cis (mathematics)

{{Short description|Alternate mathematical notation for cos x + i sin x}}

{{Redir|Cisoidal oscillation|the plane curve generated from two curves|Cissoid}}

{{math|cis}} is a mathematical notation defined by {{math|1=cis x = cos x + i sin x}}, where {{math|cos}} is the cosine function, {{mvar|i}} is the imaginary unit and {{math|sin}} is the sine function. {{mvar|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, {{nowrap|{{math|e^ix}},}} which offers an even shorter notation for {{nowrap|{{math|cos x + i sin x}},}} but cis(x) is widely used as a name for this function in software libraries.

Overview

The {{math|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 {{math|i² {{=}} −1}}. So,

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

i.e. "{{math|cis}}" is an acronym for "{{math|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 {{math|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, 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.

Mathematical identities

= Derivative =

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

= Integral =

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

= Other properties =

These follow directly from Euler's formula.

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

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

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

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

The identities above hold if {{mvar|x}} and {{mvar|y}} are any complex numbers. If {{mvar|x}} and {{mvar|y}} are real, then

: $|\operatorname{cis} x - \operatorname{cis} y| \le |x-y|.$

History

The {{math|cis}} notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham (who also called it "sector of {{math|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).

{{anchor|cas}}In 1942, inspired by the {{math|cis}} notation, Ralph V. L. Hartley introduced the {{math|cas}} (for cosine-and-sine) function for the real-valued Hartley kernel, a meanwhile established shortcut in conjunction with Hartley transforms:

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

Motivation

The {{math|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 the same; exponential notation emphasizes the whole, whereas {{math|cis x}} and {{math|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 {{math|cos + i sin}}).

The {{math|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 {{math|e^ix}}. The usual proof that {{math|1=cis x = e^ix}} requires calculus, which the student may not have studied before encountering the expression {{math|cos x + i sin x}}.

This notation was more common when typewriters were used to convey mathematical expressions.{{cn|date=September 2024}}

Notes

{{reflist|group="nb"|refs=

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

}}

References

{{reflist|refs=

{{cite book |title=Computational Noncommutative Algebra and Applications |editor-first=Jim |editor-last=Byrnes |author-first1=Ekaterina |author-last1=L.-Rundblad |author-first2=Alexei |author-last2=Maidan |author-first3=Peter |author-last3=Novak |author-first4=Valeriy |author-last4=Labunets |chapter=Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing |series=NATO Science Series II: Mathematics, Physics and Chemistry (NAII) |volume=136 |publisher=Springer Science + Business Media, Inc. |location=Prometheus Inc., Newport, USA |publication-place=Dordrecht, Netherlands |date=2004 |isbn=978-1-4020-1982-1 |issn=1568-2609 |doi=10.1007/1-4020-2307-3 |pages=401–411 |url=http://web.cecs.pdx.edu/~mperkows/CAPSTONES/Quaternion/katya2.pdf |access-date=2017-10-28 |url-status=live |archive-url=https://web.archive.org/web/20171028130812/http://web.cecs.pdx.edu/~mperkows/CAPSTONES/Quaternion/katya2.pdf |archive-date=2017-10-28}}

{{cite web |title=Cis |author-first=Bruce |author-last=Simmons |date=2014-07-28 |orig-date=2004 |work=Mathwords: Terms and Formulas from Algebra I to Calculus |publisher=Clackamas Community College, Mathematics Department |location=Oregon City, Oregon, USA |url=http://www.mathwords.com/c/cis.htm |access-date=2016-01-15 |url-status=live |archive-url=https://web.archive.org/web/20230716150005/http://www.mathwords.com/c/cis.htm |archive-date=2023-07-16}}

{{cite book |title=A First Course in Fourier Analysis |author-first=David W. |author-last=Kammler |publisher=Cambridge University Press |date=2008-01-17 |edition=2 |isbn=978-1-13946903-6 |url=https://books.google.com/books?id=znP-ADtE8ZQC |access-date=2017-10-28}}

{{cite book |title=The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science |author-first1=Carl F. |author-last1=Lorenzo |author-first2=Tom T. |author-last2=Hartley |publisher=John Wiley & Sons |date=2016-11-14 |isbn=978-1-11913942-3 |url=https://books.google.com/books?id=LdyADQAAQBAJ |access-date=2017-10-28}}

{{cite book |title=Elements of Quaternions |author-first=William Rowan |author-last=Hamilton |author-link=William Rowan Hamilton |date=1866-01-01 |edition=1 |editor-first=William Edwin |editor-last=Hamilton |editor-link=William Edwin Hamilton |publisher=Longmans, Green & Co., University Press, Michael Henry Gill |publication-place=London, UK |location=Dublin, Irland |chapter=Book II, Chapter II. Fractional powers, General roots of unity |pages=250–257, 260, 262–263 |chapter-url=https://archive.org/details/elementsquaterni00hamirich/page/n323 |access-date=2016-01-17 |quote-pages=250, 252 |quote=[...] {{nowrap|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 {{nowrap|cos + i sin}} [...]}} ([https://archive.org/stream/elementsquaterni00hamirich#page/n0/mode/1up], [https://archive.org/details/elementsquatern02hamigoog][https://books.google.com/books?id=b2stAAAAYAAJ]) (NB. This work was published posthumously, Hamilton died in 1865.)

{{cite book |title=Elements of Quaternions |author-first=William Rowan |author-last=Hamilton |author-link=William Rowan Hamilton |editor-first1=William Edwin |editor-last1=Hamilton |editor-link1=William Edwin Hamilton |editor-first2=Charles Jasper |editor-last2=Joly |editor-link2=Charles Jasper Joly |date=1899 |orig-date=1866-01-01 |edition=2 |volume=I |publisher=Longmans, Green & Co. |location=London, UK |page=262 |url=https://archive.org/details/117770258_001 |access-date=2019-08-03 |quote-page=262 |quote=[...] recent abridgment cis for {{nowrap|cos + i sin}} [...]}} (NB. This edition was reprinted by Chelsea Publishing Company in 1969.)

{{anchor|Stringham-1893}}{{cite book |title=Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis |author-first=Irving |author-last=Stringham |author-link=Irving Stringham |date=1893-07-01 |orig-date=1891 |edition=1 |volume=1 |publisher=The Berkeley Press |others=C. A. Mordock & Co. (printer) |location=San Francisco, California, USA |pages=71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135 |url=https://archive.org/details/uniplanaralgebra00stri |access-date=2016-01-18 |quote-page=71 |quote=As an abbreviation for {{nowrap|cos θ + i sin θ}} it is convenient to use cis θ, which may be read: sector of θ.}}

{{cite book |author-first=Florian |author-last=Cajori |author-link=Florian Cajori |title=A History of Mathematical Notations |volume=2 |orig-date=March 1929 |publisher=Open court publishing company |location=Chicago, Illinois, USA |date=1952 |edition=3rd corrected printing of 1929 issue, 2nd |page=133 |isbn=978-1-60206-714-1 |url=https://books.google.com/books?id=bT5suOONXlgC |access-date=2016-01-18 |quote-page=133 |quote=Stringham denoted {{nowrap|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.)

{{anchor|Harkness-Morley-1898}}{{cite book |title=Introduction to the Theory of Analytic Functions |author-first1=James |author-last1=Harkness |author-link1=James Harkness (mathematician) |author-first2=Frank |author-last2=Morley |author-link2=Frank Morley |date=1898 |edition=1 |publisher=Macmillan and Company |location=London, UK |pages=[https://archive.org/details/cu31924059412910/page/n234 18], 22, 48, 52, 170 |url=https://archive.org/details/cu31924059412910 |access-date=2016-01-18 |isbn=978-1-16407019-1}} (NB. ISBN for reprint by Kessinger Publishing, 2010.)

{{cite book |title=Precalculus: Functions and Graphs |series=Precalculus Series |author-first1=Earl William |author-last1=Swokowski |author-link1=:d:Q59629142 |author-first2=Jeffery |author-last2=Cole |edition=12 |publisher=Cengage Learning |date=2011 |isbn=978-0-84006857-6 |url=https://books.google.com/books?id=8GB2Udf8wnoC |access-date=2016-01-18}}

{{cite book |title=Developer Reference for Intel Math Kernel Library (Intel MKL) 2017 - C |series=MKL documentation; IDZ Documentation Library |id=671504 |date=2016-09-06 |chapter=v?CIS |publisher=Intel Corporation |page=1799 |url=https://www.intel.com/content/www/us/en/content-details/671504/developer-reference-for-intel-math-kernel-library-intel-mkl-2017-c.html |access-date=2016-01-15}}

{{cite web |title=Intel C++ Compiler Reference |date=2007 |orig-date=1996 |publisher=Intel Corporation |pages=34, 59–60 |id=307777-004US |url=http://supercomputer.susu.ru/upload/users/instructions/C_Compiler_Reference.pdf |access-date=2016-01-15 |url-status=live |archive-url=https://web.archive.org/web/20230716150413/http://supercomputer.susu.ru/upload/users/instructions/C_Compiler_Reference.pdf |archive-date=2023-07-16}}

{{cite web |title=CIS |work=Common Lisp Hyperspec |date=1996 |publisher=The Harlequin Group Limited |url=http://www.ai.mit.edu/projects/iiip/doc/CommonLISP/HyperSpec/Body/fun_cis.html |access-date=2016-01-15 |url-status=live |archive-url=https://web.archive.org/web/20230716150421/http://www.ai.mit.edu/projects/iiip/doc/CommonLISP/HyperSpec/Body/fun_cis.html |archive-date=2023-07-16}}

{{cite web |title=CIS |publisher=LispWorks, Ltd. |date=2005 |orig-date=1996 |url=http://clhs.lisp.se/Body/f_cis.htm#cis |access-date=2016-01-15 |url-status=live |archive-url=https://web.archive.org/web/20230716150740/http://clhs.lisp.se/Body/f_cis.htm#cis |archive-date=2023-07-16}}

{{cite web |title=std.math: expi |work=D programming language |date=2016-01-11 |orig-date=2000 |publisher=Digital Mars |url=http://dlang.org/phobos/std_math.html#.expi |access-date=2016-01-14 |url-status=live |archive-url=https://web.archive.org/web/20230716150623/https://dlang.org/phobos/std_math.html#.expi |archive-date=2023-07-16}}

{{cite web |url=https://docs.julialang.org/en/v1.0/base/math/#Base.cis |title=Mathematics; Mathematical Operators |work=The Julia Language |access-date=2019-12-05 |url-status=live |archive-url=https://web.archive.org/web/20200819172952/https://docs.julialang.org/en/v1.0/base/math/ |archive-date=2020-08-19}}

{{cite web |title=Struct num_complex::Complex |url=https://docs.rs/num-complex/latest/num_complex/struct.Complex.html#method.cis |access-date=2022-08-05 |url-status=live |archive-url=https://web.archive.org/web/20230716150903/https://docs.rs/num-complex/latest/num_complex/struct.Complex.html#method.cis |archive-date=2023-07-16}}

{{cite web |title=CIS |work=Haskell reference |publisher=ZVON |url=http://zvon.org/other/haskell/Outputcomplex/cis_f.html |access-date=2016-01-15 |url-status=live |archive-url=https://web.archive.org/web/20230716151029/http://zvon.org/other/haskell/Outputcomplex/cis_f.html |archive-date=2023-07-16}}

{{cite web |title=Rationale for International Standard - Programming Languages - C |version=5.10 |date=April 2003 |pages=114, 117, 183, 186–187 |url=http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf |access-date=2010-10-17 |url-status=live |archive-url=https://web.archive.org/web/20160606072228/http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf |archive-date=2016-06-06}}

{{cite web |author-first=Eric Wolfgang |author-last=Weisstein |author-link=Eric Wolfgang Weisstein |title=Cis |work=MathWorld |publisher=Wolfram Research, Inc. |orig-date=2000 |date=2015 |url=http://mathworld.wolfram.com/Cis.html |access-date=2016-01-09 |url-status=live |archive-url=https://web.archive.org/web/20160127061403/http://mathworld.wolfram.com/Cis.html |archive-date=2016-01-27}}

{{cite book |title=Analysis I |language=German |author-first=Martin |author-last=Fuchs |date=2011 |publisher=Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´|chapter=Chapter 11: Differenzierbarkeit von Funktionen |edition=WS 2011/2012 |pages=3, 13 |url=http://www.math.uni-sb.de/ag/fuchs/Ana1/Paragraph11.pdf |access-date=2016-01-15 |url-status=live |archive-url=https://web.archive.org/web/20230716161241/https://www.math.uni-sb.de/ag/fuchs/Ana1/Paragraph11.pdf |archive-date=2023-07-16}}

{{cite book |title=Analysis I |language=German |author-first=Martin |author-last=Fuchs |date=2011 |publisher=Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany |chapter=Chapter 8.IV: Spezielle Funktionen – Die trigonometrischen Funktionen |edition=WS 2011/2012 |pages=16–20 |url=http://www.math.uni-sb.de/ag/fuchs/Ana1/Paragraph8.pdf |access-date=2016-01-15 |url-status=live |archive-url=https://web.archive.org/web/20230716151357/https://www.math.uni-sb.de/ag/fuchs/Ana1/Paragraph8.pdf |archive-date=2023-07-16}}

{{cite journal |title=A More Symmetrical Fourier Analysis Applied to Transmission Problems |author-link=Ralph Hartley |author-last=Hartley |author-first=Ralph V. L. |journal=Proceedings of the IRE |publisher=Institute of Radio Engineers |volume=30 |issue=3 |date=March 1942 |doi=10.1109/JRPROC.1942.234333 |s2cid=51644127 |pages=144–150 |url=https://www.researchgate.net/publication/3468825 |access-date=2023-07-16 |url-status=live |archive-url=https://web.archive.org/web/20190405073552/https://www.researchgate.net/publication/3468825_A_More_Symmetrical_Fourier_Analysis_Applied_to_Transmission_Problems |archive-date=2019-04-05}}

{{cite book |author-link=Ronald N. Bracewell |author-last=Bracewell |author-first=Ronald N. |title=The Fourier Transform and Its Applications |publisher=McGraw-Hill |edition=3 |orig-date=1985, 1978, 1965 |date=June 1999 |isbn=978-0-07303938-1}}

{{cite book |title=Abstract Algebra: An Introduction to Groups, Rings and Fields |author-first=Clive |author-last=Reis |date=2011 |edition=1 |publisher=World Scientific Publishing Co. Pte. Ltd. |isbn=978-9-81433564-5 |pages=434–438}}

{{cite web |title=The fundamental theorem of algebra - a visual proof |author-first=Edmund |author-last=Weitz |author-link=:de:Edmund Weitz |date=2016 |publisher=Hamburg University of Applied Sciences (HAW), Department Medientechnik |location=Hamburg, Germany |url=http://weitz.de/fund/ |access-date=2019-08-03 |url-status=live |archive-url=https://archive.today/20190803122756/http://weitz.de/fund/ |archive-date=2019-08-03}}

{{cite book |title=Komplexe Zahlen: Ein Leitprogramm in Mathematik |language=de |author-first1=Christina |author-last1=Diehl |author-first2=Marcel |author-last2=Leupp |location=Basel & Herisau, Switzerland |publisher=Eidgenössische Technische Hochschule Zürich (ETH) |date=January 2010 |page=41 |url=https://ethz.ch/content/dam/ethz/special-interest/dual/educeth-dam/documents/Unterrichtsmaterialien/mathematik/Komplexe%20Zahlen%20(Leitprogramm)/Leitprogramm.pdf |url-status=live |archive-url=https://web.archive.org/web/20170827073841/https://www.ethz.ch/content/dam/ethz/special-interest/dual/educeth-dam/documents/Unterrichtsmaterialien/mathematik/Komplexe%20Zahlen%20(Leitprogramm)/Leitprogramm.pdf |archive-date=2017-08-27 |quote-page=41 |quote=[...] 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)

{{cite journal |title=Chapter XXX. On loaded lines in telephonic transmission |author-link=George Ashley Campbell |author-first=George Ashley |author-last=Campbell |date=1903 |orig-date=1901-06-07 |volume=5 |issue=27 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |series=Series 6 |publisher=Taylor & Francis |doi=10.1080/14786440309462928 |pages=313–330 |url=https://jontalle.web.engr.illinois.edu/Public/Campbell-LoadedLines.03.pdf |access-date=2023-07-16 |url-status=live |archive-url=https://web.archive.org/web/20230716113406/https://jontalle.web.engr.illinois.edu/Public/Campbell-LoadedLines.03.pdf |archive-date=2023-07-16}} (2+18 pages)

{{cite journal |title=Cisoidal oscillations |author-link=George Ashley Campbell |author-first=George Ashley |author-last=Campbell |journal=Proceedings of the American Institute of Electrical Engineers |publisher=American Institute of Electrical Engineers |volume=XXX |issue=1–6 |date=April 1911 |doi=10.1109/PAIEE.1911.6659711 |s2cid=51647814 |pages=789–824 |url=https://ia800708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1109%252Fpaiee.1910.6660428.zip&file=10.1109%252Fpaiee.1911.6659711.pdf |access-date=2023-06-24}} (37 pages)

{{cite journal |title=The Practical Application of the Fourier Integral |author-link=George Ashley Campbell |author-first=George Ashley |author-last=Campbell |journal=The Bell System Technical Journal |publisher=American Telephone and Telegraph Company |volume=7 |issue=4 |date=1928-10-01 |orig-date=1927-09-13 |doi=10.1002/j.1538-7305.1928.tb00347.x |s2cid=53552671 |pages=639–707 [641] |url=https://ia803204.us.archive.org/11/items/bstj7-4-639/bstj7-4-639_text.pdf |access-date=2023-06-24 |quote-page=641 |quote=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)

{{cite book |title=Analysis I |language=de |author-first1=Herbert |author-last1=Amann |author-link1=:d:Q102078329 |author-first2=Joachim |author-last2=Escher |author-link2=:de:Joachim Escher (Mathematiker) |series=Grundstudium Mathematik |publisher=Birkhäuser Verlag |location=Basel, Switzerland |date=2006 |edition=3 |isbn=978-3-76437755-7 |id={{ISBN|3-76437755-0}} |pages=292, 298}} (445 pages)

{{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter 15.2. Complex absolute value |date=2017-08-22 |location=Salt Lake City, Utah, USA |publisher=Springer International Publishing AG |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |s2cid=30244721 |page=443}}

{{cite book |title=A Course in Complex Analysis in One Variable |chapter=Chapter 1. First Concepts |author-first=Martin A. |author-last=Moskowitz |publisher=World Scientific Publishing Co. Pte. Ltd. |location=City University of New York Graduate Center, New York, USA |publication-place=Singapore |date=2002

|isbn=981-02-4780-X |page=7 |chapter-url=https://books.google.com/books?id=Acw5DwAAQBAJ&pg=PA7}} (ix+149 pages)

}}

Category:Trigonometry

Category:Mathematical identities