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 andThe 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
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