Hyperbolic Functions
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Hyperbolic Functions
Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic
hyperbolic functions are the hyperbolic sine "sinh" ( / s
nt
/ or / a
n/
), and the hyperbolic cosine "cosh"
( / k
/
), from which are derived the hyperbolic tangent "tanh" ( / taent /), and so on, corresponding to
the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine
"arsinh" (also called "asinh" or sometimes "arcsinh") and so on. 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.
Hyperbolic functions occur in the solutions of some important linear differential equations, for example
the equation defining a catenary, of some cubic equations, and of Laplace's equation in Cartesian
coordinates. The latter is important in many areas of physics, including electromagnetic theory, heat
transfer, fluid dynamics, and special relativity.
The hyperbolic functions take real values for a real argument called a hyperbolic angle. In complex
analysis, they are simply rational functions of exponentials, and so are meromorphic. Hyperbolic
functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich
Lambert.[3] 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.
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The hyperbolic functions arise in many problems of mathematics and mathematical physics in which
integrals involving arise (whereas the circular functions involve ). For instance, the hyperbolic sine
arises in the gravitational potential of a cylinder and the calculation of the Roche limit. The hyperbolic
cosine function is the shape of a hanging cable (the socalled catenary). The hyperbolic tangent arises in
the calculation of and rapidity of special relativity. All three appear in the Schwarzschild metric using
external isotropic Kruskal coordinates in general relativity. The hyperbolic secant arises in the profile of
a laminar jet. The hyperbolic cotangent arises in the Langevin function for magnetic polarization.
What is calculus?
Calculus in Latin means stones used for counting. It is the branch of mathematics deals with change it
is the study of how things change. The fundamental idea of calculus is to study change by studying
"instantaneous" change that is we can say changes over tiny intervals of time.
Who invented Calculus?
The notion of derivative, integral was developed in 17th century but the actual development was done
by Isaac Newton and Leibniz. The development of calculus and its applications to physics and
engineering is probably the most significant factor in the development of modern science The
development of calculus is also responsible for the industrial revolution and everything that has
followed from it including almost all the major advances of the last few centuries
When to use Calculus? : Calculus is a very versatile and valuable tool. It is used in
 finding the slope of a curve
 finding the area of a irregular shape
 visualizing graph
 calculating optimal values
 finding the average of a function
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