cis (mathematics)

$cis$ is a rarely used mathematical notation in which $cis(x) = cos(x) + i sin(x)$ ,^[1]^[2]^[3]^[4]^[5] where $cos$ is the cosine function, $i$ is the imaginary unit and $sin$ is the sine. The notation is redundant, as Euler's formula offers an even shorter and more general short-hand notation for $cos(x) + i sin(x)$ .

Overview

The $cis$ notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866)^[6] and subsequently used by Irving Stringham in works such as Uniplanar Algebra (1893),^[7]^[8] or by James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898).^[8]^[9] 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^[6]^[7]^[2] 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)^[10]), available for many compilers, programming languages (including C, C++,^[11] Common Lisp,^[12]^[13] D,^[14] Fortran,^[15] Haskell^[16]), and operating systems (including Windows, Linux,^[15] OS X and HP-UX^[17]). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually.^[14]^[18]

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]^[3]^[18]

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.

Mathematical identities

Derivative

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

^[1]^[19]

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)

^[20]

\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|.

^[20]

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 $cis(x 2)$ versus $exp(ix 2)$ . For many readers, $cis(x 2)$ is the clearest, easiest to read of the three.

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 doesn't 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 proof that $cis(x) = e ix$ requires calculus, which the student may not have studied before they encountered the expression $cos(x) + i sin(x)$ .

References

1 2 3 4 5 Weisstein, Eric W. (2015) [2000]. "Cis". MathWorld. Wolfram Research, Inc. Archived from the original on 2016-01-27. Retrieved 2016-01-09.
1 2 Swokowski, Earl; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 0840068573. 9780840068576. Retrieved 2016-01-18.
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.
↑ 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.
↑ Pierce, Rod (2016-01-04) [2000]. "Complex Number Multiplication". Maths Is Fun. Retrieved 2016-01-15.
1 2 Hamilton, William Rowan (1866-01-01). "II. Fractional powers, General roots of unity". Written at Dublin. In Hamilton, William Edwin. 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, 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.
1 2 Cajori, Florian (1952) [1929]. A History of Mathematical Notations. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, US: Open court publishing company. p. 133. ISBN 978-1-60206-714-1. 1602067147. 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 978-1164070191. 1164070193. 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.
1 2 "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Retrieved 2016-01-14.
1 2 "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.
↑ "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 performace tuneups". Hewlett-Packard Development Company, L.P. 2007. Retrieved 2016-01-15.
1 2 "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.
↑ 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.
1 2 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.

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