tan^(-1)(1/x)

1/2 i log(1 - i/x) - 1/2 i log(1 + i/x)

(no roots exist)

R (all real numbers)

{y element R : -π/2

injective (one-to-one)

odd

1/2 π sqrt(1/x^2) x - x + x^3/3 - x^5/5 + O(x^6) (generalized Puiseux series)

(1/4 (-2 i log(x + i) + 2 i log(2) + π) - (x + i)/4 + 1/16 i (x + i)^2 + 1/48 (x + i)^3 - 1/128 i (x + i)^4 - 1/320 (x + i)^5 + O((x + i)^6)) + π floor(-(arg(1/x) + arg(x + i) - π)/(2 π))

(1/4 (2 i log(x - i) - 2 i log(2) + π) - (x - i)/4 - 1/16 i (x - i)^2 + 1/48 (x - i)^3 + 1/128 i (x - i)^4 - 1/320 (x - i)^5 + O((x - i)^6)) - π floor(-(arg(1/x) + arg(x - i) - π)/(2 π))

1/x - 1/(3 x^3) + 1/(5 x^5) + O((1/x)^6) (Laurent series)

d/dx(cot^(-1)(x)) = -1/(x^2 + 1)

integral cot^(-1)(x) dx = 1/2 log(x^2 + 1) + x cot^(-1)(x) + constant

max{cot^(-1)(x)} = π/2 at x = 0

lim_(x-> ± ∞) cot^(-1)(x) = 0

cot^(-1)(x) = cs^(-1)(x|0)

cot^(-1)(x) = sc^(-1)(1/x|0)

cot^(-1)(x) = i coth^(-1)(i x)

cot^(-1)(x) = sum_(k=0)^∞ ((-1)^k x^(-1 - 2 k))/(1 + 2 k) for abs(x)>1

cot^(-1)(x) = 1/2 π sqrt(1/x^2) x - sum_(k=0)^∞ ((-1)^k x^(1 + 2 k))/(1 + 2 k) for abs(x)<1

cot^(-1)(x) = 1/2 i log(-1/2 i (-i + x)) + 1/2 i sum_(k=1)^∞ ((i/2)^k (-i + x)^k)/k for abs(-i + x)<2

cot^(-1)(x) = x integral_1^∞ 1/(1 + t^2 x^2) dt for Re(x)>0

cot^(-1)(x) = -π + x integral_1^∞ 1/(1 + t^2 x^2) dt for Re(x)<0

cot^(-1)(x) = -i/(4 π^(3/2) x) integral_(-i ∞ + γ)^(i ∞ + γ) (1 + 1/x^2)^(-s) Γ(1/2 - s) Γ(1 - s) Γ(s)^2 ds for (0<γ<1/2 and abs(arg(1 + 1/x^2))<π)

cot^(-1)(x) = -i/(4 π x) integral_(-i ∞ + γ)^(i ∞ + γ) ((1/x^2)^(-s) Γ(1/2 - s) Γ(1 - s) Γ(s))/Γ(3/2 - s) ds for (0<γ<1/2 and abs(arg(1/x^2))<π)

cot^(-1)(x) = 1/(x + x (Κ_(k=1)^∞ (k^2/x^2)/(1 + 2 k))) = 1/(x + x 1/((3 + 4/((5 + 9/((7 + 16/((9 + ...) x^2)) x^2)) x^2)) x^2)) for (not (i x element R and -1<=i x<=1))