Coshxax

Coshxax is a term encountered in some informal mathematical writings to refer to a function built from the hyperbolic cosine with a linear argument. In typical usage coshxax denotes the function f(x; a) = cosh(a x), where a is a real parameter. The name combines “cosh” with the symbols “ax,” suggesting the argument is scaled by both a and x, though the exact convention can vary between sources.

Definition and basic properties: For real a, the function f(x) = cosh(a x) is defined for all real x and takes real values. It is an even function: f(-x) = f(x). Its derivative is f'(x) = a sinh(a x), and the second derivative is f''(x) = a^2 cosh(a x). The range is [1, ∞) when a ≠ 0, with f(0) = 1; if a = 0, f(x) ≡ 1. As |x| grows, cosh(a x) grows approximately like (1/2) e^{|a x|}.

Relation and typical uses: Coshxax is closely related to the standard hyperbolic cosine cosh. The function cosh(a x) is common in solutions to linear differential equations with constant coefficients, in problems involving hyperbolic geometry, and in symmetric exponential models. When a is small, the function behaves similarly to 1 + (a x)^2/2 near x = 0.

Notation caveat: The concatenated form coshxax is not a standard, universally accepted notation. Many texts simply write cosh(a x). When encountering coshxax, it is important to clarify the intended argument and parameter to avoid ambiguity.

See also: cosh, hyperbolic functions, cosh(ax).