cis (mathematics)

cis is a rarely used mathematical notation in which $\operatorname{cis}(x) = \cos(x) + i\sin(x)$,^{[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]

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 i² = −1.

This can also be expressed using the following notation

$\operatorname{cis}(x) = \cos(x) + i\sin(x),\,$^[1][3]

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][18]

Integral

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

Other properties

These follow directly from Euler's formula.

$\operatorname{cis}(x+y) = \operatorname{cis}(x)\,\operatorname{cis}(y)$^[19]

$\operatorname{cis}(x-y) = {\operatorname{cis}(x) \over \operatorname{cis}(y)}$

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

$|\operatorname{cis}(x) - \operatorname{cis}(y)| \le |x-y|.$^[19]

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^ix2 versus cis(x²) versus exp(ix²). For many readers, cis(x²) 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).

See also

• De Moivre's formula
• Euler's formula
• Complex number
• Ptolemaios' theorem
• Phasor

References

1. Weisstein, Eric W. (2015) [2000]. "Cis". MathWorld. Wolfram Research, Inc. Archived from the original on 2015-09-08. Retrieved 2016-01-09.
2. Swokowski, Earl; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.) (Cengage Learning). ISBN 0840068573. 9780840068576. Retrieved 2016-01-18.
3. 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.
4. 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.
5. Pierce, Rod (2016-01-04) [2000]. "Complex Number Multiplication". Maths Is Fun. Retrieved 2016-01-15.
6. 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 […] (, )
7. 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.
8. 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.)
9. 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.)
10. Intel. "v?CIS". Intel Developer Zone. Retrieved 2016-01-15.
11. "Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Retrieved 2016-01-15.
12. "CIS". Common Lisp Hyperspec. The Harlequin Group Limited. 1996. Retrieved 2016-01-15.
13. "CIS". LispWorks, Ltd. 2005 [1996]. Retrieved 2016-01-15.
14. "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Retrieved 2016-01-14.
15. "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.
16. "CIS". Haskell reference. ZVON. Retrieved 2016-01-15.
17. "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.
18. 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.
19. 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.