cis is a mathematical notation defined by cis x = cos x + i sin x , where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number (angle between line to point and x-axis in polar form ). The notation is less commonly used in mathematics than Euler's formula , e , which offers an even shorter notation for cos x + i sin x , but cis(x) is widely used as a name for this function in software libraries .
12-539: [REDACTED] Look up cis or cis- in Wiktionary, the free dictionary. Cis or cis- may refer to: Places [ edit ] Cis, Trentino , in Italy In Poland: Cis, Świętokrzyskie Voivodeship , south-central Cis, Warmian-Masurian Voivodeship , north Math, science and biology [ edit ] cis (mathematics) (cis( θ )),
24-621: A trigonometric mathematical function related to Euler's formula Cis (beetle) , genus Cis–trans isomerism , in chemistry cis-regulatory element , regions of non-coding DNA which regulate the transcription of nearby genes Other uses [ edit ] Cisgender , a descriptor for somebody whose gender identity matches their assigned gender at birth C♯ (musical note) , known as cis in some European notations See also [ edit ] CIS (disambiguation) All pages with titles beginning with Cis Ciss (disambiguation) Csi (disambiguation) Topics referred to by
36-621: A trigonometric mathematical function related to Euler's formula Cis (beetle) , genus Cis–trans isomerism , in chemistry cis-regulatory element , regions of non-coding DNA which regulate the transcription of nearby genes Other uses [ edit ] Cisgender , a descriptor for somebody whose gender identity matches their assigned gender at birth C♯ (musical note) , known as cis in some European notations See also [ edit ] CIS (disambiguation) All pages with titles beginning with Cis Ciss (disambiguation) Csi (disambiguation) Topics referred to by
48-537: Is different from Wikidata All article disambiguation pages All disambiguation pages cis [REDACTED] Look up cis or cis- in Wiktionary, the free dictionary. Cis or cis- may refer to: Places [ edit ] Cis, Trentino , in Italy In Poland: Cis, Świętokrzyskie Voivodeship , south-central Cis, Warmian-Masurian Voivodeship , north Math, science and biology [ edit ] cis (mathematics) (cis( θ )),
60-434: Is different from Wikidata All article disambiguation pages All disambiguation pages Cis (mathematics) The cis notation is a shorthand for the combination of functions on the right-hand side of Euler's formula : where i = −1 . So, i.e. " cis " is an acronym for " Cos i Sin ". It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While
72-405: The cis notation, Ralph V. L. Hartley introduced the cas (for cosine-and-sine ) function for the real-valued Hartley kernel , a meanwhile established shortcut in conjunction with Hartley transforms : The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. The mathematics of trigonometry and exponentials are related but not exactly
84-626: The domain of definition is usually x ∈ R {\displaystyle x\in \mathbb {R} } , complex values z ∈ C {\displaystyle z\in \mathbb {C} } are possible as well: so the cis function can be used to extend Euler's formula to a more general complex version . The function is mostly used as a convenient shorthand notation to simplify some expressions, for example in conjunction with Fourier and Hartley transforms , or when exponential functions shouldn't be used for some reason in math education. In information technology,
96-572: The function sees dedicated support in various high-performance math libraries (such as Intel 's Math Kernel Library (MKL) or MathCW ), available for many compilers and programming languages (including C , C++ , Common Lisp , D , Haskell , Julia , and Rust ). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually. These follow directly from Euler's formula . The identities above hold if x and y are any complex numbers. If x and y are real, then The cis notation
108-442: The same term [REDACTED] This disambiguation page lists articles associated with the title Cis . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Cis&oldid=1241959137 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description
120-442: The same term [REDACTED] This disambiguation page lists articles associated with the title Cis . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Cis&oldid=1241959137 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description
132-464: The same; exponential notation emphasizes the whole, whereas cis x and cos x + i sin x notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for cos + i sin ). The cis notation is convenient for math students whose knowledge of trigonometry and complex numbers permit this notation, but whose conceptual understanding does not yet permit
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#1732855298231144-656: Was first coined by William Edwin Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham (who also called it " sector of x ") in works such as Uniplanar Algebra (1893), James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898), or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928). In 1942, inspired by
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