cosh^2x-1

Input

cosh^2(x) - 1

Plot

Plot
Plot

Alternate form

sinh^2(x)
(cosh(x) - 1) (cosh(x) + 1)
1/4 (2 cosh(2 x) - 2)

Roots

x = i π n, n element Z

Integer root

x = 0

Properties as a real function

Domain

R (all real numbers)

Range

{y element R : y>=0} (all non-negative real numbers)

Parity

even

Periodicity

periodic in x with period i π

Series expansion at x=0

x^2 + x^4/3 + (2 x^6)/45 + x^8/315 + O(x^9) (Taylor series)
Big‐O notation »

Derivative

d/dx(cosh^2(x) - 1) = 2 sinh(x) cosh(x)

Indefinite integral

integral(cosh^2(x) - 1) dx = 1/4 (sinh(2 x) - 2 x) + constant

Global minimum

min{cosh^2(x) - 1} = 0 at x = 0

Alternative representation

cosh^2(x) - 1 = -1 + cos^2(i x)
cosh^2(x) - 1 = -1 + cos^2(-i x)
cosh^2(x) - 1 = -1 + (1/sec(i x))^2
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Series representation

cosh^2(x) - 1 = sum_(k=1)^∞ (2^(-1 + 2 k) x^(2 k))/((2 k)!)
cosh^2(x) - 1 = -1 + ( sum_(k=0)^∞ x^(2 k)/((2 k)!))^2
cosh^2(x) - 1 = -1 - 1/2 sum_(k=1)^∞ (-i π + 2 x)^(2 k)/((2 k)!)
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Definite integral over a half-period

integral_0^((i π)/2) (-1 + cosh^2(x)) dx = -(i π)/4≈-0.785398 i

Definite integral over a period

integral_0^(i π) (-1 + cosh^2(x)) dx = -(i π)/2≈-1.5708 i

Definite integral mean square

integral_0^(i π)-(i (-1 + cosh^2(x))^2)/π dx = 3/8 = 0.375
POWERED BY THE WOLFRAM LANGUAGE