Trigonmetric hyparabolic function vs general function
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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
The basic hyperbolic functions are the hyperbolic sine "sinh" (/sɪntʃ, ʃaɪn/),[1] and the hyperbolic cosine "cosh" (/kɒʃ, koʊʃ/),[2] from which are derived the hyperbolic tangent"tanh" (/tæntʃ, θæn/),[3] hyperbolic cosecant"csch" or "cosech" (/ˈkoʊʃɛk/[2] or /ˈkoʊsɛtʃ/), hyperbolic secant "sech" (/ʃɛk, sɛtʃ/),[4] and hyperbolic cotangent "coth" (/koʊθ, kɒθ/),[5][6]corresponding to the derived trigonometric functions.
The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")[7][8][9]and so on.

A ray through the unit hyperbola {\displaystyle x^{2}-y^{2}=1} in the point {\displaystyle (\cosh a,\sinh a),} where {\displaystyle a} is twice the area between the ray, the hyperbola, and the {\displaystyle x}-axis. For points on the hyperbola below the {\displaystyle x}-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
Hyperbolic functions occur in the solutions of many linear differential equations (for example, the equation defining a catenary), of some cubic equations, in calculations of angles and distances in hyperbolic geometry, and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence holomorphic.
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.[10] Riccati used Sc.and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]sinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.[11] The abbreviations sh and ch are still used in some other languages, like French and Russian.
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
The basic hyperbolic functions are the hyperbolic sine "sinh" (/sɪntʃ, ʃaɪn/),[1] and the hyperbolic cosine "cosh" (/kɒʃ, koʊʃ/),[2] from which are derived the hyperbolic tangent"tanh" (/tæntʃ, θæn/),[3] hyperbolic cosecant"csch" or "cosech" (/ˈkoʊʃɛk/[2] or /ˈkoʊsɛtʃ/), hyperbolic secant "sech" (/ʃɛk, sɛtʃ/),[4] and hyperbolic cotangent "coth" (/koʊθ, kɒθ/),[5][6]corresponding to the derived trigonometric functions.
The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")[7][8][9]and so on.

A ray through the unit hyperbola {\displaystyle x^{2}-y^{2}=1} in the point {\displaystyle (\cosh a,\sinh a),} where {\displaystyle a} is twice the area between the ray, the hyperbola, and the {\displaystyle x}-axis. For points on the hyperbola below the {\displaystyle x}-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
Hyperbolic functions occur in the solutions of many linear differential equations (for example, the equation defining a catenary), of some cubic equations, in calculations of angles and distances in hyperbolic geometry, and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence holomorphic.
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.[10] Riccati used Sc.and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]sinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.[11] The abbreviations sh and ch are still used in some other languages, like French and Russian.
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