Hyperbolic Function - Standard Algebraic Expressions

Standard Algebraic Expressions

(b) sinh(x) is half the difference of ex and e−x Hyperbolic functions (a) cosh and (b) sinh obtained using exponential functions and

The hyperbolic functions are:

  • Hyperbolic sine:
  • Hyperbolic cosine:
  • Hyperbolic tangent:
  • Hyperbolic cotangent:
  • Hyperbolic secant:
  • Hyperbolic cosecant:

Hyperbolic functions can be introduced via imaginary circular angles:

  • Hyperbolic sine:
  • Hyperbolic cosine:
  • Hyperbolic tangent:
  • Hyperbolic cotangent:
  • Hyperbolic secant:
  • Hyperbolic cosecant:

where i is the imaginary unit defined by i2 = −1.

The complex forms in the definitions above derive from Euler's formula.

Note that, by convention, sinh2 x means (sinh x)2, not sinh(sinh x), and similarly for the other hyperbolic functions when used with positive exponents. In contrast, sinh−1 x refers to the inverse function arsinh x and not to a reciprocal (and again likewise for the other hyperbolic functions).

Read more about this topic:  Hyperbolic Function

Famous quotes containing the words standard, algebraic and/or expressions:

    “As long as male behavior is taken to be the norm, there can be no serious questioning of male traits and behavior. A norm is by definition a standard for judging; it is not itself subject to judgment.”
    —Myriam Miedzian, U.S. author. Boys Will Be Boys, ch. 1 (1991)

    “I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
    —Henry David Thoreau (1817–1862)

    “Those expressions are omitted which can not with propriety be read aloud in the family.”
    —Thomas Bowdler (1754–1825)