## Inverse Cotangent

The function , also denoted arccot(), where is the Cotangent and the superscript denotes an Inverse Function and not the multiplicative inverse. The Maclaurin Series is given by

 (1)

and Laurent Series by
 (2)

Euler derived the Infinite series

 (3)

(Wetherfield 1996).

The inverse cotangent satisfies

 (4)

for Positive and Negative , and
 (5)

for . The inverse cotangent is given in terms of other inverse trigonometric functions by
 (6) (7) (8)

for Positive or Negative , and
 (9) (10) (11) (12)

for .

A number

 (13)

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
 (14)

where
 (15)

with the Floor Function, and
 (16) (17)

with and , and where the recurrence is continued until . If an Inverse Tangent sum is written as
 (18)

then equation (14) becomes
 (19)

where
 (20)

Inverse cotangent sums can be used to generate Machin-Like Formulas.

An interesting inverse cotangent identity attributed to Charles Dodgson (Lewis Carroll) by Lehmer (1938b; Bromwich 1965, Castellanos 1988ab) is

 (21)

where
 (22)

Other inverse cotangent identities include

 (23)

 (24)

as well as many others (Bennett 1926, Lehmer 1938b).

References

Abramowitz, M. and Stegun, C. A. (Eds.). Inverse Circular Functions.'' §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.

Bennett, A. A. The Four Term Diophantine Arccotangent Relation.'' Ann. Math. 27, 21-24, 1926.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.

Bromwich, T. J. I. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991.

Castellanos, D. The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988a.

Castellanos, D. The Ubiquitous Pi. Part II.'' Math. Mag. 61, 148-163, 1988b.

Lehmer, D. H. A Cotangent Analogue of Continued Fractions.'' Duke Math. J. 4, 323-340, 1938a.

Lehmer, D. H. On Arccotangent Relations for .'' Amer. Math. Monthly 45, 657-664, 1938b.

Weisstein, E. W. Arccotangent Series.'' Mathematica notebook CotSeries.m.

Wetherfield, M. The Enhancement of Machin's Formula by Todd's Process.'' Math. Gaz., 333-344, July 1996.