The inverse tangent is also called the arctangent and is denoted either or arctan . It has the
Maclaurin Series

(1) |

(2) |

(3) |

(4) |

(5) | |||

(6) | |||

(7) |

for Positive or Negative , and

(8) | |||

(9) | |||

(10) | |||

(11) |

for .

In terms of the Hypergeometric Function,

(12) | |||

(13) |

(Castellanos 1988). Castellanos (1986, 1988) also gives some curious formulas in terms of the Fibonacci Numbers,

(14) | |||

(15) | |||

(16) |

where

(17) | |||

(18) |

and is the largest Positive Root of

(19) |

The inverse tangent satisfies the addition Formula

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

(28) | |||

(29) |

Then compute

(30) | |||

(31) |

and the inverse tangent is given by

(32) |

An inverse tangent with integral is called reducible if it is expressible as a finite sum of the form

(33) |

(34) |

(35) |

(36) |

(37) |

(38) |

(39) |

Arndt and Gosper give the remarkable inverse tangent identity

(40) |

**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.

Acton, F. S. ``The Arctangent.'' In *Numerical Methods that Work, upd. and rev.* Washington, DC:
Math. Assoc. Amer., pp. 6-10, 1990.

Arndt, J. ``Completely Useless Formulas.'' http://www.jjj.de/hfloat/hfloatpage.html#formulas.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT
Artificial Intelligence Laboratory, Memo AIM-239, Item 137, Feb. 1972.

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

Castellanos, D. ``Rapidly Converging Expansions with Fibonacci Coefficients.'' *Fib. Quart.* **24**, 70-82, 1986.

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

Conway, J. H. and Guy, R. K. ``Størmer's Numbers.'' *The Book of Numbers.* New York: Springer-Verlag, pp. 245-248, 1996.

Todd, J. ``A Problem on Arc Tangent Relations.'' *Amer. Math. Monthly* **56**, 517-528, 1949.

© 1996-9

1999-05-26