In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire …

The two basic hyperbolic functions are sinh and cosh: sinh (x) = ex - e-x2. (pronounced shine or sinch). cosh (x) = ex + e-x2.

Feb 14, 2026 · The hyperbolic sine is defined as sinhz=1/2 (e^z-e^ (-z)). (1) The notation shz is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram …

Nov 25, 2024 · Learn the different hyperbolic trigonometric functions, including sine, cosine, and tangent, with their formulas, examples, and diagrams. Also, learn their identities.

The function defines the hyperbolic sine of an angle. It’s similar to how a sine works for a unit circle, except here we’re working with a hyperbola instead of the unit circle.

Today, we’re going to unravel the concept of the hyperbolic sine (sinh), exploring its formula, applications, and practical examples. Hyperbolic Sine (sinh) is a mathematical function defined as …

Jan 27, 2026 · Hyperbolic functions, the hyperbolic sine of z (written sinh z); the hyperbolic cosine of z (cosh z); the hyperbolic tangent of z (tanh z); and the hyperbolic cosecant, secant, and cotangent of z.

We can similarly define functions on the unit hyperbola x 2 y 2 = 1, call x = cosh (t) the function hyperbolic cosine and y = sinh (t) the function hyperbolic sine.

Here are two graphics showing the real and imaginary parts of the hyperbolic sine function over the complex plane.

Definition 4.11.1 The hyperbolic cosine is the function cosh x = e x + e x 2, and the hyperbolic sine is the function sinh x = e x e x 2 . Notice that cosh is even (that is, cosh (x) = cosh (x)) while sinh is odd …