tanh^(-1)(1/x)

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

(no roots exist)

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

{y element R : y!=0}

injective (one-to-one)

odd

(1/2 i (-i log(x + 1) + i log(2) + π) + (x + 1)/4 + 1/16 (x + 1)^2 + 1/48 (x + 1)^3 + 1/128 (x + 1)^4 + 1/320 (x + 1)^5 + O((x + 1)^6)) + i π floor(arg(1 + 1/x)/(2 π)) + i π floor(-(arg(1/x) + arg(x + 1) - π)/(2 π))

-1/2 π sqrt(-1/x^2) x + x + x^3/3 + x^5/5 + O(x^6) (generalized Puiseux series)

(1/2 (log(2) - log(x - 1)) + (x - 1)/4 - 1/16 (x - 1)^2 + 1/48 (x - 1)^3 - 1/128 (x - 1)^4 + 1/320 (x - 1)^5 + O((x - 1)^6)) - i π floor(-(arg(x - 1) + arg(1/x) - π)/(2 π))

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

d/dx(coth^(-1)(x)) = 1/(1 - x^2)

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

lim_(x-> ± ∞) coth^(-1)(x) = 0

coth^(-1)(x) = ns^(-1)(x|1)

coth^(-1)(x) = tanh^(-1)(1/x)

coth^(-1)(x) = sn^(-1)(1/x|1)

integral_0^1 coth^(-1)(x) dx≈2.2639435074...

integral_(-1)^0 coth^(-1)(x) dx≈-2.2639435074...

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

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

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

coth^(-1)(x) = x integral_1^∞ 1/(-1 + t^2 x^2) dt for ((Im(x) = 0 and (x<-1 or 00)

coth^(-1)(x) = x integral_(-∞)^1 1/(-1 + t^2 x^2) dt for ((Im(x) = 0 and (x<-1 or 00)

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

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