complex exponential

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number :

where is the base of the natural logarithm, is the imaginary unit, and and are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted ("cosine plus i sine"). The formula is still valid if is a complex number, and so some authors refer to the more general complex version as Euler's formula.