Hyperbolic geometry | Exponentials | Hyperbolic functions | Analytic functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. 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 unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and 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. The basic hyperbolic functions are: * hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/), * hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/), from which are derived: * hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/), * hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/) * hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/), * hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/), corresponding to the derived trigonometric functions. The inverse hyperbolic functions are: * area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") * area hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh") * and so on. 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. In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today. The abbreviations sh, ch, th, cth are also currently used, depending on personal preference. (Wikipedia).
Introduction to Hyperbolic Functions
This video provides a basic overview of hyperbolic function. The lesson defines the hyperbolic functions, shows the graphs of the hyperbolic functions, and gives the properties of hyperbolic functions. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com
From playlist Differentiation of Hyperbolic Functions
Introduction to Hyperbolic Functions
This video provides a basic overview of hyperbolic function. The lesson defines the hyperbolic functions, shows the graphs of the hyperbolic functions, and gives the properties of hyperbolic functions.
From playlist Using the Properties of Hyperbolic Functions
Calculus 2: Hyperbolic Functions (1 of 57) What is a Hyperbolic Function? Part 1
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what are hyperbolic functions and how it compares to trig functions. Next video in the series can be seen at: https://youtu.be/c8OR8iJ-aUo
From playlist CALCULUS 2 CH 16 HYPERBOLIC FUNCTIONS
Ling Long - Hypergeometric Functions, Character Sums and Applications - Lecture 2
Title: Hypergeometric Functions, Character Sums and Applications Speaker: Prof. Ling Long, Louisiana State University Abstract: Hypergeometric functions form a class of special functions satisfying a lot of symmetries. They are closely related to the arithmetic of one-parameter families of
From playlist Hypergeometric Functions, Character Sums and Applications
Hyperbolic Functions: Definitions, Identities, Derivatives, and Inverses
We've learned about trigonometric functions, which relate to the unit circle. So what are hyperbolic functions? Why, those relate to the hyperbola of course! They are a little strange, but once we go through some details they will start to make sense a little bit. Watch the whole Mathemat
From playlist Mathematics (All Of It)
Ling Long - Hypergeometric Functions, Character Sums and Applications - Lecture 6
Title: Hypergeometric Functions, Character Sums and Applications Speaker: Prof. Ling Long, Louisiana State University Abstract: Hypergeometric functions form a class of special functions satisfying a lot of symmetries. They are closely related to the arithmetic of one-parameter families of
From playlist Hypergeometric Functions, Character Sums and Applications
Ling Long - Hypergeometric Functions, Character Sums and Applications - Lecture 5
Title: Hypergeometric Functions, Character Sums and Applications Speaker: Prof. Ling Long, Louisiana State University Abstract: Hypergeometric functions form a class of special functions satisfying a lot of symmetries. They are closely related to the arithmetic of one-parameter families of
From playlist Hypergeometric Functions, Character Sums and Applications
Ling Long - Hypergeometric Functions, Character Sums and Applications - Lecture 4
Title: Hypergeometric Functions, Character Sums and Applications Speaker: Prof. Ling Long, Louisiana State University Abstract: Hypergeometric functions form a class of special functions satisfying a lot of symmetries. They are closely related to the arithmetic of one-parameter families of
From playlist Hypergeometric Functions, Character Sums and Applications
Ling Long - Hypergeometric Functions, Character Sums and Applications - Lecture 3
Title: Hypergeometric Functions, Character Sums and Applications Speaker: Prof. Ling Long, Louisiana State University Abstract: Hypergeometric functions form a class of special functions satisfying a lot of symmetries. They are closely related to the arithmetic of one-parameter families of
From playlist Hypergeometric Functions, Character Sums and Applications
Worldwide Calculus: Integration using Hyperbolic Sine and Cosine
Lecture on 'Integration using Hyperbolic Sine and Cosine' from 'Worldwide Integral Calculus'. For more lecture videos and $10 digital textbooks, visit www.centerofmath.org.
From playlist Worldwide Integral Calculus
Hyperbolic Functions II: Graphs of all the functions
Channel social media: Instagram: @whatthehectogon https://www.instagram.com/whatthehectogon/ Twitter: @whatthehectogon https://twitter.com/whatthehectogon Check out my friend Bill's DnD channel: Marching West https://www.youtube.com/channel/UCFNd... Twitter: @WestMarching https://twitter.
From playlist Complete Hyperbole
Reverse, Reverse III: Inverse Hyperbolic Graphs
Channel social media: Instagram: @whatthehectogon https://www.instagram.com/whatthehectogon/ Twitter: @whatthehectogon https://twitter.com/whatthehectogon Check out my friend Bill's DnD channel: Marching West https://www.youtube.com/channel/UCFNd... Twitter: @WestMarching https://twitter.
From playlist Complete Hyperbole
Hyperbolic Functions Introduction 6 Ex Calculus 1 PLEASE READ DESCRIPTION
Please note the copy error on the green board, cosh^2(x)=1/2(1+cosh(2x)) Evaluating Hyperbolic Functions for a given value at 7:37 and 8:30 Find values of other 5 Hyperbolic Functions from a given Hyperbolic Function at 12:50 Verifying a Hyperbolic Identity 3 examples at 16:02 20:53 25:21
From playlist Calculus
Hyperbolic Functions Derivative & Integrals 5 Examples Calculus 1
I work through 5 examples of finding derivatives and integrals of hyperbolic functions Derivative of a hyperbolic function examples at 1:25 and 7:05 Integral of a hyperbolic function examples at 14:45 17:25 and 22:01 Hyperbolic Functions Introduced 6 Examples at https://www.youtube.com/wa
From playlist Calculus
Inverse Hyperbolic Functions Derivative and Integral Calculus 1
Evaluating an Inverse Hyperbolic Expression at 2:42 Derivative of an Inverse Hyperbolic Function at 4:25 8:38 Integration examples at 13:35 18:00 Hyperbolic Function Introduction 6 Examples https://www.youtube.com/watch?v=TEdED74yWPY Hyperbolic Functions Derivative & Integrals https://www
From playlist Calculus
Ling Long - Hypergeometric Functions, Character Sums and Applications - Lecture 7
Title: Hypergeometric Functions, Character Sums and Applications Speaker: Prof. Ling Long, Louisiana State University Abstract: Hypergeometric functions form a class of special functions satisfying a lot of symmetries. They are closely related to the arithmetic of one-parameter families of
From playlist Hypergeometric Functions, Character Sums and Applications
A. Zorich - Counting simple closed geodesics and volumes of moduli spaces (Part 1)
In the first two lectures I will try to tell (or, rather, to give an idea) of how Maryam Mirzakhani has counted simple closed geodesics on hyperbolic surfaces. I plan to briefly mention her count of Weil-Peterson volumes and her proof of Witten's conjecture, but only on the leve
From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications