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, , which offers an even shorter and more general notation for cos(x) + i sin(x).
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 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,OS X and HP-UX). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually.
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