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:

    “An indirect quotation we can usually expect to rate only as better or worse, more or less faithful, and we cannot even hope for a strict standard of more and less; what is involved is evaluation, relative to special purposes, of an essentially dramatic act.”
    —Willard Van Orman Quine (b. 1908)

    “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)

    “Its idea of “production value” is spending a million dollars dressing up a story that any good writer would throw away. Its vision of the rewarding movie is a vehicle for some glamour-puss with two expressions and eighteen changes of costume, or for some male idol of the muddled millions with a permanent hangover, six worn-out acting tricks, the build of a lifeguard, and the mentality of a chicken-strangler.”
    —Raymond Chandler (1888–1959)