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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 x, one has = + , where e is the base of the natural logarithm, i is the ...
Complex exponential and trigonometric functions are strongly related by Euler's formula: = + (). This formula provides the decomposition of complex exponential into real and imaginary parts : e x + i y = e x cos y + i e x sin y . {\displaystyle e^{x+iy}=e^{x}\,\cos y+ie^{x}\,\sin y.}
Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts , and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies e i φ = cos φ + i sin φ {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi }
Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describing uniform circular motion in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.
He introduced scientific notation. He discovered what is now known as Euler's formula, that for any real number, the complex exponential function satisfies = + . Richard Feynman called this "the most remarkable formula in mathematics". [3] Euler's identity is a special case of this:
Euler's identity is a direct result of Euler's formula, published in his monumental 1748 work of mathematical analysis, Introductio in analysin infinitorum, [16] but it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.
the matrix exponential reduces to a plain product of the exponentials of the two respective pieces. This is a formula often used in physics, as it amounts to the analog of Euler's formula for Pauli spin matrices, that is rotations of the doublet representation of the group SU(2).