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, eix, 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.
Overviewedit
The cis notation is a shorthand for the combination of functions on the right-hand side of Euler's formula:
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.
The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another.[36] 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).[32]
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 eix. As students learn concepts that build on prior knowledge, it is important not to force them into levels of math for which they are not yet prepared: the usual proof that cis x = eix 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.
^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.
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^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)
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^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.
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^ ab"std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Archived from the original on 2023-07-16. Retrieved 2016-01-14.
^ ab"Installation Guide and Release Notes" (PDF). Intel Fortran Compiler Professional Edition 11.0 for Linux (11.0 ed.). 2008-11-06. Retrieved 2016-01-15.[permanent dead link]
^"CIS". Haskell reference. ZVON. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
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^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.
^ abFuchs, 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 [...] ([1], [2][3]) (NB. This work was published posthumously, Hamilton died in 1865.)
^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.
^ abCajori, 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.)
^ abCampbell, George Ashley (1928-10-01) [1927-09-13]. "The Practical Application of the Fourier Integral" (PDF). The Bell Systems 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 ei2πft. 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)
^Ahangar, Reza R. (2017-02-17) [2016-12-30, 2017-02-14]. "The Relativistic Geometry of the Complex Matter Space" (PDF). Journal of Applied Mathematics and Physics. 5 (2). Texas A&M University, Kingsville, Texas, USA: Scientific Research Publishing: 422–438 [428–431]. doi:10.4236/jamp.2017.52037. eISSN 2327-4379. ISSN 2327-4352. S2CID 86868092. Archived from the original (PDF) on 2023-07-27. Retrieved 2023-07-27. p. 431: [...] the complex number z in [Complex Matter Space] can be described symbolically by z = r (cosh(Φ) + i sinh(Φ)) = r * CISH(Φ) if |x| > |y| [...] z = r (sinh(Φ) + i cosh(Φ)) = r * SICH(Φ) if |x| < |y| [...] [4] (17 pages)
^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 eiφ 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 eiφ verwendet. [...] (109 pages)