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).

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