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Cotangent

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Cot
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The cotangent function cotz is defined for z not in piZ by

cotz=(cosz)/(sinz)
(1)
=(i(e^(iz)+e^(-iz)))/(e^(iz)-e^(-iz))
(2)
=(i(e^(2iz)+1))/(e^(2iz)-1).
(3)

Equivalently, cotz=1/tanz wherever the tangent tanz is finite and nonzero. The cotangent is implemented in the Wolfram Language as Cot[z].

The notations ctnz (Erdélyi et al. 1981, p. 7; Jeffrey 2000, p. 111) and ctgz (Gradshteyn and Ryzhik 2000, p. xxix) are sometimes used in place of cotz. Note that the cotangent is not in as widespread use in Europe as are sinz, cosz, and tanz, although it does appear explicitly in various German and Russian handbooks (e.g., Gradshteyn and Ryzhik 2000, p. 28). Interestingly, cotz is treated on par with the other trigonometric functions in most tabulations (Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, p. 28), while secz and cscz are sometimes not (Gradshteyn and Ryzhik 2000, p. 28).

An important identity connecting the cotangent with the cosecant is given by

1+cot^2theta=csc^2theta.
(4)

The cotangent has smallest positive fixed point x such that cotx=x at 0.8603335890... (OEIS A069855; Bertrand 1865, p. 285).

The derivative is given by

d/(dz)cotz=-csc^2z
(5)

and the indefinite integral by

intcotzdz=ln(sinz)+C,
(6)

where C is a constant of integration and a local branch of the natural logarithm is chosen. On a real interval between consecutive multiples of pi, a real antiderivative is ln|sinx|+C.

Definite integrals include

int_(pi/4)^(pi/2)cotxdx=1/2ln2
(7)
int_0^(pi/4)ln(cotx)dx=K
(8)
int_0^(pi/4)xcotxdx=1/8(piln2+4K)
(9)
int_0^(pi/2)xcotxdx=1/2piln2
(10)
int_(pi/4)^(pi/2)xcotxdx=1/8(3piln2-4K)
(11)
int_0^(pi/4)x^2cotxdx=1/(64)[16piK+2pi^2ln2-35zeta(3)]
(12)
int_0^(pi/2)x^2cotxdx=1/8[2pi^2ln2-7zeta(3)],
(13)

where K is Catalan's constant, ln2 is the natural logarithm of 2, and zeta(3) is Apéry's constant. These integrals include ones considered by Glaisher (1893). Additional integrals include

int_0^(pi/4)cot^nxdx=1/4[psi_0(1/4(3-n))-psi_0(1/4(1-n))]
(14)

for Re[n]<1, where psi_0(z) is the digamma function, and

int_0^(pi/2)cot^nxdx=1/2pisec[1/2(pin)]
(15)

for -1<Re[n]<1.

The Laurent series for cotz about the origin is

cotz=1/z-1/3z-1/(45)z^3-2/(945)z^5-1/(4725)z^7-...
(16)
=sum_(n=0)^(infty)((-1)^n2^(2n)B_(2n))/((2n)!)z^(2n-1),
(17)

(OEIS A002431 and A036278), where B_n is a Bernoulli number and 0<|z|<pi.

A sum identity for the cotangent, valid for z not in Z, is given by

picot(piz)=1/z+2zsum_(n=1)^infty1/(z^2-n^2).
(18)

For an integer n>=3, cot(pi/n) is rational only for n=4. In particular, the degrees of the algebraic numbers cot(pi/n) for n=2, 3, ... are 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 2, ... (OEIS A089929).

See also

Hyperbolic Cotangent, Inverse Cotangent, Lehmer's Constant, Tangent

Related Wolfram sites

https://functions.wolfram.com/ElementaryFunctions/Cot/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Circular Functions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972.Bertrand, J. Exercise II in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie de L. Hachette et Cie, p. 285, 1865.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215, 1987.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 6, 1981.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989.Glaisher, J. W. L. "On Certain Numerical Products in Which the Exponents Depend upon the Numbers." Messenger Math. 23, 145-175, 1893.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Jeffrey, A. "Trigonometric Identities." §2.4 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 111-117, 2000.Sloane, N. J. A. Sequences A002431/M0124, A036278, A069855, and A089929 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Tangent tan(x) and Cotangent cot(x) Functions." Ch. 34 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 319-330, 1987.Tropfke, J. Teil IB, §2. "Die Begriffe von Tangens und Kotangens eines Winkels." In Geschichte der Elementar-Mathematik in systematischer Darstellung mit besonderer Berücksichtigung der Fachwörter, fünfter Band, zweite aufl. Berlin, Germany and Leipzig, Germany: de Gruyter, pp. 23-28, 1923.Zwillinger, D. (Ed.). "Trigonometric or Circular Functions." §6.1 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 452-460, 1995.

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Cotangent

Cite this as:

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

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