gives the arc cotangent of the complex number .
- 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 .
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.
Examplesopen allclose all
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:
Numerical Evaluation (6)
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:
ArcCot can deal with real-valued intervals:
ArcCot threads elementwise over lists and matrices:
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:
Plot the ArcCot function:
Plot the real part of :
Plot the imaginary part of :
Polar plot with :
Function Properties (5)
ArcCot is defined for all real values:
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 :
Formula for the derivative:
Indefinite integral of ArcCot:
Definite integral of ArcCot over an interval centered at the origin is 0:
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:
Function Identities and Simplifications (3)
Function Representations (5)
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 (3)