## Inverse Tangent

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

 (1)

A more rapidly converging form due to Euler is given by
 (2)

(Castellanos 1988). The inverse tangent satisfies
 (3)

for Positive and Negative , and
 (4)

for . The inverse tangent is given in terms of other inverse trigonometric functions by
 (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)

as well as the more complicated Formulas
 (21)

 (22)

 (23)

the latter of which was known to Euler. The inverse tangent Formulas are connected with many interesting approximations to Pi

 (24)

Euler gave
 (25)

where
 (26)

The inverse tangent has Continued Fraction representations

 (27)

To find numerically, the following Arithmetic-Geometric Mean-like Algorithm can be used. Let
 (28) (29)

Then compute
 (30) (31)

and the inverse tangent is given by
 (32)

(Acton 1990).

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

 (33)

where are Positive or Negative Integers and are Integers . is reducible Iff all the Prime factors of occur among the Prime factors of for , ..., . A second Necessary and Sufficient condition is that the largest Prime factor of is less than . Equivalent to the second condition is the statement that every Gregory Number can be uniquely expressed as a sum in terms of s for which is a Størmer Number (Conway and Guy 1996). To find this decomposition, write
 (34)

so the ratio
 (35)

is a Rational Number. Equation (35) can also be written
 (36)

Writing (33) in the form
 (37)

allows a direct conversion to a corresponding Inverse Cotangent Formula
 (38)

where
 (39)

Todd (1949) gives a table of decompositions of for . Conway and Guy (1996) give a similar table in terms of Størmer Numbers.

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.