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Inverse Cotangent


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The inverse cotangent is the multivalued function cot^(-1)z (Zwillinger 1995, p. 465), also denoted arccotz (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 311; Jeffrey 2000, p. 124) or arcctgz (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127), that is the inverse function of the cotangent. The variants Arccotz (e.g., Beyer 1987, p. 141; Bronshtein and Semendyayev, 1997, p. 70) and Cot^(-1)z 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). Worse yet, the notation arccotz is sometimes used for the principal value, with Arccotz being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation cot^(-1)z (commonly used in North America and in pocket calculators worldwide), cotz is the cotangent and the superscript -1 denotes an inverse function, not the multiplicative inverse.

The principal value of the inverse cotangent is implemented in the Wolfram Language as ArcCot[z].

InverseCotangentBranchCut

There are at least two possible conventions for defining the inverse cotangent. This work follows the convention of Abramowitz and Stegun (1972, p. 79) and the Wolfram Language, taking cot^(-1)x to have range (-pi/2,pi/2], a discontinuity at x=0, and the branch cut placed along the line segment (-i,i). This definition can be expressed in terms of the natural logarithm by

 cot^(-1)z=i/2[ln((z-i)/z)-ln((z+i)/z)].
(1)

This definition is also consistent, as it must be, with the Wolfram Language's definition of ArcTan, so ArcCot[z] is equal to ArcTan[1/z].

A different but common convention (e.g., Zwillinger 1995, p. 466; Bronshtein and Semendyayev, 1997, p. 70; Jeffrey 2000, p. 125) defines the range of cot^(-1)x as (0,pi), thus giving a function that is continuous on the real line R. Extreme care should be taken where examining identities involving inverse trigonometric functions, since their range of applicability or precise form may differ depending on the convention being used.

The derivative of cot^(-1)z is given by

 d/(dz)cot^(-1)z=-1/(1+z^2)
(2)

and the integral by

 intcot^(-1)zdz=zcot^(-1)z+1/2ln(1+z^2)+C.
(3)

The Maclaurin series of the inverse cotangent for x>0 is given by

cot^(-1)x=pi/2-sum_(k=0)^(infty)((-1)^kx^(2k+1))/(2k+1)
(4)
=pi/2-x+1/3x^3-1/5x^5+1/7x^7-1/9x^9+...
(5)

(OEIS A005408). The Laurent series about z=infty is given by

cot^(-1)z=sum_(k=0)^(infty)((-1)^kz^(-(2k+1)))/(2k+1)
(6)
=z^(-1)-1/3z^(-3)+1/5z^(-5)-1/7z^(-7)+1/9z^(-9)+...
(7)

for |z|>1.

Euler derived the infinite series

 cot^(-1)z=zsum_(n=1)^infty((2n-2)!!)/((2n-1)!!(z^2+1)^n)
(8)

(Wetherfield 1996).

The inverse cotangent satisfies

 cot^(-1)z=tan^(-1)(1/z)
(9)

for z!=0,

 cot^(-1)z=-cot^(-1)(-z)
(10)

for all z in C^*, and

cot^(-1)x={sec^(-1)((sqrt(x^2+1))/x)-pi for x<0; sec^(-1)((sqrt(x^2+1))/x) for x>0
(11)
={-1/2pi-tan^(-1)x for x<0; 1/2pi-tan^(-1)x for x>=0
(12)
={-sin^(-1)(1/(sqrt(x^2+1))) for x<0; sin^(-1)(1/(sqrt(x^2+1))) for x>0
(13)
={-1/2pi-cot^(-1)(1/x) for x<0; 1/2pi-cot^(-1)(1/x) for x>0
(14)
={-csc^(-1)(sqrt(x^2+1)) for x<0; csc^(-1)(sqrt(x^2+1)) for x>0
(15)
={cos^(-1)(x/(sqrt(x^2+1)))-pi for x<0; cos^(-1)(x/(sqrt(x^2+1))) for x>0
(16)
={-1/2pi-sin^(-1)(x/(sqrt(x^2+1))) for x<0; 1/2pi-sin^(-1)(x/(sqrt(x^2+1))) for x>0.
(17)

Analytic sums of cotangents include the beautiful result

 sum_(n=1)^inftycot^(-1)n^2=cot^(-1)((1+t)/(1-t))=1.42474...,
(18)

(OEIS A091007), where

 t=cot(1/2pisqrt(2))tanh(1/2pisqrt(2))
(19)

(H. S. Wilf, pers. comm., May 21, 2002).

A number

 t_x=cot^(-1)x,
(20)

where x is an integer or rational number, is sometimes called a Gregory number. Lehmer (1938a) showed that cot^(-1)(a/b) can be expressed as a finite sum of inverse cotangents of integer arguments

 cot^(-1)(a/b)=sum_(i=1)^k(-1)^(i-1)cot^(-1)n_i,
(21)

where

 n_i=|_(a_i)/(b_i)_|,
(22)

with |_x_| the floor function, and

a_(i+1)=a_in+i+b_i
(23)
b_(i+1)=a_i-n_ib_i,
(24)

with a_0=a and b_0=b, and where the recurrence is continued until b_(k+1)=0. If an inverse tangent sum is written as

 tan^(-1)n=sum_(k=1)f_ktan^(-1)n_k+ftan^(-1)1,
(25)

then equation (◇) becomes

 cot^(-1)n=sum_(k=1)f_kcot^(-1)n_k+ccot^(-1)1,
(26)

where

 c=2-f-2sum_(k=1)f_k.
(27)

Inverse cotangent sums can be used to generate Machin-like formulas.

Other inverse cotangent identities include

2cot^(-1)(2x)-cot^(-1)x=cot^(-1)(4x^3+3x)
(28)
3cot^(-1)(3x)-cot^(-1)x=cot^(-1)((27x^4+18x^2-1)/(8x)),
(29)

as well as many others (Bennett 1926, Lehmer 1938b). Note that for equation (29), the choice of convention for cot^(-1)z is significant, since it holds for all complex z in the [0,pi] convention, but holds only outside a lens-shaped region centered on the origin in the [-pi/2,pi/2] convention.


See also

Cotangent, Inverse Cosecant, Inverse Cosine, Inverse Secant, Inverse Sine, Inverse Tangent, Inverse Trigonometric Functions, Lehmer Cotangent Expansion, Machin's Formula, Machin-Like Formulas, Tangent

Related Wolfram sites

http://functions.wolfram.com/ElementaryFunctions/ArcCot/

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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.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.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, p. 70, 1997.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.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 311, 1998.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.Lehmer, D. H. "A Cotangent Analogue of Continued Fractions." Duke Math. J. 4, 323-340, 1938a.Lehmer, D. H. "On Arccotangent Relations for pi." Amer. Math. Monthly 45, 657-664, 1938b.Sloane, N. J. A. Sequences A005408/M2400 and A091007 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.Wetherfield, M. "The Enhancement of Machin's Formula by Todd's Process." Math. Gaz. 80, 333-344, 1996.Zwillinger, D. (Ed.). "Inverse Circular Functions." §6.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 465-467, 1995.

Referenced on Wolfram|Alpha

Inverse Cotangent

Cite this as:

Weisstein, Eric W. "Inverse Cotangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseCotangent.html

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