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 values in ![(-pi/2,pi/2]](/images/equations/InverseCotangent/Inline12.svg)
, a discontinuity at 
, and the branch cut placed
along the line segment 
. For 
, this definition can be expressed in terms of the natural
logarithm by
![cot^(-1)z=i/2[ln((z-i)/z)-ln((z+i)/z)].](/images/equations/InverseCotangent/NumberedEquation1.svg) |
(1)
|
At 
,
the value is 
.
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
 |
(2)
|
and the integral by
 |
(3)
|
For 
,
the one-sided Maclaurin series of the inverse
cotangent is given by
 |  |  |
(4)
|
 |  |  |
(5)
|
(OEIS A005408). The Laurent series about 
is given by
 |  |  |
(6)
|
 |  |  |
(7)
|
for 
.
For 
,
the infinite series derived
by Euler is
 |
(8)
|
(Wetherfield 1996).
The inverse cotangent satisfies
 |
(9)
|
for 
,
 |
(10)
|
for all 
,
and
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
Analytic sums of cotangents include the beautiful result
 |
(18)
|
(OEIS A091007), where
 |
(19)
|
(H. S. Wilf, pers. comm., May 21, 2002).
A number
 |
(20)
|
where 
is an integer or rational
number, is sometimes called a Gregory number.
For positive integers 
and 
, Lehmer (1938a) showed that 
can be expressed as a finite sum of inverse cotangents of integer
arguments
 |
(21)
|
where
 |
(22)
|
with 
the floor function, and
 |  |  |
(23)
|
 |  |  |
(24)
|
with 
and 
,
and where the recurrence relation is continued
until 
.
For positive integer arguments 
and 
, if a finite inverse tangent sum is written as
 |
(25)
|
then equation (21) becomes
 |
(26)
|
where
 |
(27)
|
Inverse cotangent sums can be used to generate Machin-like formulas.
For a real number 
, an inverse cotangent identity
is
 |
(28)
|
For a real number 
, the related identity in MathWorld's
principal value convention is
 |
(29)
|
where 
.
The correction is necessary because inverse cotangent identities
depend on the principal value convention (Bennett
1926, Lehmer 1938b).
." Amer. Math. Monthly 45, 657-664, 1938b.Sloane,
N. J. A. Sequences