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Viviparus georgianus, commonly known as the banded mystery snail, is a species of large freshwater snail in the family Viviparidae, the river snails. It is native to North America, generally found from the northeastern United States to Florida and the Gulf of Mexico, and thrives in eutrophic lentic environments such as lakes, ponds and some low ...
A representation of the 3D structure of the protein myoglobin showing turquoise α-helices. This protein was the first to have its structure solved by X-ray crystallography. Toward the right-center among the coils, a prosthetic group called a heme group (shown in gray) with a bound oxygen molecule (red). Proteins are large biomolecules and ...
Testosterone is the primary male sex hormone and androgen in males. [3] In humans, testosterone plays a key role in the development of male reproductive tissues such as testicles and prostate, as well as promoting secondary sexual characteristics such as increased muscle and bone mass, and the growth of body hair.
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci ...
- 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 ...
American retail corporation Walmart has been the world's largest company by revenue since 2014. [1] The list is limited to the largest 50 companies, all of which have annual revenues exceeding US$130 billion. This list is incomplete, as not all companies disclose their information to the media and/or general public. [3]
Died. April 7, 2024. (2024-04-07) (aged 62) Philadelphia, Pennsylvania, US. Lori and George Schappell (born as Lori Schappell and Dori Schappell; September 18, 1961 – April 7, 2024) were American conjoined twins. George performed as a country singer. As of 2020, they were the oldest living conjoined twins in the world.
