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Guallatiri is a stratovolcano in Chile that is 6,060–6,071 m (19,880–19,918 ft) high. It is located southwest of, or possibly within, the Nevados de Quimsachata volcanic group. The summit, surrounded by numerous fumaroles, may be a lava dome or volcanic plug, while the lower flanks of the volcano are covered by lava flows and lava domes.
ISO 3166-1 ( Codes for the representation of names of countries and their subdivisions – Part 1: Country codes) is a standard defining codes for the names of countries, dependent territories, and special areas of geographical interest. It is the first part of the ISO 3166 standard published by the International Organization for Standardization .
The Chinese Communist Party ( CCP ), [3] officially the Communist Party of China ( CPC ), [4] is the founding and sole ruling party of the People's Republic of China (PRC). Under the leadership of Mao Zedong, the CCP emerged victorious in the Chinese Civil War against the Kuomintang. In 1949, Mao proclaimed the establishment of the People's ...
- Characterizing Properties
- Useful Relations
- Second Derivatives
- Standard Integrals
- Taylor Series Expressions
- Infinite Products and Continued Fractions
- Comparison with Circular Functions
- Relationship to The Exponential Function
- Hyperbolic Functions For Complex Numbers
- See Also
Hyperbolic cosine
It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc lengthcorresponding to that interval:
Hyperbolic tangent
The hyperbolic tangent is the (unique) solution to the differential equation f ′ = 1 − f 2, with f (0) = 0.
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for θ {\\displaystyle \\theta } , 2 θ {\\displaystyle 2\\theta } , 3 θ {\\displaystyle 3\\theta } or θ {...
Each of the functions sinh and cosh is equal to its second derivative, that is: All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x {\\displaystyle e^{x}} and e − x {\\displaystyle e^{-x}} .
The following integrals can be proved using hyperbolic substitution: where C is the constant of integration.
It is possible to express explicitly the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions. The sum of the sinh and cosh series is the infinite series expression of the exponential function. The following series are followed by a description of a subset of their domain of convergence, where...
The following expansions are valid in the whole complex plane: 1. sinh x = x ∏ n = 1 ∞ ( 1 + x 2 n 2 π 2 ) = x 1 − x 2 2 ⋅ 3 + x 2 − 2 ⋅ 3 x 2 4 ⋅ 5 + x 2 − 4 ⋅ 5 x 2 6 ⋅ 7 + x 2 − ⋱ {\\displaystyle \\sinh x=x\\prod _{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{n^{2}\\pi ^{2}}}\\right)={\\cfrac {x}{1-{\\cfrac {x^{2}}{2\\cdot 3+x^{2}-{\\cfrac {2\\cdot 3x^{2}}{4\\cd...
The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle. Since the area of a circular sector with radius r and angle u (in radians) is r2u/2, it will be equal to u when r = √2. In the diagram, such a circle is tangent to the hyperbola ...
The decomposition of the exponential function in its even and odd partsgives the identities Additionally,
Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh z and cosh z are then holomorphic. Relationships to ordinary trigonometric functions are given by Euler's formulafor complex numbers: Thus, hyperbolic functions are periodic ...
according to International Monetary Fund estimates [n 1] [1] Countries by nominal GDP in 2019 [n 2] > $20 trillion. $10–20 trillion. $5–10 trillion. $1–5 trillion. $750 billion – $1 trillion. $500–750 billion. $250–500 billion.
Gone with the Wind is a 1939 American epic historical romance film adapted from the 1936 novel by Margaret Mitchell.The film was produced by David O. Selznick of Selznick International Pictures and directed by Victor Fleming.Set in the American South against the backdrop of the Civil War and the Reconstruction era, the film tells the story of Scarlett O'Hara (Vivien Leigh), the strong-willed ...
Phase Four 's group of films began with Black Widow (2021), and was followed by Shang-Chi and the Legend of the Ten Rings (2021), Eternals (2021), Spider-Man: No Way Home (2021), Doctor Strange in the Multiverse of Madness (2022), Thor: Love and Thunder (2022), and Black Panther: Wakanda Forever (2022).
