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Wikipedia Topic: Logistic Function with Selected References Selected Articles These are entries printed using the PDF feature of Wikipedia. The title of each article is listed in bookmarks. Excerpts Wikipedia (pronounced /wɪkɨpidi.ə/ WIK-i-PEE-dee-ə) is a multilingual, web- based, free-content encyclopedia project based on an openly editable model. The name "Wikipedia" is a portmanteau of the words wiki (a technology for creating collaborative websites, from the Hawaiian word wiki, meaning "quick") and encyclopedia. Wikipedia's articles provide links to guide the user to related pages with additional information. Wikipedia is written collaboratively by largely anonymous Internet volunteers who write without pay. Anyone with Internet access can write and make changes to Wikipedia articles (except in certain cases where editing is restricted to prevent disruption or vandalism). Users can contribute anonymously, under a pseudonym, or with their real identity, if they choose. People of all ages, cultures and backgrounds can add or edit article prose, references, images and other media here. What is contributed is more important than the expertise or qualifications of the contributor. What will remain depends upon whether it fits within Wikipedia's policies, including being verifiable against a published reliable source, so excluding editors' opinions and beliefs and unreviewed research, and is free of copyright restrictions and contentious material about living people. Contributions cannot damage Wikipedia because the software allows easy reversal of mistakes and many experienced editors are watching to help and ensure that edits are cumulative improvements. Wikipedia is a live collaboration differing from paper-based reference sources in important ways. Unlike printed encyclopedias, Wikipedia is continually created and updated, with articles on historic events appearing within minutes, rather than months or years. Older articles tend to grow more comprehensive and balanced; newer articles may contain misinformation, unencyclopedic content, or vandalism. Awareness of this aids obtaining valid information and avoiding recently added misinformation (see Researching with Wikipedia). From/en.wikipedia.org 16 January 2011 Wikipedia: Researching with Wikipedia From Wikipedia, the free encyclopedia Wikipedia can be a great tool for learning and researching information. However, as with all reference works, not everything in Wikipedia is accurate, comprehensive, or unbiased. Many of the general rules of thumb for conducting research apply to Wikipedia, including: • Always be wary of any one single source (in any medium — web, print, television or radio), or of multiple works that derive from a single source. • Where articles have references to external sources (whether online or not) read the references and check whether they really do support what the article says. • In most academic institutions, major references to Wikipedia, along with most encyclopedias, are unacceptable for a research paper. Other encyclopedias, such as Encyclopædia Britannica, have notable authors working for them and may be cited as a secondary source in most cases. For example, Cornell University has a guide on how to cite encyclopedias. However, because of Wikipedia's unique nature, there are also some rules for conducting research that are special to Wikipedia, and some general rules that do not apply to Wikipedi From http://en.wikipedia.org/wiki/Wikipedia:Researching_with_Wikipedia From/en.wikipedia.org 16 January 2011 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 so-called 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. The hyperbolic functions are defined by (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) For arguments multiplied by , (17) (18) The hyperbolic functions satisfy many identities analogous to the trigonometric identities (which can be inferred using Osborn's rule) such as (19) From mathworld.wolfram.com/HyperbolicFunctions.html 16 January 2011 (20) (21) See also Beyer (1987, p. 168). Some half-angle formulas are (22) (23) where . Some double-angle formulas are (24) (25) (26) Identities for complex arguments include (27) (28) The absolute squares for complex arguments are (29) (30) SEE ALSO: Double-Angle Formulas, Fibonacci Hyperbolic Functions, Half-Angle Formulas, Hyperbolic Cosecant, Hyperbolic Cosine, Hyperbolic Cotangent, Generalized Hyperbolic Functions, Hyperbolic Secant, Hyperbolic Sine, Hyperbolic Tangent, Inverse Hyperbolic Functions, Osborn's Rule REFERENCES: Abramowitz, M. and Stegun, I. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83-86, 1972. From mathworld.wolfram.com/HyperbolicFunctions.html 16 January 2011 Anderson, J. W. "Trigonometry in the Hyperbolic Plane." §5.7 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 146-151, 1999. Beyer, W. H. "Hyperbolic Function." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 168-186 and 219, 1987. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 126-131, 1967. Harris, J. W. and Stocker, H. "Hyperbolic Functions." Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 245-262, 1998. Jeffrey, A. "Hyperbolic Identities." §2.5 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 117-122, 2000. Yates, R. C. "Hyperbolic Functions." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 113-118, 1952. Zwillinger, D. (Ed.). "Hyperbolic Functions." §6.7 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 476-481 1995. CITE THIS AS: Weisstein, Eric W. "Hyperbolic Functions." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperbolicFunctions.html From mathworld.wolfram.com/HyperbolicFunctions.html 16 January 2011 Hyperbolic function 1 Hyperbolic function In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" (typically pronounced /ˈsɪntʃ/ or English pronunciation: /ˈʃaɪn/), and the hyperbolic cosine "cosh" (typically pronounced /ˈkɒʃ/), from which are derived the hyperbolic tangent "tanh" (typically pronounced /ˈtæntʃ/ or English pronunciation: /ˈθæn/), and so on, corresponding to the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh", or sometimes by the misnomer of "arcsinh"[1] ) and so on. A ray through the origin intercepts the hyperbola in the point Just as the points (cos t, sin t) form a circle , where is twice the area between the ray and the -axis. with a unit radius, the points (cosh t, sinh t) For points on the hyperbola below the -axis, the area is considered negative form the right half of the equilateral (see animated version with comparison with the trigonometric (circular) functions). hyperbola. Hyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary, and 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.[2] 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.[3] Hyperbolic function 2 Standard algebraic expressions The hyperbolic functions are: • Hyperbolic sine: sinh, cosh and tanh csch, sech and coth • Hyperbolic cosine: • Hyperbolic tangent: • Hyperbolic cotangent: • Hyperbolic secant: • Hyperbolic cosecant: Hyperbolic function 3 Hyperbolic functions can be introduced via imaginary circular angles: • Hyperbolic sine: • Hyperbolic cosine: • Hyperbolic tangent: • Hyperbolic cotangent: • Hyperbolic secant: • Hyperbolic cosecant: where i is the imaginary unit defined as i2 = -1. The complex forms in the definitions above derive from Euler's formula. Note that, by convention, sinh2 x means (sinh x)2, not sinh(sinh x); similarly for the other hyperbolic functions when used with positive exponents. Another notation for the hyperbolic cotangent function is ctnh x, though coth x is far more common. Useful relations Hence: It can be seen that cosh x and sech x are even functions; the others are odd functions. Hyperbolic sine and cosine satisfy the identity which is similar to the Pythagorean trigonometric identity. One also has for the other
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