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) |

for Positive or Negative , and

(9) | |||

(10) | |||

(11) | |||

(12) |

for .

A number

(13) |

(14) |

(15) |

(16) | |||

(17) |

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

(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) |

**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

1999-05-26