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