搜尋結果
Typographical symbols and punctuation marks are marks and symbols used in typography with a variety of purposes such as to help with legibility and accessibility, or to identify special cases. This list gives those most commonly encountered with Latin script. For a far more comprehensive list of symbols and signs, see List of Unicode characters.
Wikipedia is written by volunteer editors and hosted by the Wikimedia Foundation, a non-profit organization that also hosts a range of other volunteer projects : Commons. Free media repository. MediaWiki. Wiki software development. Meta-Wiki. Wikimedia project coordination. Wikibooks. Free textbooks and manuals.
CJK Unified Ideographs is a Unicode block containing the most common CJK ideographs used in modern Chinese, Japanese, Korean and Vietnamese characters. When contrasted with other blocks containing CJK Unified Ideographs, it is also referred to as the Unified Repertoire and Ordering (URO).[3] The block has hundreds of variation sequences ...
International Organization for Standardization. ISO 3166-1. ISO 3166-2. ISO 3166-3. Country code. Comparison of alphabetic country codes. List of IOC country codes. List of FIFA country codes. International vehicle registration code. List of aircraft registration prefixes. List of GS1 country codes.
- 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 ...
Taiwan,[II][k] officially the Republic of China (ROC),[I][l] is a country[27] in East Asia.[o] It is located at the junction of the East and South China Seas in the northwestern Pacific Ocean, with the People's Republic of China (PRC) to the northwest, Japan to the northeast, and the Philippines to the south. The territories controlled by the ...
Wikipedia is a free online encyclopedia that anyone can edit, and millions already have . Wikipedia's purpose is to benefit readers by presenting information on all branches of knowledge. Hosted by the Wikimedia Foundation, it consists of freely editable content, whose articles also have numerous links to guide readers towards more information.
