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Cotangent

Plots

Alternate forms

cos(x)/sin(x)

-(i (e^(-i x) + e^(i x)))/(e^(-i x) - e^(i x))

Alternate form assuming x is real

-sin(2 x)/(cos(2 x) - 1)

Roots

x = 1/2 (2 π n + π), n element Z

Properties as a real function

{x element R : x/π not element Z}

R (all real numbers)

periodic in x with period π

surjective onto R

odd

Series expansion at x = 0

1/x - x/3 - x^3/45 - (2 x^5)/945 + O(x^6)
(Laurent series)

Derivative

d/dx(cot(x)) = -csc^2(x)

Indefinite integral

integral cot(x) dx = log(sin(x)) + constant
(assuming a complex-valued logarithm)

Identities

cot(x) = cot(m π + x) for m element Z

cot(x) = cot(2 x) + csc(2 x)

cot(x) = (1 + cos(2 x)) csc(2 x)

cot(x) = 1/2 (cot(x/2) - tan(x/2))

cot(x) = 1/2 (-1 + cot^2(x)) tan(2 x)

cot(x) = sin(2 x)/(1 - cos(2 x))

cot(x) = csc(x) sec(x) - tan(x)

cot(x) = 1/2 (-1 + cot^2(x/2)) tan(x/2)

Alternative representations

cot(x) = 1/tan(x)

cot(x) = i coth(i x)

cot(x) = -i coth(-i x)

Series representations

cot(x) = -i - 2 i sum_(k=1)^∞ q^(2 k) for q = e^(i x)

cot(x) = -i sum_(k=-∞)^∞ e^(2 i k x) sgn(k)

cot(x) = i + 2 i sum_(k=0)^∞ e^(-2 i (1 + k) x) for Im(x)<0