# cis (mathematics)

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, $e^{ix}$ , which offers an even shorter and more general notation for cos(x) + i sin(x).

## Overview

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.

It is mostly used as a convenient shorthand notation to simplify some expressions, for example in conjunction with Fourier and Hartley transforms, 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)), available for many compilers, programming languages (including C, C++, Common Lisp, D, Fortran, Haskell, Julia), and operating systems (including Windows, Linux, macOS 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

$e^{ix}=\cos(x)+i\sin(x),$ $e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin(x)$ $e^{i\pi }=-1$ $\cos(x)={\frac {e^{ix}+e^{-ix}}{2}}$ $\sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}$ where i2 = −1.

This can also be expressed using the following notation

$\operatorname {cis} (x)=\cos(x)+i\sin(x),$ 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, such as being directly tied to the polar form of a complex number (and being easier to grasp).

## Mathematical identities

### Derivative

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

$\int \operatorname {cis} (z)\,\mathrm {d} z=-i\operatorname {cis} (z)=-ie^{iz}$ ### Other properties

These follow directly from Euler's formula.

$\operatorname {cis} (x+y)=\operatorname {cis} (x)\,\operatorname {cis} (y)$ $\operatorname {cis} (x-y)={\operatorname {cis} (x) \over \operatorname {cis} (y)}$ The identities above hold if x and y are any complex numbers. If x and y are real, then

$|\operatorname {cis} (x)-\operatorname {cis} (y)|\leq |x-y|.$ ## 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, eix2 versus cis(x2) or exp(ix2). For many readers, cis(x2) is the clearest, easiest to read of the three.[citation needed]

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 does not 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 for which they are not yet prepared: 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).

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:

cas(x) = cos(x) + sin(x).