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, e^ix, 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.

Overview

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

${\displaystyle e^{ix}=\cos x+i\sin x,}$

where i² = −1. So,

${\displaystyle \operatorname {cis} x=\cos x+i\sin x,}$^[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 ${\displaystyle x\in \mathbb {R} }$, complex values ${\displaystyle z\in \mathbb {C} }$ are possible as well:

${\displaystyle \operatorname {cis} z=\cos z+i\sin z,}$

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, programming languages (including C, C++,^[14] Common Lisp,^[15][16] D,^[17] Fortran,^[18] Haskell,^[19] Julia,^[20] and Rust^[21]), and operating systems (including Windows, Linux,^[18] macOS and HP-UX^[22]). 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

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z))\operatorname {cis} z=i\operatorname {cis} z=ie^{iz))$^[1][23]

Integral

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

Other properties

These follow directly from Euler's formula.

${\displaystyle \cos(x)={\frac {\operatorname {cis} (x)+\operatorname {cis} (-x)}{2))={\frac {e^{ix}+e^{-ix)){2))}$

${\displaystyle \sin(x)={\frac {\operatorname {cis} (x)-\operatorname {cis} (-x)}{2i))={\frac {e^{ix}-e^{-ix)){2i))}$

${\displaystyle \operatorname {cis} (x+y)=\operatorname {cis} x\,\operatorname {cis} y}$^[24]

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

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

${\displaystyle |\operatorname {cis} x-\operatorname {cis} y|\leq |x-y|.}$^[24]

History

The cis notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866)^[25][26] and subsequently used by Irving Stringham (who also called it "sector of x") in works such as Uniplanar Algebra (1893),^[27][28] James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898),^[28][29] or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928).^[30][31][32]

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:^[33][34]

${\displaystyle \operatorname {cas} x=\cos x+\sin x.}$

Two hyperbolic function shortcuts have been defined as:^[35]

${\displaystyle \operatorname {cish} x=\cosh x+i\sinh x}$

${\displaystyle \operatorname {sich} x=\sinh x+i\cosh x}$

Motivation

The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another.^[36] 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).^[32]

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 for which they are not yet prepared: the usual proof that cis x = e^ix 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.