Cis (mathematics)

is a 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 function. The notation is less commonly used in mathematics than Euler's formula, eix, 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. The cis notation is a shorthand for the combination of functions on the right-hand side of Euler's formula: where i2 = −1. So, 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 , complex values are possible as well: so the cis 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, 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. These follow directly from Euler's formula. The identities above hold if x and y are any complex numbers. If x and y are real, then The cis 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 x") 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).