Cis (mathematics)

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cis is a mathematical notation defined by cis x = cos x + i sin x, [nb 1] where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, eix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries.

Contents

Overview

The cis notation is a shorthand for the combination of functions on the right-hand side of Euler's formula:

where i2 = −1. So,

[1] [2] [3] [4]

i.e. "cis" is an acronym for "Cos i Sin".

It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the domain of definition is usually , complex values are possible as well:

so the cis function can be used to extend Euler's formula to a more general complex version. [5]

The function is mostly used as a convenient shorthand notation to simplify some expressions, [6] [7] [8] for example in conjunction with Fourier and Hartley transforms, [9] [10] [11] 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) [12] or MathCW [13] ), available for many compilers and programming languages (including C, C++, [14] Common Lisp, [15] [16] D, [17] Haskell, [18] Julia, [19] and Rust [20] ). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually. [17] [3]

Mathematical identities

Derivative

[1] [21]

Integral

[1]

Other properties

These follow directly from Euler's formula.

[22]

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

[22]

History

The cis notation was first coined by William Edwin Hamilton in Elements of Quaternions (1866) [23] [24] and subsequently used by Irving Stringham (who also called it "sector of x") in works such as Uniplanar Algebra (1893), [25] [26] James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898), [26] [27] or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928). [28] [29] [30]

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: [31] [32]

Motivation

The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. [33] 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). [30]

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 eix. The usual proof that cis x = eix requires calculus, which the student may not have studied before encountering the expression cos x + i sin x.

This notation was more common when typewriters were used to convey mathematical expressions.[ citation needed ]

See also

Notes

  1. Here, i refers to the imaginary unit in mathematics. Since i is commonly used to denote electric current in electrical engineering and control systems engineering, the imaginary unit is alternatively denoted by j instead of i in these contexts. Regardless of context, this does not affect the established name of the function as cis.

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<span class="mw-page-title-main">Complex number</span> Number with a real and an imaginary part

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number ,a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.

<span class="mw-page-title-main">Euler's formula</span> Complex exponential in terms of sine and cosine

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x. The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case.

<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

The exponential function is a mathematical function denoted by or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments , including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".

<span class="mw-page-title-main">Gamma function</span> Extension of the factorial function

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In mathematics, the Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable .

<span class="mw-page-title-main">Polar coordinate system</span> Coordinates comprising a distance and an angle

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

<span class="mw-page-title-main">Trigonometric functions</span> Functions of an angle

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

<span class="mw-page-title-main">Euler's identity</span> Mathematical equation linking e, i and pi

In mathematics, Euler's identity is the equality where

<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.

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<span class="mw-page-title-main">Fourier series</span> Decomposition of periodic functions into sums of simpler sinusoidal forms

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<span class="mw-page-title-main">Exponentiation</span> Arithmetic operation

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<span class="mw-page-title-main">Inverse trigonometric functions</span> Inverse functions of sin, cos, tan, etc.

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<span class="mw-page-title-main">Airy function</span> Special function in the physical sciences

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<span class="mw-page-title-main">Clausen function</span> Transcendental single-variable function

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<span class="mw-page-title-main">Sinc function</span> Special mathematical function defined as sin(x)/x

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<span class="mw-page-title-main">Sine and cosine</span> Fundamental trigonometric functions

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References

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  23. Hamilton, William Rowan (1866-01-01). "Book II, Chapter II. Fractional powers, General roots of unity". Written at Dublin, Irland. 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. pp. 250, 252: [...] 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.)
  24. Hamilton, William Rowan (1899) [1866-01-01]. Hamilton, William Edwin; Joly, Charles Jasper (eds.). Elements of Quaternions. Vol. I (2 ed.). London, UK: Longmans, Green & Co. p. 262. Retrieved 2019-08-03. p. 262: [...] recent abridgment cis for cos + i sin [...] (NB. This edition was reprinted by Chelsea Publishing Company in 1969.)
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  29. Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers . XXX (1–6). American Institute of Electrical Engineers: 789–824. doi:10.1109/PAIEE.1911.6659711. S2CID   51647814 . Retrieved 2023-06-24. (37 pages)
  30. 1 2 Campbell, George Ashley (1928-10-01) [1927-09-13]. "The Practical Application of the Fourier Integral" (PDF). The Bell System Technical Journal . 7 (4). American Telephone and Telegraph Company: 639–707 [641]. doi:10.1002/j.1538-7305.1928.tb00347.x. S2CID   53552671 . Retrieved 2023-06-24. p. 641: It has been recognized, almost from the start, however, that the form which best combines mathematical simplicity and complete generality makes use of the exponential oscillating function eift. More recently the overwhelming advantage of using this oscillating function in the discussion of sinusoidal oscillatory systems has been generally recognized. It is, therefore, plain that this oscillating function should be adopted as the basic oscillation for both of the proposed tables. A name for this oscillation, associating it with sines and cosines, rather than with the real exponential function, seems desirable. The abbreviation cis x for (cos x + i sin x) suggests that we name this function a cis or a cisoidal oscillation. (69 pages)
  31. Hartley, Ralph V. L. (March 1942). "A More Symmetrical Fourier Analysis Applied to Transmission Problems". Proceedings of the IRE . 30 (3). Institute of Radio Engineers: 144–150. doi:10.1109/JRPROC.1942.234333. S2CID   51644127. Archived from the original on 2019-04-05. Retrieved 2023-07-16.
  32. Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN   978-0-07303938-1.
  33. 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. p. 41: [...] Bitte vergessen Sie aber nicht, dass e 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 verwendet. [...] (109 pages)