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

x = 0

R (all real numbers)

{y element R : -π/2

injective (one-to-one)

odd

x - x^3/3 + x^5/5 + O(x^6) (Taylor 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(3/4 - arg(x + i)/(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((π - 2 arg(x - i))/(4 π))

π/2 - 1/x + 1/(3 x^3) - 1/(5 x^5) + O((1/x)^6) (Laurent series)

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

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

lim_(x->-∞) tan^(-1)(x) = -π/2≈-1.5708

lim_(x->∞) tan^(-1)(x) = π/2≈1.5708

tan^(-1)(x) = sc^(-1)(x|0)

tan^(-1)(x) = cot^(-1)(1/x)

tan^(-1)(x) = tan^(-1)(1, x)

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

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

tan^(-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

tan^(-1)(x) = x integral_0^1 1/(1 + t^2 x^2) dt

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

tan^(-1)(x) = -(i x)/(4 π) integral_(-i ∞ + γ)^(i ∞ + γ) ((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(x^2))<π)

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

tan^(-1)(x) = x - x^3/(3 + Κ_(k=1)^∞ ((1 - (-1)^k + k)^2 x^2)/(3 + 2 k)) = x - x^3/(3 + (9 x^2)/(5 + (4 x^2)/(7 + (25 x^2)/(9 + (16 x^2)/(11 + ...))))) for (i x not element R or ((not 1<=i x<∞) and (not -∞

tan^(-1)(x) = x/(1 + Κ_(k=1)^∞ ((1 - 2 k)^2 x^2)/(1 + x^2 - 2 k (-1 + x^2))) = x/(1 + x^2/(1 + x^2 - 2 (-1 + x^2) + (9 x^2)/(1 + x^2 - 4 (-1 + x^2) + (25 x^2)/(1 + x^2 - 6 (-1 + x^2) + (49 x^2)/(1 + ... + x^2 - 8 (-1 + x^2)))))) for (i x not element R or ((not 1<=i x<∞) and (not -∞

tan^(-1)(x) = x/(1 + x^2 + Κ_(k=1)^∞ (2 x^2 (1 - 2 floor((1 + k)/2)) floor((1 + k)/2))/((1 + 2 k) (1 + 1/2 (1 + (-1)^k) x^2))) = x/(1 + x^2 + -(2 x^2)/(3 - (2 x^2)/(5 (1 + x^2) - (12 x^2)/(7 - (12 x^2)/(9 (1 + x^2) + ... ))))) for (i x not element R or ((not 1<=i x<∞) and (not -∞