Cis (mathematics)

Template:Short description Template:Redir Template:Lowercase title Template:Use dmy dates In mathematics, Template:Math is a function defined by Template:Math, where Template:Math is the cosine function, Template:Mvar is the imaginary unit and Template:Math is the sine function. Template:Mvar is the argument of the complex number (angle between line to point and x-axis in polar form). It is mathematically equivalent to Euler's formula, Template:Nowrap which offers an even shorter notation for Template:Nowrap but cis(x) is widely used as a name for this function in software libraries.

Overview

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

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

where Template:Math. So,

$\mathrm{cis}x=\cos x+i\sin x,$

i.e. "Template:Math" is an acronym for "Template:Math".

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

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

so the Template:Math function can be used to extend Euler's formula to a more general complex version.

The function 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) or MathCW), available for many compilers and programming languages (including C, C++, Common Lisp, D, Haskell, Julia, and Rust). Depending on the platform, the fused operation is about twice as fast as calling the sine and cosine functions individually.

Mathematical identities

Derivative

$\frac{d}{dz}\mathrm{cis}z=i\mathrm{cis}z=i{e}^{iz}$

Integral

$\int \mathrm{cis}zdz=-i\mathrm{cis}z=-i{e}^{iz}$

Other properties

These follow directly from Euler's formula.

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

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

$\mathrm{cis}(x+y)=\mathrm{cis}x\mathrm{cis}y$

$\mathrm{cis}(x-y)=\frac{\mathrm{cis}x}{\mathrm{cis}y}$

$\mathrm{cis}(nx)=\mathrm{cis}(x{)}^n$

$\mathrm{cis}\left(\frac{x}{n}\right)=\sqrt[n]{\mathrm{cis}(x)}$

The identities above hold if Template:Mvar and Template:Mvar are any complex numbers. If Template:Mvar and Template:Mvar are real, then

$|\mathrm{cis}x-\mathrm{cis}y|\le |x-y|.$

History

The Template:Math notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham (who also called it "sector of Template:Math") in works such as Uniplanar Algebra (1893), James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898), or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928).

Template:AnchorIn 1942, inspired by the Template:Math notation, Ralph V. L. Hartley introduced the Template:Math (for cosine-and-sine) function for the real-valued Hartley kernel, a meanwhile established shortcut in conjunction with Hartley transforms:

$\mathrm{cas}x=\cos x+\sin x.$

Motivation

The Template:Math 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 Template:Math and Template:Math notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for Template:Math).

The Template:Math 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 Template:Math. The usual proof that Template:Math requires calculus, which the student may not have studied before encountering the expression Template:Math.

This notation was more common when typewriters were used to convey mathematical expressions.Template:Cn

See also

• Complex number
• De Moivre's formula
• Euler's formula
• Phasor
• Ptolemy's theorem
• Versor