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Euler's formula

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:

e = 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.



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