The hyperbolic trigonometric functions extend the notion of the parametric Circle; Hyperbolic Trigonometric Identities; Shape of a Suspension Bridge; See Also. In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are. Comparing Trig and Hyperbolic Trig Functions. By the Maths Hyperbolic Trigonometric Functions. Definition using unit Double angle identities sin(2 ) .

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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

### Hyperbolic Functions

Exploration for Theorem 5. There are various equivalent ways for defining the hyperbolic functions. The complex forms in the definitions above derive from Euler’s formula.

The inverse hyperbolic functions are:.

### Mathematics reference: Hyperbolic trigonometry identities

The hyperbolic sine and the identitjes cosine are entire functions. Note that we often write sinh n x instead of the correct [sinh x ] nsimilarly for the other hyperbolic functions. The inverse functions are called argument of hyperbolic sinedenoted argsinh xargument of hyperbolic cosinedenoted argcosh xargument of hyperbolic tangentdenoted argtanh xand argument of hyperbolic cotangentdenoted argcoth x. Identitied following integrals can be proved using hyperbolic substitution:.

Hyperbolic functions occur in the solutions of many linear differential equations for example, the equation defining a catenaryof some cubic equationsin calculations of angles and distances in hyperbolic geometryand of Laplace’s equation in Hyperbolix coordinates. A series exploration i.

Absolute value Back to Theory – Elementary functions. For all complex numbers. Thus it is an even functionthat is, symmetric with respect to the y -axis.

The following identities are very similar to trig identities, but they are tricky, since once in a while a sign is the other way around, which can mislead an unwary student. Limits at endpoints of the domain are. It can be seen that cosh x and sech x are even functions ; the others are odd functions. There is no zero point and no point of inflection, there are no local extrema. Additionally, it is easy to show that are entire functions.

## Hyperbolic function

Technical mathematics with calculus 3rd ed. Exploration for Definition 5.

Both types depend on an argumenteither circular angle or hyperbolic angle. The sum of the sinh and cosh series is the infinite series expression of the exponential function. The hyperbolic sine and hyperbolic cosine are defined by.

Similarly, the yellow and red sectors together depict an area and hyperbolic angle magnitude. This page was last edited on 19 Decemberat We actually have “nice” formulas for the inverses:.

It is possible to express the above functions as Taylor series:. How should we define the complex hyperbolic functions? Just as the points cos tsin t form a circle with a unit radius, the points cosh tsinh t form the right half of the equilateral hyperbola.

We begin with some periodic results. Lambert adopted the names but altered the abbreviations to what they are today. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. Exploration for the identities. What does the mapping look like? For a full list, see list of integrals of hyperbolic functions. Retrieved 24 January Retrieved 18 March D’Antonio, Charles Edward Sandifer.

Exploration for the real and imaginary parts of Sin and Cos. The hyperbolic functions take a real argument called a hyperbolic angle. With these definitions in place, it is now easy to create the other complex trigonometric functions, provided the denominators in the following expressions do not equal zero.