Cotangent

Min Max Re Im

The cotangent function is the function defined by

where is the tangent. The cotangent is implemented in the Wolfram Language as Cot[z].

The notations (Erdélyi et al. 1981, p. 7; Jeffrey 2000, p. 111) and (Gradshteyn and Ryzhik 2000, p. xxix) are sometimes used in place of . Note that the cotangent is not in as widespread use in Europe as are , , and , although it does appear explicitly in various German and Russian handbooks (e.g., Gradshteyn and 2000, p. 28). Interestingly, 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 and are sometimes not (Gradshteyn and Ryzhik 2000, p. 28).

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

The cotangent has smallest real fixed point such at 0.8603335890... (OEIS A069855; Bertrand 1865, p. 285).

The derivative is given by

and the indefinite integral by

where is a constant of integration.

Definite integrals include

where is Catalan's constant, is the natural logarithm of 2, and is Apéry's constant. Integrals (9) and (10) were considered by Glaisher (1893). Additional integrals include

for , where is the digamma function, and

for .

The Laurent series for about the origin is

(OEIS A002431 and A036278), where is a Bernoulli number.

A nice sum identity for the cotangent is given by

For an integer , is rational only for . In particular, the algebraic degrees of for , 3, ... are 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 2, ... (OEIS A089929).