Expi (mathematics)

is a mathematical notation defined by , where is the cosine function, is the imaginary unit and is the sine function. The notation is less commonly used in mathematics than Euler's formula, which offers an even shorter notation for but cis(x) is widely used as a name for this function in software libraries.

Overview

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

:$e^ = \cos x + i\sin x,$

where . So,

:$\operatorname x = \cos x + i\sin x,$

i.e. "" is an acronym for "".

The 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 Rust), 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.

Mathematical identities

Derivative

:$\frac\operatorname z = i\operatorname z = ie^$

Integral

:$\int\operatorname z \,\mathrmz = -i\operatorname z = -ie^$

Other properties

These follow directly from Euler's formula.

:$\operatorname(x+y) = \operatorname x\,\operatorname y$

:$\operatorname(x-y) =$

The identities above hold if and are any complex numbers. If and are real, then
:$, \operatorname x - \operatorname y, \le , x-y, .$

History

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

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

The 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 . 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 requires calculus, which the student may not have studied before encountering the expression .

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

: $\operatorname x = \cos x + \sin x.$

See also

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

References

{{DEFAULTSORT:Cis
Trigonometry
Mathematical identities