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) |
(2) |
(3) |
The inverse cotangent satisfies
(4) |
(5) |
(6) | |||
(7) | |||
(8) |
(9) | |||
(10) | |||
(11) | |||
(12) |
A number
(13) |
(14) |
(15) |
(16) | |||
(17) |
(18) |
(19) |
(20) |
An interesting inverse cotangent identity attributed to Charles Dodgson (Lewis Carroll) by Lehmer (1938b; Bromwich 1965,
Castellanos 1988ab) is
(21) |
(22) |
Other inverse cotangent identities include
(23) |
(24) |
See also Cotangent, Inverse Tangent, Machin's Formula, Machin-Like Formulas, Tangent
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.
© 1996-9 Eric W. Weisstein