cis is a less commonly used 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. The notation is less commonly used than Euler's formula, , which offers an even shorter and more general notation for cos(x) + i sin(x).
The cis notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham in works such as Uniplanar Algebra (1893), or by James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898). It connects trigonometric functions with exponential functions in the complex plane via Euler's formula.
In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL)), available for many compilers, programming languages (including C, C++, Common Lisp, D, Fortran, Haskell, Julia), and operating systems (including Windows, Linux, OS X and HP-UX). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually.
Relation to the complex exponential function
The complex exponential function can be expressed
where i2 = −1.
This can also be expressed using the following notation
i.e. "cis" abbreviates "cos + i sin".
Though at first glance this notation is redundant, being equivalent to eix, its use is rooted in several advantages.
These follow directly from Euler's formula.
The identities above hold if x and y are any complex numbers. If x and y are real, then
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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, eix2 versus cis(x2) or 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 eix. 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) = eix requires calculus, which the student may not have studied before they encountered the expression cos(x) + i sin(x).
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- Swokowski, Earl; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 0840068573. 9780840068576. Retrieved 2016-01-18.
- Simmons, Bruce (2014-07-28) . "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) . "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) . "Complex Number Multiplication". Maths Is Fun. Retrieved 2016-01-15.
- 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 […](, )
- Stringham, Irving (1893-07-01) . 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) . 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.)
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- 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, 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.