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

Relation to the complex exponential function

The complex exponential function can be expressed

e^{ix}=\cos(x)+i\sin(x),

^[1]

e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin(x)

e^{i\pi }=-1

\cos(x)={\frac {e^{ix}+e^{-ix}}{2}}

\sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}

where $i 2 = -1$ .

This can also be expressed using the following notation

\operatorname {cis} (x)=\cos(x)+i\sin(x),

^[1]^[4]^[22]

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

Though at first glance this notation is redundant, being equivalent to $e ix$ , 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

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

^[1]^[23]

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)

^[24]

\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|.

^[24]

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, $e ix 2$ versus $cis(x 2)$ or $exp(ix 2)$ . For many readers, $cis(x 2)$ 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. 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 they are not yet prepared for: the usual proof that $cis(x) = e ix$ requires calculus, which the student may not have studied before they encountered 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:^[25]^[26]

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

References

^ ^a ^b ^c ^d ^e Weisstein, Eric W. (2015) [2000]. "Cis". MathWorld. Wolfram Research, Inc. Archived from the original on 2016-01-27. Retrieved 2016-01-09. {{cite web}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)
^ ^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). Vol. 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. p. 401-411. doi:10.1007/1-4020-2307-3. ISBN 978-1-4020-1982-1. ISSN 1568-2609. Archived from the original (PDF) on 2017-10-28. Retrieved 2017-10-28. {{cite book}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)
^ ^a ^b Swokowski, Earl; Cole, Jeffery (2011). Precalculus: Functions and Graphs (12 ed.). Cengage Learning. ISBN 978-0-84006857-6. ISBN 0-84006857-3. Retrieved 2016-01-18. {{cite book}}: |work= ignored (help)
^ ^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.
^ 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.
^ ^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.
^ ^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.
^ Pierce, Rod (2016-01-04) [2000]. "Complex Number Multiplication". Maths Is Fun. Retrieved 2016-01-15.
^ ^a ^b Hamilton, William Rowan (1866-01-01). "II. Fractional powers, General roots of unity". Written at Dublin. In Hamilton, William Edwin (ed.). Elements of Quaternions. University Press, Michael Henry Gill, Dublin (printer) (1 ed.). London, UK: Longmans, Green & Co. 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])
^ ^a ^b 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, 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.
^ ^a ^b Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (2 (3rd corrected printing of 1929 issue) 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.)
^ 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.)
^ 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.
^ ^a ^b "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Retrieved 2016-01-14.
^ ^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.
^ "CIS". Haskell reference. ZVON. Retrieved 2016-01-15.
^ "Julia documentation – Standard library – Mathematics". Retrieved 2017-03-03.
^ "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.
^ ^a ^b "Rationale for International Standard - Programming Languages - C" (PDF). 5.10. April 2003. pp. 114, 117, 183, 186–187. Archived from the original (PDF) on 2016-06-06. Retrieved 2010-10-17. {{cite web}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)
^ Fuchs, Martin (2011). "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.
^ ^a ^b Fuchs, Martin (2011). "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.
^ 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.
^ Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.

[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. {{cite web}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)

[Byrnes_2004-2] 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). Vol. 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. p. 401-411. doi:10.1007/1-4020-2307-3. ISBN 978-1-4020-1982-1. ISSN 1568-2609. Archived from the original (PDF) on 2017-10-28. Retrieved 2017-10-28. {{cite book}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)

[Swokowski_2011-3] Swokowski, Earl; Cole, Jeffery (2011). Precalculus: Functions and Graphs (12 ed.). Cengage Learning. ISBN 978-0-84006857-6. ISBN 0-84006857-3. Retrieved 2016-01-18. {{cite book}}: |work= ignored (help)

[Simmons_2014_1-4] 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.

[Simmons_2014_2-5] 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.

[Kammler_2008-6] 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.

[Lorenzo-Hartley_2016-7] 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.

[Pierce_2016-8] Pierce, Rod (2016-01-04) [2000]. "Complex Number Multiplication". Maths Is Fun. Retrieved 2016-01-15.

[Hamilton_1866-9] Hamilton, William Rowan (1866-01-01). "II. Fractional powers, General roots of unity". Written at Dublin. In Hamilton, William Edwin (ed.). Elements of Quaternions. University Press, Michael Henry Gill, Dublin (printer) (1 ed.). London, UK: Longmans, Green & Co. 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])

[Stringham_1893-10] 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, 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_1929-11] Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (2 (3rd corrected printing of 1929 issue) 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.)

[Harkness_Morley_1898-12] 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.)

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

[Intel_2007_C++-14] "Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Retrieved 2016-01-15.

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

[Lisp_2005-16] "CIS". LispWorks, Ltd. 2005 [1996]. Retrieved 2016-01-15.

[D_2011-17] "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Retrieved 2016-01-14.

[Intel_2008_Fortran-18] "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-19] "CIS". Haskell reference. ZVON. Retrieved 2016-01-15.

[Julia-20] "Julia documentation – Standard library – Mathematics". Retrieved 2017-03-03.

[HP_2007_UX-21] "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.

[Rationale_2003_C-22] "Rationale for International Standard - Programming Languages - C" (PDF). 5.10. April 2003. pp. 114, 117, 183, 186–187. Archived from the original (PDF) on 2016-06-06. Retrieved 2010-10-17. {{cite web}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)

[Fuchs_2011_2-23] Fuchs, Martin (2011). "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_1-24] Fuchs, Martin (2011). "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.

[Hartley_1942-25] 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.

[Bracewell_1999-26] Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.

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[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]