cis (mathematics)

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Short description: Alternate mathematical notation for cos x + i sin x

"Cisoidal oscillation" redirects here. For the plane curve generated from two curves, see Cissoid.

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.

Overview

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

[math]\displaystyle{ e^{ix} = \cos x + i\sin x, }[/math]

where i2 = −1. So,

[math]\displaystyle{ \operatorname{cis} x = \cos x + i\sin x, }[/math][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 [math]\displaystyle{ x \in \mathbb{R} }[/math], complex values [math]\displaystyle{ z \in \mathbb{C} }[/math] are possible as well:

[math]\displaystyle{ \operatorname{cis} z = \cos z + i\sin z, }[/math]

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, programming languages (including C, C++,[14] Common Lisp,[15][16] D,[17] Fortran,[18] Haskell,[19] Julia,[20] and Rust[21]), and operating systems (including Windows, Linux,[18] macOS and HP-UX[22]). 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

[math]\displaystyle{ \frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cis} z = i\operatorname{cis} z = ie^{iz} }[/math][1][23]

Integral

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

Other properties

These follow directly from Euler's formula.

[math]\displaystyle{ \cos(x) = \frac{\operatorname{cis}(x) + \operatorname{cis}(-x)}{2} = \frac{e^{ix} + e^{-ix}}{2} }[/math]
[math]\displaystyle{ \sin(x) = \frac{\operatorname{cis}(x) - \operatorname{cis}(-x)}{2i}= \frac{e^{ix} - e^{-ix}}{2i} }[/math]
[math]\displaystyle{ \operatorname{cis}(x+y) = \operatorname{cis} x\,\operatorname{cis} y }[/math][24]
[math]\displaystyle{ \operatorname{cis}(x-y) = {\operatorname{cis} x \over \operatorname{cis} y} }[/math]

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

[math]\displaystyle{ |\operatorname{cis} x - \operatorname{cis} y| \le |x-y|. }[/math][24]

History

The cis notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866)[25][26] and subsequently used by Irving Stringham (who also called it "sector of x") in works such as Uniplanar Algebra (1893),[27][28] James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898),[28][29] or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928).[30][31][32]

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:[33][34]

[math]\displaystyle{ \operatorname{cas} x = \cos x + \sin x. }[/math]

In 2016, Reza R. Ahangar, a mathematics professor at TAMUK, defined two hyperbolic function shortcuts as:[35]

[math]\displaystyle{ \operatorname{cish} x = \cosh x + i\sinh x }[/math]
[math]\displaystyle{ \operatorname{sich} x = \sinh x + i\cosh x }[/math]

Motivation

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.

See also

Notes

  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.

References

  1. ↑ 1.0 1.1 1.2 "Cis". MathWorld. Wolfram Research, Inc.. 2015. http://mathworld.wolfram.com/Cis.html. 
  2. ↑ "Cis". Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, Oregon, USA: Clackamas Community College, Mathematics Department. 2014-07-28. http://www.mathwords.com/c/cis.htm. 
  3. ↑ 3.0 3.1 "Rationale for International Standard - Programming Languages - C". April 2003. pp. 114, 117, 183, 186–187. http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf. 
  4. ↑ (in de) Analysis I. Grundstudium Mathematik (3 ed.). Basel, Switzerland: Birkhäuser Verlag. 2006. pp. 292, 298. ISBN:3-76437755-0. ISBN 978-3-76437755-7.  (445 pages)
  5. ↑ "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.. 2002. p. 7. ISBN 981-02-4780-X. https://books.google.com/books?id=Acw5DwAAQBAJ&pg=PA7.  (ix+149 pages)
  6. ↑ Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. 2011. ISBN 978-0-84006857-6. https://books.google.com/books?id=8GB2Udf8wnoC. Retrieved 2016-01-18. 
  7. ↑ Abstract Algebra: An Introduction to Groups, Rings and Fields (1 ed.). World Scientific Publishing Co. Pte. Ltd.. 2011. pp. 434–438. ISBN 978-9-81433564-5. 
  8. ↑ "The fundamental theorem of algebra - a visual proof". Hamburg, Germany: Hamburg University of Applied Sciences (HAW), Department Medientechnik. 2016. http://weitz.de/fund/. 
  9. ↑ Byrnes, Jim, ed (2004). "Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing". written at Prometheus Inc., Newport, USA. Computational Noncommutative Algebra and Applications. 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. http://web.cecs.pdx.edu/~mperkows/CAPSTONES/Quaternion/katya2.pdf. Retrieved 2017-10-28. 
  10. ↑ A First Course in Fourier Analysis (2 ed.). Cambridge University Press. 2008-01-17. ISBN 978-1-13946903-6. https://books.google.com/books?id=znP-ADtE8ZQC. Retrieved 2017-10-28. 
  11. ↑ The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science. John Wiley & Sons. 2016-11-14. ISBN 978-1-11913942-3. https://books.google.com/books?id=LdyADQAAQBAJ. Retrieved 2017-10-28. 
  12. ↑ "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. https://www.intel.com/content/www/us/en/content-details/671504/developer-reference-for-intel-math-kernel-library-intel-mkl-2017-c.html. Retrieved 2016-01-15. 
  13. ↑ "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. 2017-08-22. p. 443. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. 
  14. ↑ "Intel C++ Compiler Reference". Intel Corporation. 2007. pp. 34, 59–60. http://supercomputer.susu.ru/upload/users/instructions/C_Compiler_Reference.pdf. 
  15. ↑ "CIS". Common Lisp Hyperspec. The Harlequin Group Limited. 1996. http://www.ai.mit.edu/projects/iiip/doc/CommonLISP/HyperSpec/Body/fun_cis.html. 
  16. ↑ "CIS". LispWorks, Ltd.. 2005. http://clhs.lisp.se/Body/f_cis.htm#cis. 
  17. ↑ 17.0 17.1 "std.math: expi". D programming language. Digital Mars. 2016-01-11. http://dlang.org/phobos/std_math.html#.expi. 
  18. ↑ 18.0 18.1 "Installation Guide and Release Notes". Intel Fortran Compiler Professional Edition 11.0 for Linux. 2008-11-06. https://www.ualberta.ca/CNS/RESEARCH/NumStatsServers/Intel/Documentation.28Apr11/Release_NotesF.pdf. [yes|permanent dead link|dead link}}]
  19. ↑ "CIS". Haskell reference. ZVON. http://zvon.org/other/haskell/Outputcomplex/cis_f.html. 
  20. ↑ "Mathematics; Mathematical Operators". The Julia Language. https://docs.julialang.org/en/v1.0/base/math/#Base.cis. 
  21. ↑ "Struct num_complex::Complex". https://docs.rs/num-complex/latest/num_complex/struct.Complex.html#method.cis. 
  22. ↑ "HP-UX 11i v2.0 non-critical impact: Changes to the IPF libm (NcEn843) – CC Impacts enhancement description – Major performance upgrades for power function and performance tuneups". Hewlett-Packard Development Company, L.P.. 2007. http://h21007.www2.hp.com/portal/download/files/unprot/STK/HPUX_STK_JPN/impacts/i843.html. [yes|permanent dead link|dead link}}]
  23. ↑ "Chapter 11: Differenzierbarkeit von Funktionen" (in German). Analysis I (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´. 2011. pp. 3, 13. http://www.math.uni-sb.de/ag/fuchs/Ana1/Paragraph11.pdf. Retrieved 2016-01-15. 
  24. ↑ 24.0 24.1 "Chapter 8.IV: Spezielle Funktionen – Die trigonometrischen Funktionen" (in German). Analysis I (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany. 2011. pp. 16–20. http://www.math.uni-sb.de/ag/fuchs/Ana1/Paragraph8.pdf. Retrieved 2016-01-15. 
  25. ↑ Hamilton, William Edwin, ed (1866-01-01). "Book II, Chapter II. Fractional powers, General roots of unity". written at Dublin, Irland. Elements of Quaternions (1 ed.). London, UK: Longmans, Green & Co., University Press, Michael Henry Gill. pp. 250–257, 260, 262–263. https://archive.org/details/elementsquaterni00hamirich/page/n323. 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.)
  26. ↑ Elements of Quaternions. I (2 ed.). London, UK: Longmans, Green & Co.. 1899. p. 262. https://archive.org/details/117770258_001. Retrieved 2019-08-03. "[…] recent abridgment cis for cos + i sin […]"  (NB. This edition was reprinted by Chelsea Publishing Company in 1969.)
  27. ↑ Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis. 1. C. A. Mordock & Co. (printer) (1 ed.). San Francisco, California, USA: The Berkeley Press. 1893-07-01. pp. 71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135. https://archive.org/details/uniplanaralgebra00stri. 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." 
  28. ↑ 28.0 28.1 A History of Mathematical Notations. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, Illinois, USA: Open court publishing company. 1952. p. 133. ISBN 978-1-60206-714-1. https://books.google.com/books?id=bT5suOONXlgC. 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.)
  29. ↑ Introduction to the Theory of Analytic Functions (1 ed.). London, UK: Macmillan and Company. 1898. pp. 18, 22, 48, 52, 170. ISBN 978-1-16407019-1. https://archive.org/details/cu31924059412910. Retrieved 2016-01-18.  (NB. ISBN for reprint by Kessinger Publishing, 2010.)
  30. ↑ "Chapter XXX. On loaded lines in telephonic transmission". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Series 6 (Taylor & Francis) 5 (27): 313–330. 1903. doi:10.1080/14786440309462928. https://jontalle.web.engr.illinois.edu/Public/Campbell-LoadedLines.03.pdf. Retrieved 2023-07-16.  (2+18 pages)
  31. ↑ "Cisoidal oscillations". Proceedings of the American Institute of Electrical Engineers (American Institute of Electrical Engineers) XXX (1–6): 789–824. April 1911. doi:10.1109/PAIEE.1911.6659711. 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. Retrieved 2023-06-24.  (37 pages)
  32. ↑ 32.0 32.1 "The Practical Application of the Fourier Integral". The Bell Systems Technical Journal (American Telephone and Telegraph Company) 7 (4): 639–707 [641]. 1928-10-01. doi:10.1002/j.1538-7305.1928.tb00347.x. https://ia803204.us.archive.org/11/items/bstj7-4-639/bstj7-4-639_text.pdf. Retrieved 2023-06-24. "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)
  33. ↑ "A More Symmetrical Fourier Analysis Applied to Transmission Problems". Proceedings of the IRE (Institute of Radio Engineers) 30 (3): 144–150. March 1942. doi:10.1109/JRPROC.1942.234333. https://www.researchgate.net/publication/3468825. Retrieved 2023-07-16. 
  34. ↑ The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. June 1999. ISBN 978-0-07303938-1. 
  35. ↑ "The Relativistic Geometry of the Complex Matter Space". Journal of Applied Mathematics and Physics (Texas A&M University, Kingsville, Texas, USA: Scientific Research Publishing) 5 (2): 422–438 [428–431]. 2017-02-17. doi:10.4236/jamp.2017.52037. ISSN 2327-4352. https://pdfs.semanticscholar.org/28b7/9cc554628d426641ee33ac26fc77a79a6a28.pdf. Retrieved 2023-07-27. "[…] 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)
  36. ↑ (in de) Komplexe Zahlen: Ein Leitprogramm in Mathematik. Basel & Herisau, Switzerland: Eidgenössische Technische Hochschule Zürich (ETH). January 2010. p. 41. https://ethz.ch/content/dam/ethz/special-interest/dual/educeth-dam/documents/Unterrichtsmaterialien/mathematik/Komplexe%20Zahlen%20(Leitprogramm)/Leitprogramm.pdf. "[…] 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)



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