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Topic: Hyperbolic Trigonometric function

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List of coshes

Hyperbola

Circle

	Trigonometric function - Wikipedia, the free encyclopedia
		The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)).
		The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin.
		The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle.
	en.wikipedia.org /wiki/Trigonometric_function (3245 words)

	Exponential function - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-14)
		The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives.
		This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions.
		The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
	xahlee.org /_p/wiki/Exponential_function.html (900 words)

	Hyperbolic function - free-definition (Site not responding. Last check: 2007-09-14)
		The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities.
		Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments.
		The functions sinh z and cosh z are then holomorphic; their Taylor series expansions are given in the Taylor series article.
	www.free-definition.com /Tanh.html (306 words)

	Hyperbolic function - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-14)
		In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
		The parameter t is not a circular angle, but rather a hyperbolic angle which represents twice the area between the x-axis, the hyperbola and the straight line which links the origin with the point (cosh t, sinh t) on the hyperbola.
		The function sinh x is an odd function, that is symmetric with respect to the origin, and sinh 0 = 0
	xahlee.org /_p/wiki/Sinh.html (305 words)

	exponential function -- Encyclopædia Britannica (Site not responding. Last check: 2007-09-14)
		the hyperbolic sine of z (written sinh z); the hyperbolic cosine of z (cosh z); the hyperbolic tangent of z (tanh z); and the hyperbolic cosecant, secant, and cotangent of z.
		These functions are most conveniently defined in terms of the exponential function, with sinh z = (e e) and cosh z = (e + e) and with the other hyperbolic trigonometric functions defined in a manner...
		In simple terms, a function f is a mathematical rule that assigns to a number x (in some number system and possibly with certain limitations on its value) another number f(x).
	www.britannica.com /eb/article?tocId=9033449&query=function (990 words)

	Hyperbolic Trigonometric Functions
		We define the Arcsinh(x) function as the principal inverse of the hyperbolic sine function.
		This is a first-order ordinary differential-equation in the radius r = 1 / p as a function of the angle theta.
		The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and
	www.rism.com /Trig/hyperbol.htm (9675 words)

	BSSA, Volume 87:1 (Site not responding. Last check: 2007-09-14)
		To answer this question, we have investigated two weighting functions: One is the hyperbolic function suggested by Brown (1994), and the other is a trigonometric function we propose.
		We found that the hyperbolic function performs accurately, particularly when a wave front is only smoothly bent, but there exists a tolerance range for the value of the coefficient.
		Because the range is limited, and because it changes for different modelings, use of the hyperbolic function can become troublesome in intensive modelings where it is impractical to have human supervision.
	www.seismosoc.org /publications/BSSA_html/bssa_87-1/96008.html (267 words)

	CVM 1.1 (VW): Eigenfunctions of the Laplacian
		Applying the Laplacian operator to a function usually results in a new function totally different from the original.
		Two familiar examples are the sine and cosine functions, but the hyperbolic trigonometric functions work too, as do simple exponential functions.
		In such a case, the hyperbolic and exponential functions would be rejected and the set of eigenvalues is discrete.
	www.geom.uiuc.edu /~dpvc/CVM/1998/01/vw/article/constructing/efl.html (258 words)

	Hyperbolic Functions
		The hyperbolic functions enjoy properties similar to the trigonometric functions; their definitions, though, are much more straightforward:
		The other hyperbolic functions are defined the same way, the rest of the trigonometric functions is defined:
		For every formula for the trigonometric functions, there is a similar (not necessary identical) formula for the hyperbolic functions:
	www.sosmath.com /trig/hyper/hyper01/hyper01.html (155 words)

	Hyperbolic tangent: Integration (subsection 21/01)
		Involving functions of the direct function, hyperbolic and trigonometric functions (12 formulas)
		Involving functions of the direct function, hyperbolic, exponential and trigonometric functions (12 formulas)
		Involving functions of the direct function, hyperbolic, exponential, trigonometric and a power functions (8 formulas)
	functions.wolfram.com /ElementaryFunctions/Tanh/21/01 (207 words)

	Hyperbolic cosine integral: Integration (subsection 21/01)
		Involving functions of the direct function, hyperbolic and trigonometric functions (56 formulas)
		Involving functions of the direct function, hyperbolic, exponential and trigonometric functions (54 formulas)
		Involving functions of the direct function, hyperbolic, exponential, trigonometric and a power functions (12 formulas)
	functions.wolfram.com /ElementaryFunctions/Cosh/21/01 (230 words)

	Earliest Uses of Symbols for Trigonometric and Hyperbolic Functions (Site not responding. Last check: 2007-09-14)
		Daniel Bernoulli was the first to use symbols for the inverse trigonometric functions.
		A full-page footnote explained his choice of notation for the inverse trigonometric functions, such as cos.
		The practice of placing the exponent beside the symbol for the trigonometric function to indicate, for example, the square of the sine of x, was used by William Jones in 1710.
	members.aol.com /jeff570/trigonometry.html (1034 words)

	Math 122: Chapter 7 Study Guide (Site not responding. Last check: 2007-09-14)
		Find the derivative of one of the inverse trigonometric functions by using an appropriate triangle and the derivative of a trigonometric function.
		Given a graph of a function, determine whether or not the function is one-to-one.
		You are expected to know the derivatives of the trigonometric and hyperbolic trigonometric functions.
	www.richland.edu /james/fall03/m122/m122-s07.html (277 words)

	functions
		The use of trigonometric functions arises from the early connection between mathematics and astronomy.
		Business Functions is a library of up to nearly 500 high-performance functions running under Microsoft Excel, for financial analysis, planning and appraisal.
		Trigonometric Functions and Calculus for Liberal Arts and Business Majors.
	www.jointctr.org /?Category=functions (448 words)

	Mathematics (Site not responding. Last check: 2007-09-14)
		Real numbers function, properties of functions, induction, sequences, limit of function, continuity of functions.Derivatives ; principles of differentiation differeuliate of Chainrule.derivatives of inverse functions.Application of differential in approximation.
		Functions of several variables, definition domain, range, neigh bour hood.
		Functions of bounded variation, Riemann integral and its properties.
	www.y.net.ye /sciencefac-taizuni/mathematics.htm (1701 words)

	Science News: Neural nets catch the ABCs of DNA (Site not responding. Last check: 2007-09-14)
		In accordance with a mathematical theorem, the net seems to have found a continuous function that describes a given set of points and then used this function to predict the next point to fit the pattern.
		The net constructed this generating function in a method somewhat analogous to the mathematical technique called Fourier analysis, which enables scientists to approximate any curve by adding together sine waves of different frequency, phase and amplitude Instead of sine waves, however, the net used another trigonometric function, hyperbolic tangents (tanh).
		The net's choice of the tanh function ismore serendipitous than surprising, because the researchers in essence "loaded the dice' by using tanh functions throughout their program.
	www.findarticles.com /p/articles/mi_m1200/is_v132/ai_5114665 (1187 words)

	The educational encyclopedia, mathematics, trigonometry (Site not responding. Last check: 2007-09-14)
		Trigonometry and basic functions trigonometry, trigonometric, elementary functions, basic functions, sine, cosine, sin, cos, Gudermannian, hyperbolic amplitude, Gamma function
		Trigonometry trigonometry, ratio and proportion, pythagorean theorem, solve right triangles, trigonometric functions, law of sines, law of cosines, unit circle, pi, radian measure, arclength, arc length, 30-60-90 triangle, trigonometric table
		Trigonometric functions angles, degrees, radians, sine function, cosine function, tangent function, cotangent function, inverse trig functions
	users.telenet.be /educypedia/education/mathematicstrigonometry.htm (272 words)

	The inverted-U function (from motivation) -- Encyclopædia Britannica
		The relationship between changes in arousal and motivation is often expressed as an inverted-U function (also known as the Yerkes-Dodson law).
		The basic concept is that, as arousal level increases, performance improves, but only to a point, beyond which increases in arousal lead to a deterioration in performance.
		Particular functions of the set had been formulated earlier by the Swiss mathematicians Daniel Bernoulli, who studied the oscillations of a chain suspended by one...
	www.britannica.com /eb/article?tocId=12694 (844 words)

	Trigonometric function - Enpsychlopedia (Site not responding. Last check: 2007-09-14)
		There is a very interesting visual proof that the tangent function satisfies this differential equation; see [1] (http://www.usfca.edu/vca/PDF/vca-preface.pdf).
		Eli Maor, Trigonometric Delights (http://www.pupress.princeton.edu/books/maor/) (Princeton Univ. Press, 1998).
		Dave's Short Trig Course (http://babbage.clarku.edu/~djoyce/java/trig/) uses interactive java applets that are very helpful for learning about trig functions on the unit circle, as well as being useful more generally for students of trigonometry.
	www.grohol.com /psypsych/Sine (3372 words)

	The MATH signature
		The signature MATH specifies basic mathematical constants, the square root function, and trigonometric, hyperbolic, exponential and logarithmic functions based on a real type.
		In a list of rules specifying the behavior of a function in special cases, the first matching rule defines the semantics.
		Note that tan will produce infinities at various finite values, roughly corresponding to the singularities of the tangent function.
	www.smlnj.org /doc/basis/pages/math.html (283 words)

	GiNaC: numeric.cpp File Reference
		Helper function to print a real number in a nicer way than is CLN's default.
		Its most important design principle is to completely hide the inner working of that other package from the user of GiNaC.
		It must either provide implementation of arithmetic operators and numerical evaluation of special functions or implement the interface to the bignum package.
	www.ginac.de /reference/numeric_8cpp.html (1017 words)

	Help for the Sing Surf program
		Generic functions like Ridge are defined which are combined with the equation of the surface to produce the equation to define the Ridges for that surface.
		Most functions correspond to the function with the same name in the C maths library.
		It also make special effort to find the singularities of the surface, which are places where all three partial derivatives of the function vanish.
	www.singsurf.org /singsurf/help.html (2097 words)

	[No title] (Site not responding. Last check: 2007-09-14)
		Equations of mathematical physics: Hyperbolic (wave equation), parabolic (diffusion or heat equation and Schroedinger equation), and elliptic (Laplace equation and Poisson equation)
		Spherical Bessel functions, Legendre polynomials and functions, and spherical harmonics.
		Analytic properties of elementary functions: square root, exponential function, trigonometric and hyperbolic functions, and logarithm
	www.phy-astr.gsu.edu /stockman/data/6520mp.html (166 words)

	The MATH signature
		specifies basic mathematical constants, the square root function, and trigonometric, hyperbolic, exponential, and logarithmic functions based on a real type.
		These return the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent, respectively, of
		This document may be distributed freely over the internet as long as the copyright notice and license terms below are prominently displayed within every machine-readable copy.
	www.standardml.org /Basis/math.html (367 words)

	EMTeachline mathematics software - problem solving - math test authoring (Site not responding. Last check: 2007-09-14)
		Covered subject areas are arithmetic, pre-algebra, algebra, trigonometry and hyperbolic trigonometry.
		Trigonometry and Hyperbolic Trigonometry: trigonometric and hyperbolic functions, arcus functions.
		Related theory: definitions and rules, trigonometric and hyperbolic formulas.
	www.emteachline.com /eng/index_main.htm (1560 words)

	Trigonometric function (SAP Library - SAP DB)
		) in every trigonometric function identify an angle in radians.
		The only exception to this is the RADIANS function.
		Arc tangent of the value a/b, where -
	sapdb.org /7.4/htmhelp/71/817368b9b311d2a97100a0c9449261/content.htm (35 words)

	[No title] (Site not responding. Last check: 2007-09-14)
		Subject: Duncan Functions Does anyone have any references to the mathematical properties of Duncan functions?
		Linear combinations of these functions arise as eigenfunctions of the Euler-Bernoulli beam equation, u''''(x) - k^4 u(x) = 0, where k>0 is an eigenvalue.
		The Duncan functions are sums or differences of a trigonometric function and a hyperbolic trigonometric function; for example, s1(kx) = sin(kx) + sinh(kx).
	www.math.ucl.ac.be /~magnus/num2/duncan.txt (370 words)

	Kyung Hee University
		Complex number, analytic function, exponential, trigonometric, hyperbolic and Logarithmic function,
		Measurable function, monotone convergence theorem, Fatou's lemma, Lebesgue dominated convergence theorem, Minkowski's inequality
		Relation and functions, denumerable sets and nondenumerable sets, cardinal number and ordinal number, Zorn's lemma
	www.kyunghee.edu /acadaemy/under_su3_02.php (540 words)

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