Inverse Cotangent

Min Max Re Im

The inverse cotangent is the multivalued function (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 311; Jeffrey 2000, p. 124) or (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127), that is the inverse function of the cotangent. The variants (e.g., Beyer 1987, p. 141; Bronshtein and Semendyayev, 1997, p. 70) and are sometimes used to refer to explicit principal values of the inverse cotangent, although this distinction is not always made (e.g., Zwillinger 1995, p. 466). Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation (commonly used in North America and in pocket calculators worldwide), is the cotangent and the superscript denotes an inverse function, not the multiplicative inverse.

The principal value of the inverse cotangent is implemented in the Wolfram Language as ArcCot[z].

There are at least two possible conventions for defining the inverse cotangent. This work follows the convention of Abramowitz and Stegun (1972, p. 79) and the Wolfram Language, taking to have range , a discontinuity at , and the branch cut placed along the line segment . This definition can be expressed in terms of the natural logarithm by

This definition is also consistent, as it must be, with the Wolfram Language's definition of ArcTan, so ArcCot[z] is equal to ArcTan[1/z].

A different but common convention (e.g., Zwillinger 1995, p. 466; Bronshtein and Semendyayev, 1997, p. 70; Jeffrey 2000, p. 125) defines the range of as , thus giving a function that is continuous on the real line . Extreme care should be taken where examining identities involving inverse trigonometric functions, since their range of applicability or precise form may differ depending on the convention being used.

The derivative of is given by

and the integral by

The Maclaurin series of the inverse cotangent for is given by

(OEIS A005408). The Laurent series about is given by

for .

Euler derived the infinite series

(Wetherfield 1996).

The inverse cotangent satisfies

for ,

for all , and

Analytic sums of cotangents include the beautiful result

(OEIS A091007), where

(H. S. Wilf, pers. comm., May 21, 2002).

A number

where is an integer or rational number, is sometimes called a Gregory number. Lehmer (1938a) showed that can be expressed as a finite sum of inverse cotangents of integer arguments

where

with the floor function, and

with and , and where the recurrence is continued until . If an inverse tangent sum is written as

then equation (◇) becomes

where

Inverse cotangent sums can be used to generate Machin-like formulas.

Other inverse cotangent identities include

as well as many others (Bennett 1926, Lehmer 1938b). Note that for equation (29), the choice of convention for is significant, since it holds for all complex in the convention, but holds only outside a lens-shaped region centered on the origin in the convention.