Mannertan

Mannertan is a hypothetical mathematical transformation used in some educational materials to illustrate how nonlinear tangent-based mappings can be combined with linear components. It is not a standard term in formal mathematics, but it appears in classroom examples and online explanations as a simple model for studying nonlinearity and signal shaping.

Definition and form. A Mannertan transform, denoted M_p, is defined for a real input x and a parameter triple p = (a, b, c) with a ≠ 0 and b ≠ 0 by M_p(x) = a tan(bx) + c. The domain of x is restricted to avoid the vertical asymptotes of tan(bx), which occur when bx = π/2 + kπ for any integer k.

Properties. The transform inherits the vertical asymptotes of the tangent function, scaled and shifted by the parameters a and c. The period of the mapped function is π/b, and M_p is strictly increasing or decreasing on each interval between successive asymptotes, depending on the sign of a. As a result, the output can become unbounded near the asymptotes, and small changes in input near these points produce large changes in output.

Applications and context. In teaching and exploratory work, Mannertan provides a simple, controllable example of nonlinear transformation, illustrating how scaling, frequency (via b), and vertical shifts (via c) influence graph shape and sensitivity. It is commonly used to demonstrate nonlinearity, to test intuition about input-output distortion, and to accompany discussions of activation-like behaviors in a purely mathematical setting.

See also. Tangent function, nonlinear transformations, signal processing.