Integral Transforms  (3)

Compute the inverse Laplace transform using InverseLaplaceTransform:

InverseFourierTransform:

MellinTransform:

Function Identities and Simplifications  (3)

Simplify expressions involving ArcCot:

Use TrigToExp to express ArcCot using Log:

Expand assuming real variables and :

Function Representations  (5)

Represent using ArcTan:

Representation through inverse Jacobi functions:

Represent using Hypergeometric2F1:

Representation in terms of MeijerG:

ArcCot can be represented as a DifferentialRoot:

Applications  (4)

Find angles of the right triangle with sides 3, 4 and hypotenuse 5:

They total to 90°:

Addition theorem for cotangent function:

Solve a differential equation:

Branch cut of ArcCot runs along the imaginary axis:

Properties & Relations  (4)

Use TrigToExp to express ArcCot using Log:

Use FullSimplify to simplify expressions with ArcCot:

ArcCot gives the angle in radians, while ArcCotDegrees gives the same angle in degrees:

Use Reduce to solve inequalities involving ArcCot:

Wolfram Research (1988), ArcCot, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCot.html (updated 2021).

Text

Wolfram Research (1988), ArcCot, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCot.html (updated 2021).

CMS

Wolfram Language. 1988. "ArcCot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCot.html.

APA

Wolfram Language. (1988). ArcCot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCot.html

BibTeX

@misc{reference.wolfram_2025_arccot, author="Wolfram Research", title="{ArcCot}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCot.html}", note=[Accessed: 21-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arccot, organization={Wolfram Research}, title={ArcCot}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcCot.html}, note=[Accessed: 21-May-2025 ]}