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Alternate names

arccot | arccotangent | arcctg

Definition

The inverse cotangent is the multivalued function cot^(-1) z, also denoted arccot z or arcctg z, that is the inverse function of the cotangent. The variants Arccot z (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. Worse yet, the notation arccot z is sometimes used for the principal value, with Arccot z being used for the multivalued function. Note that in the notation cot^(-1) z (commonly used in North America and in pocket calculators worldwide), cot z 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]. There are at least two possible conventions for defining the inverse cotangent. This work follows the convention of Abramowitz and Stegun and the Wolfram Language, taking cot^(-1) x to have range (-π/2, π/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)]. 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, π), 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/(d z) cot^(-1) z = - 1/(1 + z^2) and the integral by integral cot^(-1) z d z = z cot^(-1) z + 1/2 ln(1 + z^2) + C. The Maclaurin series of the inverse cotangent for x>0 is given by cot^(-1) x | = | π/2 - sum_(k = 0)^∞ ((-1)^k x^(2k + 1))/(2k + 1) | = | π/2 - x + 1/3 x^3 - 1/5 x^5 + 1/7 x^7 - 1/9 x^9 + ... (OEIS A005408). The Laurent series about z = ∞ is given by cot^(-1) z | = | sum_(k = 0)^∞ ((-1)^k z^(-(2k + 1)))/(2k + 1) | = | z^(-1) - 1/3 z^(-3) + 1/5 z^(-5) - 1/7 z^(-7) + 1/9 z^(-9) + ... for left bracketing bar z right bracketing bar >1. Euler derived the infinite series cot^(-1) z = z sum_(n = 1)^∞ ((2n - 2)!!)/((2n - 1)!!(z^2 + 1)^n) (Wetherfield 1996). The inverse cotangent satisfies cot^(-1) z = tan^(-1)(1/z) for z!=0, cot^(-1) z = - cot^(-1)(-z) for all z element C^*, and cot^(-1) x | = | {sec^(-1)(sqrt(x^2 + 1)/x) - π | for x<0 sec^(-1)(sqrt(x^2 + 1)/x) | for x>0 auto right match | = | {-1/2 π - tan^(-1) x | for x<0 1/2 π - tan^(-1) x | for x>=0 auto right match | = | {-sin^(-1)(1/sqrt(x^2 + 1)) | for x<0 sin^(-1)(1/sqrt(x^2 + 1)) | for x>0 auto right match | = | {-1/2 π - cot^(-1)(1/x) | for x<0 1/2 π - cot^(-1)(1/x) | for x>0 auto right match | = | {-csc^(-1)(sqrt(x^2 + 1)) | for x<0 csc^(-1)(sqrt(x^2 + 1)) | for x>0 auto right match | = | {cos^(-1)(x/sqrt(x^2 + 1)) - π | for x<0 cos^(-1)(x/sqrt(x^2 + 1)) | for x>0 auto right match | = | {-1/2 π - sin^(-1)(x/sqrt(x^2 + 1)) | for x<0 1/2 π - sin^(-1)(x/sqrt(x^2 + 1)) | for x>0. auto right match Analytic sums of cotangents include the beautiful result sum_(n = 1)^∞ cot^(-1) n^2 = cot^(-1)((1 + t)/(1 - t)) = 1.42474..., (OEIS A091007), where t congruent cot(1/2 πsqrt(2)) tanh(1/2 πsqrt(2)) (H. S. Wilf, pers. comm., May 21, 2002). A number t_x = cot^(-1) x, 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, where n_i = ⌊a_i/b_i ⌋, with ⌊x⌋ the floor function, and a_(i + 1) | = | a_i n + i + b_i b_(i + 1) | = | a_i - n_i b_i, 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_k tan^(-1) n_k + f tan^(-1) 1, then equation (◇) becomes cot^(-1) n = sum_(k = 1) f_k cot^(-1) n_k + c cot^(-1) 1, where c = 2 - f - 2 sum_(k = 1) f_k. 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) 3cot^(-1)(3x) - cot^(-1) x | = | cot^(-1)((27x^4 + 18x^2 - 1)/(8x)), as well as many others (Bennett 1926, Lehmer 1938b). Note that for equation (-1), the choice of convention for cot^(-1) z is significant, since it holds for all complex z in the [0, π] convention, but holds only outside a lens-shaped region centered on the origin in the [-π/2, π/2] convention.