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Inverse Secant

Plots

Plots

Plots

Alternate forms

cos^(-1)(1/x)

1/2 (π - 2 sin^(-1)(1/x))

π/2 + i log(sqrt(1 - 1/x^2) + i/x)

Root

x = 1

Properties as a real function

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

{y element R : 0<=y<π/2 or π/2<y<=π}

injective (one-to-one)

Series expansion at x = -1

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)))

Series expansion at x = 0

π/2 + (-1)^(floor(arg(x)/π) + 1) (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)))

Series expansion at x = 1

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) + π/2

Series expansion at x = ∞

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

Derivative

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

Indefinite integral

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)

Global maximum

max{sec^(-1)(x)} = π at x = -1

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