# MACLAURIN SERIES HYPERBOLIC COSINE

Hyperbolic functions may also be deduced from trigonometric functions with complex arguments:. There are various equivalent ways for defining the hyperbolic functions. It can be seen that cosh x and sech x are even functions ; the others are odd functions. The complex forms in the definitions above derive from Euler’s formula. This page was last edited on 25 February , at The decomposition of the exponential function in its even and odd parts gives the identities. Laplace’s equations are important in many areas of physics , including electromagnetic theory , heat transfer , fluid dynamics , and special relativity. Haskell , “On the introduction of the notion of hyperbolic functions”, Bulletin of the American Mathematical Society 1:

Similarly, the yellow and red sectors together depict an area and hyperbolic angle magnitude. In other projects Wikimedia Commons. The first one is analogous to Euler’s formula. The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. The inverse hyperbolic functions are:. Thus it is an even function , that is, symmetric with respect to the y -axis. From Wikipedia, the free encyclopedia. Sinh and cosh are both equal to their second derivative , that is:.

It can be shown that the area under the curve of the hyperbolic cosine over a finite interval is always equal to the arc length corresponding to that interval: Retrieved from ” https: The hyperbolic functions may be defined as hypeebolic of differential equations: Lambert adopted the names but altered the abbreviations to what they are today.

There are various equivalent ways for defining the hyperbolic functions. Both types depend on an argumenteither circular angle or hyperbolic angle. The first one is analogous to Euler’s formula.

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Similarly, the yellow and red sectors together depict an area and hyperbolic angle magnitude. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.

Maclaurrin a result, the other hyperbolic functions are meromorphic in the whole complex plane.

## Hyperbolic function

Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments.

Hyperbolic functions Exponentials Hyperbolic geometry Analytic functions. The hyperbolic sine and the hyperbolic cosine are entire functions.

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 coskne Cartesian coordinates. Mathematical Association of America.

The hyperbolic angle is an invariant measure with respect to the squeeze mappingjust as the circular angle is invariant under rotation.

For a full list, see list of integrals of hyperbolic functions. The hyperbolic sine and cosine are the unique solution s msclaurin, c of the system.

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. Mathematical Association of America, This yields for hypedbolic the addition theorems. Hyperbolic functions were introduced in the s independently by Vincenzo Riccati and Johann Heinrich Lambert. In fact, Osborn’s rule [19] states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of 2, 6, 10, 14, The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

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### Hyperbolic function – Wikipedia

D’Antonio, Charles Edward Sandifer. The inverse hyperbolic functions are:. This page was last edited on 25 Februaryat Wikimedia Commons has media related to Hyperbolic macalurin. In mathematicshyperbolic functions are analogs of the ordinary trigonometricor circularfunctions.

The sum of the sinh and cosh series is the infinite series expression of the exponential function. The yellow sector depicts an area and angle magnitude. Views Read Edit View history. The hyperbolic functions take a real argument called a hyperbolic angle. Laplace’s equations are important in many areas of physicsincluding electromagnetic theoryheat seiesfluid dynamicsand special relativity.

The hyperbolic functions represent an hyprbolic of trigonometry beyond the circular functions.

## Taylor series of hyperbolic functions

Sinh and cosh are both equal to their second derivativethat is:. The foundations of geometry and the non-euclidean plane 1st corr. It can be seen that cosh x and sech x are even functions ; the others madlaurin odd functions.

Retrieved 24 January The following integrals can be proved using hyperbolic substitution:. In complex analysisthe hyperbolic functions arise as the imaginary parts of sine and cosine.

By using this site, you agree to the Terms of Use and Privacy Policy. The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. The decomposition of the exponential function in its even and odd parts gives the identities. It is possible to express the above functions as Taylor series:. In other projects Wikimedia Commons.