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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 ,

phi={-1/2pi if y<0; undefined if y=0; 1/2pi if y>0.

(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

		${-1/2pi-tan^(-1)(1/x) for x<0; 1/2pi-tan^(-1)(1/x) for x>0$	(23)
		${-1/2pi+cot^(-1)(-x) for x<0; 1/2pi+cot^(-1)(-x) for x>0$	(24)
		${-1/2pi-cot^(-1)x for x<0; 1/2pi-cot^(-1)x for x>0$	(25)
		${-cos^(-1)(1/(sqrt(x^2+1))) for x<0; cos^(-1)(1/(sqrt(x^2+1))) for x>0$	(26)
		${-sec^(-1)(sqrt(x^2+1)) for x<0; sec^(-1)(sqrt(x^2+1)) for x>0.$	(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

tan^(-1)x=x/(1+(x^2)/(3+(4x^2)/(5+(9x^2)/(7+(16x^2)/(9+...)))))

(41)

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

tan^(-1)x=x/(1+(x^2)/(3-x^2+(9x^2)/(5-3x^2+(25x^2)/(7-5x^2+...))))

(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