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Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are ...
The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for any integrable function. In particular, the function could be nowhere differentiable. (For example, f (x) could be a Weierstrass function.) The convergence of both series has very different properties.
In another poll of readers that was conducted by Physics World in 2004, Euler's identity tied with Maxwell's equations (of electromagnetism) as the "greatest equation ever". [12] At least three books in popular mathematics have been published about Euler's identity: Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, by Paul Nahin (2011 ...
The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This ...
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact valid even for complex values of x , because both sides are entire (that is, holomorphic on the whole complex plane ) functions of x , and two such functions that ...