Coshxa

Coshxa is the parametric family of hyperbolic cosine functions defined by f_a(x) = cosh(a x) for real numbers x and a. The parameter a controls horizontal scaling of the standard cosh function, allowing a single expression to model differently stretched profiles.

Definition: coshxa(x, a) = (e^{a x} + e^{-a x})/2.

Properties: It is an even function in x; its derivative with respect to x is f'_a(x) = a sinh(a x); the second derivative is f''_a(x) = a^2 cosh(a x). It satisfies the identity cosh^2(u) - sinh^2(u) = 1 with u = a x. For a > 0, the function grows roughly exponentially as |x| increases; for a = 0 the function is identically 1.

Special cases and relation to standard cosh: f_1(x) = cosh(x). When a ≠ 1, the graph is a horizontal stretch or compression by a factor of 1/|a|. Because cosh is even, cosh(a x) = cosh(-a x), so the sign of a does not change the function's values.

Applications: The coshxa form is used in problems with scaled spatial variables, in solving differential equations via substitution, and in physics and engineering contexts where hyperbolic growth appears, such as beam propagation, optical lens design, and models involving nonuniform media. It provides a compact way to represent scaled hyperbolic behavior without rewriting the exponential form each time.

See also: cosh, hyperbolic functions, scaling transformations.