cis (mathematics)

$cis$ is a mathematical notation defined by $cis x = cos x + i sin x$ , 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, $e ix$ , which offers an even shorter and more general notation for $cos x + i sin x$ .

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]

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

The $cis$ notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866)[4][5] and subsequently used by Irving Stringham in works such as Uniplanar Algebra (1893),[6][7] or by James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898).[7][8] 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,[9][10][11] for example in conjunction with Fourier and Hartley transforms,[12][13][14] 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)[15]), available for many compilers, programming languages (including C, C++,[16] Common Lisp,[17][18] D,[19] Fortran,[20] Haskell,[21] Julia[22]), and operating systems (including Windows, Linux,[20] macOS and HP-UX[23]). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually.[19][3]

Mathematical identities

Derivative

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

[1][24]

Integral

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

[1]

Other properties

These follow directly from Euler's formula.

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

[25]

\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|.

[25]

History

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

The $cis$ notation is sometimes used to emphasize one method of viewing and dealing with a problem over another.[26] 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 $e ix$ . 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 $cis x = e ix$ 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:[27][28]

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

References

Weisstein, Eric W. (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, OR, US: Clackamas Community College, Mathematics Department. Retrieved 2016-01-15.
"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.
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 […] (, ) (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. 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. in 1969.)
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.
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. 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.)
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.)
Swokowski, Earl; 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 (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). 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.
Intel. "v?CIS". Intel Developer Zone. Retrieved 2016-01-15.
"Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Retrieved 2016-01-15.
"CIS". Common Lisp Hyperspec. The Harlequin Group Limited. 1996. Retrieved 2016-01-15.
"CIS". LispWorks, Ltd. 2005 [1996]. Retrieved 2016-01-15.
"std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Retrieved 2016-01-14.
"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.
"CIS". Haskell reference. ZVON. Retrieved 2016-01-15.
"Mathematics; Mathematical Operators". The Julia Language. Archived from the original on 2020-08-19. Retrieved 2019-12-05.
"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. Retrieved 2016-01-15.
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.
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.
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^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)
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. S2CID 51644127.
Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

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

[Simmons_2014_1-2] 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.

[Rationale_2003_C-3] "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.

[Hamilton_1866-4] 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 […] (, ) (NB. This work was published posthumously, Hamilton died in 1865.)

[Hamilton_1899-5] 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. in 1969.)

[Stringham_1893-6] 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.

[Cajori_1929-7] 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. 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.)

[Harkness_Morley_1898-8] 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.)

[Swokowski_2011-9] Swokowski, Earl; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 978-0-84006857-6. Retrieved 2016-01-18.

[Reis_2011-10] 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_2016-11] 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.

[Byrnes_2004-12] 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.

[Kammler_2008-13] 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_2016-14] 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-15] Intel. "v?CIS". Intel Developer Zone. Retrieved 2016-01-15.

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

[Lisp-17] "CIS". Common Lisp Hyperspec. The Harlequin Group Limited. 1996. Retrieved 2016-01-15.

[Lisp_2005-18] "CIS". LispWorks, Ltd. 2005 [1996]. Retrieved 2016-01-15.

[D_2011-19] "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Retrieved 2016-01-14.

[Intel_2008_Fortran-20] "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.

[Haskell-21] "CIS". Haskell reference. ZVON. Retrieved 2016-01-15.

[Julia-22] "Mathematics; Mathematical Operators". The Julia Language. Archived from the original on 2020-08-19. Retrieved 2019-12-05.

[HP_2007_UX-23] "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. Retrieved 2016-01-15.

[Fuchs_2011_2-24] 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.

[Fuchs_2011_1-25] 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.

[Diehl-Leupp_2010-26] 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^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)

[Hartley_1942-27] 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. S2CID 51644127.

[Bracewell_1999-28] Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.