cos^(-1)(1/x)

1/2 (π - 2 sin^(-1)(1/x))

π/2 + i log(sqrt(1 - 1/x^2) + i/x)

x = 1

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

{y element R : 0<=y<π/2 or π/2

injective (one-to-one)

π + (-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)))

π/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)^floor(-(arg(x - 1) + arg(1/x) - π)/(2 π)) (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)))

π/2 - 1/x - 1/(6 x^3) - 3/(40 x^5) + O((1/x)^6) (Laurent series)

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

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

min{sec^(-1)(x)} = 0 at x = 1

max{sec^(-1)(x)} = π at x = -1

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

sec^(-1)(x) = dc^(-1)(x|0)

sec^(-1)(x) = nc^(-1)(x|0)

sec^(-1)(x) = cos^(-1)(1/x)

integral_0^1 sec^(-1)(x) dx≈-1.5707963268...

integral_(-1)^0 sec^(-1)(x) dx≈4.7123889804...

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

sec^(-1)(x) = 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

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

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

sec^(-1)(x) = integral_1^x 1/(sqrt(1 - 1/t^2) t^2) dt for (x not element R or ((not 1<=x<∞) and (not -∞

sec^(-1)(x) = π/2 + i/(4 π^(3/2) x) integral_(-i ∞ + γ)^(i ∞ + γ) (1 - 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 - 1/x^2))<π)

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

sec^(-1)(x) = π/2 - 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))) = π/2 - 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))