# cis (mathematics)

cis is a less commonly used mathematical notation defined by cis(x) = cos(x) + i sin(x),[1][2][3][4][5][6][7][8][9] where cos is the cosine function, i is the imaginary unit and sin is the sine function. The notation is less commonly used than Euler's formula, ${\displaystyle e^{ix}}$, which offers an even shorter and more general notation for cos(x) + i sin(x).

## Overview

The cis notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866)[10][11] and subsequently used by Irving Stringham in works such as Uniplanar Algebra (1893),[12][13] or by James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898).[13][14] It connects trigonometric functions with exponential functions in the complex plane via Euler's formula.

It is mostly used as a convenient shorthand notation to simplify some expressions,[10][12][3][15][16] for example in conjunction with Fourier and Hartley transforms,[2][6][7] 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)[17]), available for many compilers, programming languages (including C, C++,[18] Common Lisp,[19][20] D,[21] Fortran,[22] Haskell,[23] Julia[24]), and operating systems (including Windows, Linux,[22] macOS and HP-UX[25]). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually.[21][26]

## Relation to the complex exponential function

The complex exponential function can be expressed

${\displaystyle e^{ix}=\cos(x)+i\sin(x),}$[1]
${\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin(x)}$
${\displaystyle e^{i\pi }=-1}$
${\displaystyle \cos(x)={\frac {e^{ix}+e^{-ix}}{2}}}$
${\displaystyle \sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}}$

where i2 = −1.

This can also be expressed using the following notation

${\displaystyle \operatorname {cis} (x)=\cos(x)+i\sin(x),}$[1][4][26]

i.e. "cis" abbreviates "cos + i sin".

Though at first glance this notation is redundant, being equivalent to eix, its use is rooted in several advantages, such as being directly tied to the polar form of a complex number (and being easier to grasp).

## Mathematical identities

### Derivative

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cis} (z)=i\operatorname {cis} (z)=ie^{iz}}$[1][27]

### Integral

${\displaystyle \int \operatorname {cis} (z)\,\mathrm {d} z=-i\operatorname {cis} (z)=-ie^{iz}}$[1]

### Other properties

These follow directly from Euler's formula.

${\displaystyle \operatorname {cis} (x+y)=\operatorname {cis} (x)\,\operatorname {cis} (y)}$[28]
${\displaystyle \operatorname {cis} (x-y)={\operatorname {cis} (x) \over \operatorname {cis} (y)}}$

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

${\displaystyle |\operatorname {cis} (x)-\operatorname {cis} (y)|\leq |x-y|.}$[28]

## History

This notation was more common in the post–World War II era, when typewriters were used to convey mathematical expressions.

Superscripts are both offset vertically and smaller than 'cis' or 'exp'; hence, they can be problematic even for hand-writing, for example, eix2 versus cis(x2) or exp(ix2). For many readers, cis(x2) is the clearest, easiest to read of the three.[citation needed]

The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another.[29] 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).

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).

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:[30][31]

cas(x) = cos(x) + sin(x).

## References

1. Weisstein, Eric W. (2015) [2000]. "Cis". MathWorld. Wolfram Research, Inc. Archived from the original on 2016-01-27. Retrieved 2016-01-09.
2. ^ a b 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). 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.
3. ^ a b Swokowski, Earl; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 978-0-84006857-6. ISBN 0-84006857-3. Retrieved 2016-01-18.
4. ^ a b Simmons, Bruce (2014-07-28) [2004]. "Cis". Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, OR, US: Clackamas Community College, Mathematics Department. Retrieved 2016-01-15.
5. ^ Simmons, Bruce (2014-07-28) [2004]. "Polar Form of a Complex Number". Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, OR, US: Clackamas Community College, Mathematics Department. Retrieved 2016-01-15.
6. ^ a b Kammler, David W. (2008-01-17). A First Course in Fourier Analysis (2 ed.). Cambridge University Press. ISBN 978-1-13946903-6. ISBN 1-13946903-7. Retrieved 2017-10-28.
7. ^ a b 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. ISBN 1-11913942-2. Retrieved 2017-10-28.
8. ^ Pierce, Rod (2016-01-04) [2000]. "Complex Number Multiplication". Maths Is Fun. Retrieved 2016-01-15.
9. ^ 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, UT, USA: Springer International Publishing AG. p. 443. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446.
10. ^ a b Hamilton, William Rowan (1866-01-01). "Chapter II. Fractional powers, General roots of unity". Written at Dublin. 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. […] 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.)
11. ^ Hamilton, William Rowan (1899) [1866-01-01]. Hamilton, William Edwin; Joly, Charles Jasper (eds.). Elements of Quaternions. I (2 ed.). London, UK: Longmans, Green & Co. p. 262. Retrieved 2019-08-03. […] recent abridgment cis for cos + i sin […] (NB. This edition was reprinted by Chelsea Publ. Co. [de] in 1969.)
12. ^ a b Stringham, Irving (1893-07-01) [1891]. Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis. 1. C. A. Mordock & Co. (printer) (1 ed.). San Francisco, US: 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. As an abbreviation for cos θ + i sin θ it is convenient to use cis θ, which may be read: sector of θ.
13. ^ a b Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, US: Open court publishing company. p. 133. ISBN 978-1-60206-714-1. ISBN 1-60206-714-7. Retrieved 2016-01-18. 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.)
14. ^ 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. ISBN 1-16407019-3. Retrieved 2016-01-18. (NB. ISBN for reprint by Kessinger Publishing, 2010.)
15. ^ 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. ISBN 9-81433564-9.
16. ^ Weitz, Edmund (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.
17. ^ Intel. "v?CIS". Intel Developer Zone. Retrieved 2016-01-15.
18. ^ "Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Retrieved 2016-01-15.
19. ^ "CIS". Common Lisp Hyperspec. The Harlequin Group Limited. 1996. Retrieved 2016-01-15.
20. ^ "CIS". LispWorks, Ltd. 2005 [1996]. Retrieved 2016-01-15.
21. ^ a b "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Retrieved 2016-01-14.
22. ^ a b "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.
23. ^ "CIS". Haskell reference. ZVON. Retrieved 2016-01-15.
24. ^ "Mathematics; Mathematical Operators". The Julia Language. Archived from the original on 2020-08-19. Retrieved 2019-12-05.
25. ^
26. ^ a b "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.
27. ^ 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. Retrieved 2016-01-15.
28. ^ a b 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. Retrieved 2016-01-15.
29. ^ 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. […] Bitte vergessen Sie aber nicht, dass e 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 verwendet. […] (109 pages)
30. ^ Hartley, Ralph V. L. (March 1942). "A More Symmetrical Fourier Analysis Applied to Transmission Problems". Proceedings of the IRE. 30 (3): 144–150. doi:10.1109/JRPROC.1942.234333.
31. ^ Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.