sin^(-1)(1/x)

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

(no roots exist)

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

{y element R : -π/2<=y<0 or 0

injective (one-to-one)

odd

-π/2 + (-1)^floor((-arg(1/x) - arg(x + 1) + π)/(2 π)) i^(2 floor(arg((x + 1)/x)/(2 π)) + 1) (sqrt(2) sqrt(x + 1) + (5 (x + 1)^(3/2))/(6 sqrt(2)) + (43 (x + 1)^(5/2))/(80 sqrt(2)) + (177 (x + 1)^(7/2))/(448 sqrt(2)) + (2867 (x + 1)^(9/2))/(9216 sqrt(2)) + O((x + 1)^(11/2)))

(-1)^floor(arg(x)/π) (π floor(arg(x)/π) + (1/2 (2 i log(x) - i log(4) + π) + (i x^2)/4 + (3 i x^4)/32 + O(x^6)))

1/2 ((-1)^floor((-arg(x - 1) - arg(1/x) + π)/(2 π)) (-2 sqrt(2) sqrt(x - 1) + (5 (x - 1)^(3/2))/(3 sqrt(2)) - (43 (x - 1)^(5/2))/(40 sqrt(2)) + (177 (x - 1)^(7/2))/(224 sqrt(2)) - (2867 (x - 1)^(9/2))/(4608 sqrt(2)) + O((x - 1)^(11/2))) + π)

1/x + 1/(6 x^3) + 3/(40 x^5) + O((1/x)^6) (Laurent series)

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

integral csc^(-1)(x) dx = (sqrt(1 - 1/x^2) x (log(x/sqrt(x^2 - 1) + 1) - log(1 - x/sqrt(x^2 - 1))))/(2 sqrt(x^2 - 1)) + x csc^(-1)(x) + constant (assuming a complex-valued logarithm)

min{csc^(-1)(x)} = -π/2 at x = -1

max{csc^(-1)(x)} = π/2 at x = 1

lim_(x-> ± ∞) csc^(-1)(x) = 0

csc^(-1)(x) = ds^(-1)(x|0)

csc^(-1)(x) = ns^(-1)(x|0)

csc^(-1)(x) = sin^(-1)(1/x)

integral_0^1 csc^(-1)(x) dx≈3.1415926536...

integral_(-1)^0 csc^(-1)(x) dx≈-3.14159265359...

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

csc^(-1)(x) = sum_(k=0)^∞ (-1)^k _3 F_2(1/2, (1 + k)/2, (2 + k)/2 ;1, 3/2 ;1/z_0^2) (x - z_0)^k z_0^(-1 - k) for (not (z_0 element R and -1<=z_0<=1))

csc^(-1)(x) = π/2 - 2 sqrt(-1 + x) sum_(k=0)^∞ ((-1)^k (-1 + x)^k 2F1(1/2, 3/2 + k, 3/2, -1) (1/2)_k)/(k!) for abs(-1 + x)<1

csc^(-1)(x) = integral_x^∞ 1/(t sqrt(-1 + t^2)) dt for Re(x)>1

csc^(-1)(x) = integral_(-∞)^x 1/(t sqrt(-1 + t^2)) dt for Re(x)<-1

csc^(-1)(x) = integral_x^∞ 1/(sqrt(1 - 1/x^2) x^2) dx for (not (x element R and -1<=x<=1))

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