Part of a series of articles on
The mathematical constant e
Natural logarithm · Exponential function
Applications in: compound interest · Euler's identity & Euler's formula · halflives & exponential growth/decay
Defining e: proof that e is irrational · representations of e · Lindemann–Weierstrass theorem
People John Napier · Leonhard Euler
Schanuel's conjecture
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x,
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 x given in radians. This complex exponential function is sometimes called cis(x). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.^{[1]}
Richard Feynman called Euler's formula "our jewel"^{[2]} and "one of the most remarkable, almost astounding, formulas in all of mathematics."^{[3]}
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