Cis (mathematics)

Last updated September 19, 2024

$cis$ is a mathematical notation defined by $cis x = cos x + i sin x$ ,^{[nb 1]} 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, $e ix$ , 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.

Overview

The $cis$ notation is a shorthand for the combination of functions on the right-hand side of Euler's formula:

e^{ix}=\cos x+i\sin x,

where $i 2 = -1$ . So,

\operatorname {cis} x=\cos x+i\sin x,

^[1]^[2]^[3]^[4]

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 the domain of definition is usually $x\in \mathbb {R}$ , complex values $z\in \mathbb {C}$ are possible as well:

\operatorname {cis} z=\cos z+i\sin z,

so the $cis$ function can be used to extend Euler's formula to a more general complex version.^[5]

The function is mostly used as a convenient shorthand notation to simplify some expressions,^[6]^[7]^[8] for example in conjunction with Fourier and Hartley transforms,^[9]^[10]^[11] or when exponential functions shouldn't be used for some reason in math education.

In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL)^[12] or MathCW^[13]), available for many compilers and programming languages (including C, C++,^[14] Common Lisp,^[15]^[16] D,^[17] Haskell,^[18] Julia,^[19] and Rust ^[20]). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually.^[17]^[3]

Mathematical identities

Derivative

{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cis} z=i\operatorname {cis} z=ie^{iz}

^[1]^[21]

Integral

\int \operatorname {cis} z\,\mathrm {d} z=-i\operatorname {cis} z=-ie^{iz}

^[1]

Other properties

These follow directly from Euler's formula.

\cos(x)={\frac {\operatorname {cis} (x)+\operatorname {cis} (-x)}{2}}={\frac {e^{ix}+e^{-ix}}{2}}

\sin(x)={\frac {\operatorname {cis} (x)-\operatorname {cis} (-x)}{2i}}={\frac {e^{ix}-e^{-ix}}{2i}}

\operatorname {cis} (x+y)=\operatorname {cis} x\,\operatorname {cis} y

^[22]

\operatorname {cis} (x-y)={\operatorname {cis} x \over \operatorname {cis} y}

The identities above hold if $x$ and $y$ are any complex numbers. If $x$ and $y$ are real, then

|\operatorname {cis} x-\operatorname {cis} y|\leq |x-y|.

^[22]

History

The $cis$ notation was first coined by William Edwin Hamilton in Elements of Quaternions (1866)^[23]^[24] and subsequently used by Irving Stringham (who also called it "sector of $x$ ") in works such as Uniplanar Algebra (1893),^[25]^[26] James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898),^[26]^[27] or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928).^[28]^[29]^[30]

In 1942, inspired by 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:^[31]^[32]

\operatorname {cas} x=\cos x+\sin x.

Motivation

The $cis$ notation is sometimes used to emphasize one method of viewing and dealing with a problem over another.^[33] The mathematics of trigonometry and exponentials are related but not exactly 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 $cos + i sin$ ).^[30]

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 the notation $e ix$ . The usual proof that $cis x = e ix$ requires calculus, which the student may not have studied before encountering the expression $cos x + i sin x$ .

This notation was more common when typewriters were used to convey mathematical expressions.^{[ citation needed ]}

Notes

↑ Here, $i$ refers to the imaginary unit in mathematics. Since $i$ is commonly used to denote electric current in electrical engineering and control systems engineering, the imaginary unit is alternatively denoted by $j$ instead of $i$ in these contexts. Regardless of context, this does not affect the established name of the function as $cis$ .

Related Research Articles

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted $i$ , called the imaginary unit and satisfying the equation $; every complex number can be expressed in the form, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number, a is called the real part, and b is called the imaginary part . The set of complex numbers is denoted by either of the symbols or C . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.$

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number $x$ , one has $where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x . The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case.$

<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

The exponential function is a mathematical function denoted by $or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".$

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer $n$ ,

In mathematics, the Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable $.$

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In mathematics, Euler's identity is the equality $where$

<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation $x 2 + 1 = 0.$ Although there is no real number with this property, $i$ can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of $i$ in a complex number is $2 + 3 i .$

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points $(cos t, sin t)$ form a circle with a unit radius, the points $(cosh t, sinh t)$ form the right half of the unit hyperbola. Also, similarly to how the derivatives of $sin(t)$ and $cos(t)$ are $cos(t)$ and $-sin(t)$ respectively, the derivatives of $sinh(t)$ and $cosh(t)$ are $cosh(t)$ and $+sinh(t)$ respectively.

<span class="mw-page-title-main">Fourier series</span> Decomposition of periodic functions into sums of simpler sinusoidal forms

A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.

<span class="mw-page-title-main">Exponentiation</span> Arithmetic operation

In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as $b n$ , where $b$ is the base and $n$ is the power; this is pronounced as " $b$ (raised) to the $n$ ". When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, $b n$ is the product of multiplying $n$ bases:

The Fresnel integrals $S (x)$ and $C (x)$ are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function ( $erf$ ). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

In the physical sciences, the Airy function (or Airy function of the first kind) $Ai(x)$ is a special function named after the British astronomer George Biddell Airy (1801–1892). The function $Ai(x)$ and the related function $Bi(x)$ , are linearly independent solutions to the differential equation $known as the Airy equation or the Stokes equation .$

In mathematics, the Clausen function, introduced by Thomas Clausen, is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.

<span class="mw-page-title-main">Tetration</span> Arithmetic operation

In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's up arrow notation $and the left-exponent x b are common.$

In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions and of being its own inverse.

In mathematics, physics and engineering, the sinc function, denoted by $sinc(x)$ , has two forms, normalized and unnormalized.

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 angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $, the sine and cosine functions are denoted as and .$

References

1 2 3 Weisstein, Eric Wolfgang (2015) [2000]. "Cis". MathWorld . Wolfram Research, Inc. Archived from the original on 2016-01-27. Retrieved 2016-01-09.
↑ Simmons, Bruce (2014-07-28) [2004]. "Cis". Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, Oregon, USA: Clackamas Community College, Mathematics Department. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
1 2 "Rationale for International Standard - Programming Languages - C" (PDF). 5.10. April 2003. pp. 114, 117, 183, 186–187. Archived (PDF) from the original on 2016-06-06. Retrieved 2010-10-17.
↑ Amann, Herbert [at Wikidata]; Escher, Joachim [in German] (2006). Analysis I. Grundstudium Mathematik (in German) (3 ed.). Basel, Switzerland: Birkhäuser Verlag. pp. 292, 298. ISBN 978-3-76437755-7. ISBN 3-76437755-0. (445 pages)
↑ Moskowitz, Martin A. (2002). "Chapter 1. First Concepts". Written at City University of New York Graduate Center, New York, USA. A Course in Complex Analysis in One Variable. Singapore: World Scientific Publishing Co. Pte. Ltd. p. 7. ISBN 981-02-4780-X. (ix+149 pages)
↑ Swokowski, Earl William [at Wikidata]; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 978-0-84006857-6 . Retrieved 2016-01-18.
↑ Reis, Clive (2011). Abstract Algebra: An Introduction to Groups, Rings and Fields (1 ed.). World Scientific Publishing Co. Pte. Ltd. pp. 434–438. ISBN 978-9-81433564-5.
↑ Weitz, Edmund [in German] (2016). "The fundamental theorem of algebra - a visual proof". Hamburg, Germany: Hamburg University of Applied Sciences (HAW), Department Medientechnik. Archived from the original on 2019-08-03. Retrieved 2019-08-03.
↑ L.-Rundblad, Ekaterina; Maidan, Alexei; Novak, Peter; Labunets, Valeriy (2004). "Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing". Written at Prometheus Inc., Newport, USA. In Byrnes, Jim (ed.). Computational Noncommutative Algebra and Applications (PDF). NATO Science Series II: Mathematics, Physics and Chemistry (NAII). Vol. 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. pp. 401–411. doi:10.1007/1-4020-2307-3. ISBN 978-1-4020-1982-1. ISSN 1568-2609. Archived (PDF) from the original on 2017-10-28. Retrieved 2017-10-28.
↑ Kammler, David W. (2008-01-17). A First Course in Fourier Analysis (2 ed.). Cambridge University Press. ISBN 978-1-13946903-6 . Retrieved 2017-10-28.
↑ Lorenzo, Carl F.; Hartley, Tom T. (2016-11-14). The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science. John Wiley & Sons. ISBN 978-1-11913942-3 . Retrieved 2017-10-28.
↑ "v?CIS". Developer Reference for Intel Math Kernel Library (Intel MKL) 2017 - C. MKL documentation; IDZ Documentation Library. Intel Corporation. 2016-09-06. p. 1799. 671504. Retrieved 2016-01-15.
↑ Beebe, Nelson H. F. (2017-08-22). "Chapter 15.2. Complex absolute value". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, Utah, USA: Springer International Publishing AG. p. 443. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721.
↑ "Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.
↑ "CIS". Common Lisp Hyperspec . The Harlequin Group Limited. 1996. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
↑ "CIS". LispWorks, Ltd. 2005 [1996]. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
1 2 "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Archived from the original on 2023-07-16. Retrieved 2016-01-14.
↑ "CIS". Haskell reference. ZVON. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
↑ "Mathematics; Mathematical Operators". The Julia Language. Archived from the original on 2020-08-19. Retrieved 2019-12-05.
↑ "Struct num_complex::Complex". Archived from the original on 2023-07-16. Retrieved 2022-08-05.
↑ Fuchs, Martin (2011). "Chapter 11: Differenzierbarkeit von Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´. pp. 3, 13. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.
1 2 Fuchs, Martin (2011). "Chapter 8.IV: Spezielle Funktionen – Die trigonometrischen Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany. pp. 16–20. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.
↑ Hamilton, William Rowan (1866-01-01). "Book II, Chapter II. Fractional powers, General roots of unity". Written at Dublin, Irland. In Hamilton, William Edwin (ed.). Elements of Quaternions (1 ed.). London, UK: Longmans, Green & Co., University Press, Michael Henry Gill. pp. 250–257, 260, 262–263. Retrieved 2016-01-17. pp. 250, 252: [...] cos [...] + i sin [...] we shall occasionally abridge to the following: [...] cis [...]. As to the marks [...], they are to be considered as chiefly available for the present exposition of the system, and as not often wanted, nor employed, in the subsequent practise thereof; and the same remark applies to the recent abrigdement cis, for cos + i sin [...] (, ) (NB. This work was published posthumously, Hamilton died in 1865.)
↑ Hamilton, William Rowan (1899) [1866-01-01]. Hamilton, William Edwin; Joly, Charles Jasper (eds.). Elements of Quaternions. Vol. I (2 ed.). London, UK: Longmans, Green & Co. p. 262. Retrieved 2019-08-03. p. 262: [...] recent abridgment cis for cos + i sin [...] (NB. This edition was reprinted by Chelsea Publishing Company in 1969.)
↑ Stringham, Irving (1893-07-01) [1891]. Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis. Vol. 1. C. A. Mordock & Co. (printer) (1 ed.). San Francisco, California, USA: The Berkeley Press. pp. 71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135. Retrieved 2016-01-18. p. 71: As an abbreviation for cos θ + i sin θ it is convenient to use cis θ, which may be read: sector of θ.
1 2 Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, Illinois, USA: Open court publishing company. p. 133. ISBN 978-1-60206-714-1 . Retrieved 2016-01-18. p. 133: Stringham denoted cos β + i sin β by "cis β", a notation also used by Harkness and Morley. (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, US, 2013.)
↑ Harkness, James; Morley, Frank (1898). Introduction to the Theory of Analytic Functions (1 ed.). London, UK: Macmillan and Company. pp. 18, 22, 48, 52, 170. ISBN 978-1-16407019-1 . Retrieved 2016-01-18. (NB. ISBN for reprint by Kessinger Publishing, 2010.)
↑ Campbell, George Ashley (1903) [1901-06-07]. "Chapter XXX. On loaded lines in telephonic transmission" (PDF). The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . Series 6. 5 (27). Taylor & Francis: 313–330. doi:10.1080/14786440309462928. Archived (PDF) from the original on 2023-07-16. Retrieved 2023-07-16. (2+18 pages)
↑ Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers . XXX (1–6). American Institute of Electrical Engineers: 789–824. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814 . Retrieved 2023-06-24. (37 pages)
1 2 Campbell, George Ashley (1928-10-01) [1927-09-13]. "The Practical Application of the Fourier Integral" (PDF). The Bell System Technical Journal . 7 (4). American Telephone and Telegraph Company: 639–707 [641]. doi:10.1002/j.1538-7305.1928.tb00347.x. S2CID 53552671 . Retrieved 2023-06-24. p. 641: It has been recognized, almost from the start, however, that the form which best combines mathematical simplicity and complete generality makes use of the exponential oscillating function e^i2πft. More recently the overwhelming advantage of using this oscillating function in the discussion of sinusoidal oscillatory systems has been generally recognized. It is, therefore, plain that this oscillating function should be adopted as the basic oscillation for both of the proposed tables. A name for this oscillation, associating it with sines and cosines, rather than with the real exponential function, seems desirable. The abbreviation cis x for (cos x + i sin x) suggests that we name this function a cis or a cisoidal oscillation. (69 pages)
↑ Hartley, Ralph V. L. (March 1942). "A More Symmetrical Fourier Analysis Applied to Transmission Problems". Proceedings of the IRE . 30 (3). Institute of Radio Engineers: 144–150. doi:10.1109/JRPROC.1942.234333. S2CID 51644127. Archived from the original on 2019-04-05. Retrieved 2023-07-16.
↑ Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.
↑ Diehl, Christina; Leupp, Marcel (January 2010). Komplexe Zahlen: Ein Leitprogramm in Mathematik (PDF) (in German). Basel & Herisau, Switzerland: Eidgenössische Technische Hochschule Zürich (ETH). p. 41. Archived (PDF) from the original on 2017-08-27. p. 41: [...] Bitte vergessen Sie aber nicht, dass e^iφ für uns bisher nur eine Schreibweise für den Einheitszeiger mit Winkel φ ist. In anderen Büchern wird dafür oft der Ausdruck cis(φ) anstelle von e^iφ verwendet. [...] (109 pages)

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[NB_ij-1] Here, $i$ refers to the imaginary unit in mathematics. Since $i$ is commonly used to denote electric current in electrical engineering and control systems engineering, the imaginary unit is alternatively denoted by $j$ instead of $i$ in these contexts. Regardless of context, this does not affect the established name of the function as $cis$ .

[Weisstein_cis-2] 1 2 3 Weisstein, Eric Wolfgang (2015) [2000]. "Cis". MathWorld . Wolfram Research, Inc. Archived from the original on 2016-01-27. Retrieved 2016-01-09.

[Simmons_2014_1-3] Simmons, Bruce (2014-07-28) [2004]. "Cis". Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, Oregon, USA: Clackamas Community College, Mathematics Department. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[Rationale_2003_C-4] 1 2 "Rationale for International Standard - Programming Languages - C" (PDF). 5.10. April 2003. pp. 114, 117, 183, 186–187. Archived (PDF) from the original on 2016-06-06. Retrieved 2010-10-17.

[Amann-Escher_2006-5] Amann, Herbert [at Wikidata]; Escher, Joachim [in German] (2006). Analysis I. Grundstudium Mathematik (in German) (3 ed.). Basel, Switzerland: Birkhäuser Verlag. pp. 292, 298. ISBN 978-3-76437755-7. ISBN 3-76437755-0. (445 pages)

[Moskowitz_2002-6] Moskowitz, Martin A. (2002). "Chapter 1. First Concepts". Written at City University of New York Graduate Center, New York, USA. A Course in Complex Analysis in One Variable. Singapore: World Scientific Publishing Co. Pte. Ltd. p. 7. ISBN 981-02-4780-X. (ix+149 pages)

[Swokowski_2011-7] Swokowski, Earl William [at Wikidata]; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 978-0-84006857-6 . Retrieved 2016-01-18.

[Reis_2011-8] Reis, Clive (2011). Abstract Algebra: An Introduction to Groups, Rings and Fields (1 ed.). World Scientific Publishing Co. Pte. Ltd. pp. 434–438. ISBN 978-9-81433564-5.

[Weitz_2016-9] Weitz, Edmund [in German] (2016). "The fundamental theorem of algebra - a visual proof". Hamburg, Germany: Hamburg University of Applied Sciences (HAW), Department Medientechnik. Archived from the original on 2019-08-03. Retrieved 2019-08-03.

[Byrnes_2004-10] L.-Rundblad, Ekaterina; Maidan, Alexei; Novak, Peter; Labunets, Valeriy (2004). "Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing". Written at Prometheus Inc., Newport, USA. In Byrnes, Jim (ed.). Computational Noncommutative Algebra and Applications (PDF). NATO Science Series II: Mathematics, Physics and Chemistry (NAII). Vol. 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. pp. 401–411. doi:10.1007/1-4020-2307-3. ISBN 978-1-4020-1982-1. ISSN 1568-2609. Archived (PDF) from the original on 2017-10-28. Retrieved 2017-10-28.

[Kammler_2008-11] Kammler, David W. (2008-01-17). A First Course in Fourier Analysis (2 ed.). Cambridge University Press. ISBN 978-1-13946903-6 . Retrieved 2017-10-28.

[Lorenzo-Hartley_2016-12] Lorenzo, Carl F.; Hartley, Tom T. (2016-11-14). The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science. John Wiley & Sons. ISBN 978-1-11913942-3 . Retrieved 2017-10-28.

[Intel_MKL-13] "v?CIS". Developer Reference for Intel Math Kernel Library (Intel MKL) 2017 - C. MKL documentation; IDZ Documentation Library. Intel Corporation. 2016-09-06. p. 1799. 671504. Retrieved 2016-01-15.

[Beebe_2017-14] Beebe, Nelson H. F. (2017-08-22). "Chapter 15.2. Complex absolute value". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, Utah, USA: Springer International Publishing AG. p. 443. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721.

[Intel_2007_C++-15] "Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.

[Lisp-16] "CIS". Common Lisp Hyperspec . The Harlequin Group Limited. 1996. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[Lisp_2005-17] "CIS". LispWorks, Ltd. 2005 [1996]. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[D_2011-18] 1 2 "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Archived from the original on 2023-07-16. Retrieved 2016-01-14.

[Haskell-19] "CIS". Haskell reference. ZVON. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[Julia-20] "Mathematics; Mathematical Operators". The Julia Language. Archived from the original on 2020-08-19. Retrieved 2019-12-05.

[Rust-21] "Struct num_complex::Complex". Archived from the original on 2023-07-16. Retrieved 2022-08-05.

[Fuchs_2011_2-22] Fuchs, Martin (2011). "Chapter 11: Differenzierbarkeit von Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´. pp. 3, 13. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.

[Fuchs_2011_1-23] 1 2 Fuchs, Martin (2011). "Chapter 8.IV: Spezielle Funktionen – Die trigonometrischen Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany. pp. 16–20. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.

[Hamilton_1866-24] Hamilton, William Rowan (1866-01-01). "Book II, Chapter II. Fractional powers, General roots of unity". Written at Dublin, Irland. In Hamilton, William Edwin (ed.). Elements of Quaternions (1 ed.). London, UK: Longmans, Green & Co., University Press, Michael Henry Gill. pp. 250–257, 260, 262–263. Retrieved 2016-01-17. pp. 250, 252: [...] cos [...] + i sin [...] we shall occasionally abridge to the following: [...] cis [...]. As to the marks [...], they are to be considered as chiefly available for the present exposition of the system, and as not often wanted, nor employed, in the subsequent practise thereof; and the same remark applies to the recent abrigdement cis, for cos + i sin [...] (, ) (NB. This work was published posthumously, Hamilton died in 1865.)

[Hamilton_1899-25] Hamilton, William Rowan (1899) [1866-01-01]. Hamilton, William Edwin; Joly, Charles Jasper (eds.). Elements of Quaternions. Vol. I (2 ed.). London, UK: Longmans, Green & Co. p. 262. Retrieved 2019-08-03. p. 262: [...] recent abridgment cis for cos + i sin [...] (NB. This edition was reprinted by Chelsea Publishing Company in 1969.)

[Stringham_1893-26] Stringham, Irving (1893-07-01) [1891]. Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis. Vol. 1. C. A. Mordock & Co. (printer) (1 ed.). San Francisco, California, USA: The Berkeley Press. pp. 71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135. Retrieved 2016-01-18. p. 71: As an abbreviation for cos θ + i sin θ it is convenient to use cis θ, which may be read: sector of θ.

[Cajori_1929-27] 1 2 Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, Illinois, USA: Open court publishing company. p. 133. ISBN 978-1-60206-714-1 . Retrieved 2016-01-18. p. 133: Stringham denoted cos β + i sin β by "cis β", a notation also used by Harkness and Morley. (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, US, 2013.)

[Harkness_Morley_1898-28] Harkness, James; Morley, Frank (1898). Introduction to the Theory of Analytic Functions (1 ed.). London, UK: Macmillan and Company. pp. 18, 22, 48, 52, 170. ISBN 978-1-16407019-1 . Retrieved 2016-01-18. (NB. ISBN for reprint by Kessinger Publishing, 2010.)

[Campbell_1901-29] Campbell, George Ashley (1903) [1901-06-07]. "Chapter XXX. On loaded lines in telephonic transmission" (PDF). The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . Series 6. 5 (27). Taylor & Francis: 313–330. doi:10.1080/14786440309462928. Archived (PDF) from the original on 2023-07-16. Retrieved 2023-07-16. (2+18 pages)

[Campbell_1911-30] Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers . XXX (1–6). American Institute of Electrical Engineers: 789–824. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814 . Retrieved 2023-06-24. (37 pages)

[Campbell_1928-31] 1 2 Campbell, George Ashley (1928-10-01) [1927-09-13]. "The Practical Application of the Fourier Integral" (PDF). The Bell System Technical Journal . 7 (4). American Telephone and Telegraph Company: 639–707 [641]. doi:10.1002/j.1538-7305.1928.tb00347.x. S2CID 53552671 . Retrieved 2023-06-24. p. 641: It has been recognized, almost from the start, however, that the form which best combines mathematical simplicity and complete generality makes use of the exponential oscillating function e^i2πft. More recently the overwhelming advantage of using this oscillating function in the discussion of sinusoidal oscillatory systems has been generally recognized. It is, therefore, plain that this oscillating function should be adopted as the basic oscillation for both of the proposed tables. A name for this oscillation, associating it with sines and cosines, rather than with the real exponential function, seems desirable. The abbreviation cis x for (cos x + i sin x) suggests that we name this function a cis or a cisoidal oscillation. (69 pages)

[Hartley_1942-32] Hartley, Ralph V. L. (March 1942). "A More Symmetrical Fourier Analysis Applied to Transmission Problems". Proceedings of the IRE . 30 (3). Institute of Radio Engineers: 144–150. doi:10.1109/JRPROC.1942.234333. S2CID 51644127. Archived from the original on 2019-04-05. Retrieved 2023-07-16.

[Bracewell_1999-33] Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.

[Diehl-Leupp_2010-34] Diehl, Christina; Leupp, Marcel (January 2010). Komplexe Zahlen: Ein Leitprogramm in Mathematik (PDF) (in German). Basel & Herisau, Switzerland: Eidgenössische Technische Hochschule Zürich (ETH). p. 41. Archived (PDF) from the original on 2017-08-27. p. 41: [...] Bitte vergessen Sie aber nicht, dass e^iφ für uns bisher nur eine Schreibweise für den Einheitszeiger mit Winkel φ ist. In anderen Büchern wird dafür oft der Ausdruck cis(φ) anstelle von e^iφ verwendet. [...] (109 pages)

[nb 1]

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]