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Beginning with the establishment of dynastic rule by Yu the Great c. 2070 BC, and ending with the abdication of the Xuantong Emperor in AD 1912, Chinese historiography came to organize itself around the succession of monarchical dynasties.
The yen ( Japanese: 円, symbol: ¥; code: JPY) is the official currency of Japan. It is the third-most traded currency in the foreign exchange market, after the United States dollar and the euro. [2] . It is also widely used as a third reserve currency after the US dollar and the euro.
- History
- Definitions of Complex Exponentiation
- Applications
- Other Special Cases
- See Also
- Further Reading
In 1714, the English mathematician Roger Cotes presented a geometrical argument that can be interpreted (after correcting a misplaced factor of − 1 {\displaystyle {\sqrt {-1}}} ) as: Around 1740 Leonhard Euler turned his attention to the exponential function and derived the equation named after him by comparing the series expansions of the exponent...
The exponential function ex for real values of x may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of ez for complex values of z simply by substituting z in place of x and using the complex algebraic operations. In particular, ...
Topological interpretation
In the language of topology, Euler's formula states that the imaginary exponential function t ↦ e i t {\displaystyle t\mapsto e^{it}} is a (surjective) morphism of topological groups from the real line R {\displaystyle \mathbb {R} } to the unit circle S 1 {\displaystyle \mathbb {S} ^{1}} . In fact, this exhibits R {\displaystyle \mathbb {R} } as a covering space of S 1 {\displaystyle \mathbb {S} ^{1}} . Similarly, Euler's identity says that the kernel of this map is τ Z {\displaystyle \tau \m...
Other applications
In differential equations, the function eix is often used to simplify solutions, even if the final answer is a real function involving sine and cosine. The reason for this is that the exponential function is the eigenfunction of the operation of differentiation. In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed...
The special casesthat evaluate to units illustrate rotation around the complex unit circle: The special case at x = τ (where τ = 2π, one turn) yields eiτ = 1 + 0. This is also argued to link five fundamental constants with three basic arithmetic operations, but, unlike Euler's identity, without rearranging the addendsfrom the general case:
Nahin, Paul J. (2006). Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press. ISBN 978-0-691-11822-2.Wilson, Robin (2018). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. Oxford: Oxford University Press. ISBN 978-0-19-879492-9. MR 3791469.What links here Related changes Upload file Special pages Permanent link Page information Cite this page Get shortened URL Download QR code In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that ...
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix. rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system.
Yasushi Akimoto ( Japanese: 秋元 康, Hepburn: Akimoto Yasushi, born May 2, 1958) is a Japanese record producer, lyricist, and television writer, best known for creating and producing some of Japan's top idol groups, Onyanko Club and the AKB48 franchise.
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.
