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This is a list of countries by external debt: it is the total public and private debt owed to nonresidents repayable in internationally accepted currencies, goods or services, where the public debt is the money or credit owed by any level of government, from central to local, and the private debt the money or credit owed by private households or private corporations based on the country under ...
Viète. de Moivre. Euler. Fourier. v. t. e. In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.
Linearity. The Schrödinger equation is a linear differential equation, meaning that if two state vectors and are solutions, then so is any linear combination. of the two state vectors where a and b are any complex numbers. [13] : 25 Moreover, the sum can be extended for any number of state vectors.
TSMC was founded in Taiwan in 1987 by Morris Chang as the world's first dedicated semiconductor foundry. It has long been the leading company in its field. [15] [16] When Chang retired in 2018, after 31 years of TSMC leadership, Mark Liu became chairman and C. C. Wei became Chief Executive.
- 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 ...
Grandparents' Day. Mother's Day is a celebration honoring the mother of the family or individual, as well as motherhood, maternal bonds, and the influence of mothers in society. It is celebrated on different days in many parts of the world, most commonly in March or May. It complements similar celebrations honoring family members, such as ...
Carlos Ray "Chuck" Norris (born March 10, 1940) is an American martial artist and actor. He is a black belt in Tang Soo Do, Brazilian jiu jitsu and judo.[1] After serving in the United States Air Force, Norris won many martial arts championships and later founded his own discipline, Chun Kuk Do. Shortly after, in Hollywood, Norris trained ...