sinh2n2

SinH2n2 is a notation occasionally used to denote the hyperbolic sine of the quantity 2n^2, i.e., Sinh(2n^2). By definition, Sinh(x) = (e^x − e^(−x))/2, so SinH2n2 = (e^(2n^2) − e^(−2n^2))/2 for real n.

The function SinH2n2 is even in n, since the inner argument 2n^2 is unchanged by n → −n, and Sinh is an odd function of its argument. Consequently, SinH2n2(n) = SinH2n2(−n). It attains its minimum value 0 at n = 0 and increases as |n| grows. The derivative is d/dn [Sinh(2n^2)] = 4n cosh(2n^2), indicating that the function decreases for negative n and increases for positive n, with a stationary point at the origin.

For large |n|, SinH2n2 grows very rapidly and is well approximated by (1/2) e^(2n^2), since e^(2n^2) dominates e^(−2n^2). In practice, SinH2n2 is used in contexts where the hyperbolic sine of a quadratic expression appears, such as certain asymptotic analyses or problem setups. It is important to note that SinH2n2 is not a universally standardized name; it should be read as Sinh(2n^2) and interpreted from the surrounding mathematical context. See also the hyperbolic sine function, Sinh, and related identities.