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Nicholas of Worcester (died 1124) was the prior of the Benedictine priory of Worcester Cathedral (crypt pictured) from about 1115 until his death. He was born around the time of the Norman Conquest.It is not known who his parents were, but William of Malmesbury wrote that he was "of exalted descent", and it has been argued that he was a son of King Harold Godwinson.
Wikipedia[note 3] is a free content online encyclopedia written and maintained by a community of volunteers, known as Wikipedians, through open collaboration and the use of the wiki-based editing system MediaWiki. Wikipedia is the largest and most-read reference work in history.[3][4] It is consistently ranked as one of the ten most popular ...
The 2020 United States presidential election was the 59th quadrennial presidential election, held on Tuesday, November 3, 2020.[a] The Democratic ticket of former vice president Joe Biden and the junior U.S. senator from California Kamala Harris defeated the incumbent Republican president, Donald Trump, and vice president, Mike Pence.[9] The ...
Apple Inc. (formerly Apple Computer, Inc.) is an American multinational corporation and technology company headquartered in Cupertino, California, in Silicon Valley.It designs, develops, and sells consumer electronics, computer software, and online services.Devices include the iPhone, iPad, Mac, Apple Watch, Vision Pro, and Apple TV; operating systems include iOS, iPadOS, and macOS; and ...
- 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.Interview with the Vampire is a 1994 American gothic horror film directed by Neil Jordan, based on Anne Rice's 1976 novel of the same name, and starring Tom Cruise and Brad Pitt. It focuses on Lestat (Cruise) and Louis (Pitt), beginning with Louis' transformation into a vampire by Lestat in 1791. The film chronicles their time together, and ...
Google Chrome is a web browser developed by Google. It was first released in 2008 for Microsoft Windows, built with free software components from Apple WebKit and Mozilla Firefox.[16] Versions were later released for Linux, macOS, iOS, and also for Android, where it is the default browser.[17] The browser is also the main component of ChromeOS ...
