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Vincent Willem van Gogh ( Dutch: [ˈvɪnsɛnt ˈʋɪləɱ‿vɑŋ‿ˈɣɔx] ⓘ; [note 1] 30 March 1853 – 29 July 1890) was a Dutch Post-Impressionist painter who is among the most famous and influential figures in the history of Western art. In just over a decade, he created approximately 2100 artworks, including around 860 oil paintings ...
Euarctos americanus. The American black bear ( Ursus americanus ), also known as the black bear, is a species of medium-sized bear endemic to North America. It is the continent's smallest and most widely distributed bear species. The American black bear is an omnivore, with a diet varying greatly depending on season and location.
- 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.David Scott Portnoy (born March 22, 1977) is an American businessman and social media personality. He is the founder and owner of sports and popular culture company Barstool Sports. Early life Portnoy grew up in Swampscott, Massachusetts, the son of Michael, a lawyer, and Linda, a high school teacher.
Grigori Yefimovich Rasputin ( / ræˈspjuːtɪn /; Russian: Григорий Ефимович Распутин [ɡrʲɪˈɡorʲɪj jɪˈfʲiməvʲɪtɕ rɐˈsputʲɪn]; 21 January [ O.S. 9 January] 1869 – 30 December [ O.S. 17 December] 1916) was a Russian mystic and faith healer. He is best known for having befriended the imperial family of ...
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