-i log(sqrt(1 - x^2) + i x)

x = 0

{x element R : -1<=x<=1}

{y element R : -π/2<=y<=π/2}

injective (one-to-one)

odd

-π/2 + sqrt(2) sqrt(x + 1) + (x + 1)^(3/2)/(6 sqrt(2)) + (3 (x + 1)^(5/2))/(80 sqrt(2)) + (5 (x + 1)^(7/2))/(448 sqrt(2)) + (35 (x + 1)^(9/2))/(9216 sqrt(2)) + O((x + 1)^5) (Puiseux series)

x + x^3/6 + (3 x^5)/40 + O(x^6) (Taylor series)

π/2 - (sqrt(2) sqrt(1 - x) sqrt(x - 1))/sqrt(x - 1) + (sqrt(1 - x) (x - 1)^(3/2))/(6 sqrt(2) sqrt(x - 1)) - (3 sqrt(1 - x) (x - 1)^(5/2))/(80 (sqrt(2) sqrt(x - 1))) + (5 sqrt(1 - x) (x - 1)^(7/2))/(448 sqrt(2) sqrt(x - 1)) - (35 sqrt(1 - x) (x - 1)^(9/2))/(9216 (sqrt(2) sqrt(x - 1))) + O((x - 1)^(11/2)) (generalized Puiseux series)

1/2 (2 i log(x) + π + 2 i log(2)) - i/(4 x^2) - (3 i)/(32 x^4) + O((1/x)^6) (generalized Puiseux series)

d/dx(sin^(-1)(x)) = 1/sqrt(1 - x^2)

integral sin^(-1)(x) dx = sqrt(1 - x^2) + x sin^(-1)(x) + constant

min{sin^(-1)(x)} = -π/2 at x = -1

max{sin^(-1)(x)} = π/2 at x = 1

sin^(-1)(x) = sd^(-1)(x|0)

sin^(-1)(x) = sn^(-1)(x|0)

sin^(-1)(x) = i sinh^(-1)(-i x)

integral_0^1 sin^(-1)(x) dx≈0.570796326795...

integral_(-1)^0 sin^(-1)(x) dx≈-0.570796326795...

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

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

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

sin^(-1)(x) = x integral_0^1 1/sqrt(1 - t^2 x^2) dt

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

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

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