sinh2u

Sinh(2u) denotes the hyperbolic sine of the quantity 2u. It is defined from the basic hyperbolic sine function, sinh x = (e^x − e^(−x))/2, so sinh(2u) = (e^(2u) − e^(−2u))/2.

A key identity is the double-angle formula for hyperbolic sine: sinh(2u) = 2 sinh(u) cosh(u). This expresses sinh(2u) directly in terms of sinh(u) and cosh(u). The hyperbolic cosine is related by cosh^2(u) − sinh^2(u) = 1, and cosh and sinh are connected through their derivatives and integrals.

Sinh(2u) is an odd function: sinh(−2u) = −sinh(2u). It is strictly increasing for all real u because its derivative, d/du [sinh(2u)] = 2 cosh(2u), is positive for all u. The range of sinh(2u) is all real numbers.

In terms of growth, sinh(2u) behaves like (1/2) e^(2u) for large positive u and tends to −∞ as u tends to −∞. It can be computed directly from sinh(u) and cosh(u) via the double-angle formula, and it also equals (e^(2u) − e^(−2u))/2, illustrating its dependence on the exponential function.

Applications of sinh(2u) appear in solving differential equations, in the description of hyperbolic geometry, and in various physics and engineering contexts where hyperbolic functions model growth or decay processes with non-periodic behavior. Note that sinh(2u) differs from sinh^2(u), which denotes (sinh(u))^2.