is a
mathematical notation
Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling ...
defined by ,
where is the
cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
function, is the
imaginary unit
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
and is the
sine function. is the
argument
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
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, 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:
:
where . So,
:
i.e. "" is an
acronym
An acronym is a type of abbreviation consisting of a phrase whose only pronounced elements are the initial letters or initial sounds of words inside that phrase. Acronyms are often spelled with the initial Letter (alphabet), letter of each wor ...
for "".
It connects
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric 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 ...
s with
exponential functions in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
via Euler's formula. While the
domain of definition
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain of ...
is usually
,
complex values
are possible as well:
:
so the 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
Intel Corporation is an American multinational corporation and technology company headquartered in Santa Clara, California, and Delaware General Corporation Law, incorporated in Delaware. Intel designs, manufactures, and sells computer compo ...
's
Math Kernel Library (MKL)
or MathCW
), available for many compilers and programming languages (including
C,
C++,
Common Lisp
Common Lisp (CL) is a dialect of the Lisp programming language, published in American National Standards Institute (ANSI) standard document ''ANSI INCITS 226-1994 (S2018)'' (formerly ''X3.226-1994 (R1999)''). The Common Lisp HyperSpec, a hyperli ...
,
D,
Haskell
Haskell () is a general-purpose, statically typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research, and industrial applications, Haskell pioneered several programming language ...
,
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
:
Integral
:
Other properties
These follow directly from
Euler's formula.
:
:
:
:
The identities above hold if and are any complex numbers. If and are real, then
:
History
The 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 ") 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 line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
s (1901) and
Fourier integrals (1928).
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:
:
Motivation
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
A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember.
It makes use of e ...
(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 . The usual proof that requires
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, which the student may not have studied before encountering the expression .
This notation was more common when typewriters were used to convey mathematical expressions.
See also
*
De Moivre's formula
*
Euler's formula
*
Complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
*
Ptolemy's theorem
*
Phasor
*
Versor
Notes
References
{{DEFAULTSORT:Cis
Trigonometry
Mathematical identities