Inverse Tangent

The inverse tangent is the multivalued function  (Zwillinger 1995, p. 465), also denoted  (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 311; Jeffrey 2000, p. 124) or  (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127), that is the inverse function of the tangent. The variants  (e.g., Bronshtein and Semendyayev, 1997, p. 70) and  are sometimes used to refer to explicit principal values of the inverse cotangent, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466).

The inverse tangent function  is plotted above along the real axis.

Worse yet, the notation  is sometimes used for the principal value, with  being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation  (commonly used in North America and in pocket calculators worldwide),  denotes the tangent and  the inverse function, not the multiplicative inverse.

The principal value of the inverse tangent is implemented as ArcTan[z] in the Wolfram Language. In the GNU C library, it is implemented as atan(double x).

The inverse tangent is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at  and . This follows from the definition of  as

(1)

In the Wolfram Language (and in this work), this branch cut definition determines the range of  for real  as . Care must be taken, however, as other branch cut definitions can give different ranges (most commonly, ).

Min   Max       Re       Im

The inverse tangent function  is plotted above in the complex plane.

 has the special values

			(2)
			(3)
			(4)
			(5)
			(6)

The derivative of  is

(7)

and the indefinite integral is

(8)

The complex argument of a complex number  is often written as

(9)

where , sometimes also denoted , corresponds to the counterclockwise angle from the positive real axis, i.e., the value of  such that  and . Plots of  are illustrated above for real values of  and .

Min   Max       Re       Im

A special kind of inverse tangent that takes into account the quadrant in which  lies and is returned by the FORTRAN command ATAN2(y, x), the GNU C library command atan2(double y, double x), and the Wolfram Language command ArcTan[x, y], and is often restricted to the range . In the degenerate case when ,

(10)

The usual  has the Maclaurin series of

			(11)
			(12)

(OEIS A033999 and A005408). A more rapidly converging form due to Euler is given by

(13)

for real  (Castellanos 1988). This is related to the formula of Euler given by

(14)

where

(15)

The inverse tangent formulas are connected with many interesting approximations to pi

			(16)
			(17)

(OEIS A075553 and A075554).

The inverse tangent satisfies

(18)

for ,

(19)

for all complex ,

			(20)
			(21)
			(22)

for all real , where equality for the last equation is understood to be in the limit as , and

			(23)
			(24)
			(25)
			(26)
			(27)

In terms of the hypergeometric function,

(28)

for complex , and

(29)

for real  (Castellanos 1988).

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

			(30)
			(31)
			(32)

where

			(33)
			(34)

and  is the largest positive root of

(35)

The inverse tangent satisfies the addition formula

(36)

for , as well as the more complicated formula

(37)

valid for all complex . An additional identity known to Euler is given by

(38)

for  or . Another interesting inverse tangent identity attributed to Charles Dodgson (Lewis Carroll) by Lehmer (1938b; Bromwich 1991, Castellanos 1988) is

(39)

where

(40)

and .

The inverse tangent has continued fraction representations

(41)

(Lambert 1770; Lagrange 1776; Wall 1948, p. 343; Olds 1963, p. 138) and

(42)

due to Euler and sometimes known as Euler's continued fraction (Borwein et al. 2004, p. 30).

To find  numerically, the following arithmetic-geometric mean-like algorithm can be used. Let

			(43)
			(44)

Then compute

			(45)
			(46)

and the inverse tangent is given by

(47)

(Acton 1990).

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

(48)

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

(49)

so the ratio

(50)

is a rational number. Equation (50) can also be written

(51)

Writing (◇) in the form

(52)

allows a direct conversion to a corresponding inverse cotangent formula

(53)

where

(54)

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

(55)

There is an amazing set of BBP-type formulas for :

			(56)
			(57)
			(58)
			(59)

the finding one of which is a given as a problem by Bailey et al. (2006, p. 225).

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 255-256, 1967.

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

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143 and 220, 1987.

Borwein, J.; Bailey, D.; and Girgensohn, R. "Euler's Continued Fraction." §1.8.2 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, p. 30, 2004.

Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, p. 70, 1997.

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

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.

GNU C Library. "Mathematics: Inverse Trigonometric Functions." http://www.gnu.org/manual/glibc-2.2.3/html_chapter/libc_19.html#SEC389.

Gosper, R. W. "Joerg Arndt kindly forwarded me." math-fun@cs.arizona.edu posting, Jan.14, 1997.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 311, 1998.

Hildebrand, J. D. "Arctan() Appreciation Home Page!" http://www.opensky.ca/~jdhildeb/arctan/.

Jeffrey, A. "Inverse Trigonometric and Hyperbolic Functions." §2.7 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 124-128, 2000.

Lagrange, J.-L. "Sur l'usage des fractions continues dans le calcul intégral." Nouv. mém. de l'académie royale des sciences et belles-lettres Berlin, 236-264, 1776. Reprinted in Oeuvres, Vol. 4, pp. 301-302.

Lambert, J. L. Beiträge zum Gebrauch der Mathematik und deren Anwendung. Theil 2. Berlin, 1770.

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

Olds, C. D. Continued Fractions. New York: Random House, 1963.

Salamin, G. Item 137 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 67-68, Feb.1972. http://www.inwap.com/pdp10/hbaker/hakmem/pi.html#item137.

Sloane, N. J. A. Sequences A005408/M2400, A033999, A075553, and A075554 in "The On-Line Encyclopedia of Integer Sequences."

Spanier, J. and Oldham, K. B. "Inverse Trigonometric Functions." Ch. 35 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 331-341, 1987.

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

Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.

Zwillinger, D.(Ed.). "Inverse Circular Functions." §6.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 465-467, 1995.