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Euler's formula, named after Leonhard Euler, is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ( Euler's identity is a special case of the Euler formula).

Euler's formula states that, for any real number x,

Here, e is the base of the natural logarithm, i is the imaginary unit and sin and cos are trigonometric functions.

1 History

Euler's formula was proved (in an obscured form) for the first time by Roger Cotes in 1714, then rediscovered and popularized by Euler in 1748. It is interesting to note that neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the plane arose only some 50 years later (see Caspar WesselCaspar Wessel ( June 8, 1745 March 25, 1818) was a Norwegian- Danish mathematician. Wessel was born in Jonsrud, Vestby, Akershus, Norway. In 1763, having completed secondary school, he went to Denmark for further studies (Norway having at the time no univ).

2 Notes

This formula can be interpreted as saying that the function e^ix traces out the unit circle in the complex number planeThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , wher as x ranges through the real numbers. Here, x is the angleThis article is about angles in geometry. For other articles, see Angle (disambiguation An angle (from the Lat. angulus a corner, a diminutive, of which the primitive form, angus does not occur in Latin; cognate are the Lat. angere, to compress into a ben that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians. The formula is valid only if sin and cos take their arguments in radians rather than in degrees.

The proof is based on the Taylor seriesIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point expansions of the exponential function e^z (where z is a complex number) and of sin x and cos x for real numbers x (see below). In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.

The formula provides a powerful connection between analysisAnalysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in g and trigonometryTrigonometry (Greek: "the measure of triangles") is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine and cosine . It has some relationship to geometry, though there is disagreement on exactly what that relati. It is used to represent complex numbers in polar coordinatesThis article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. The coordinates of a point are the components and allows the definition of the logarithmIn mathematics, the logarithm functions are the inverses of the exponential functions. Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are r for complex arguments. By using the exponential laws

and

(which are valid for all complex numbers a and b), one can also readily derive several trigonometric identities as well as de Moivre's formula from it. Euler's formula also allows one to interpret the sine and cosine functions as mere variations of the exponential function:

These formulas can even serve as the definition of the trigonometric functions for complex arguments x. You can derive the two equations above simply by adding Euler's formulas:

and solving for either cosine or sine.

The formulae above can also be used to relate the hyperbolic sine and hyperbolic cosine functions to the usual trigonometric functions.

In differential equations, the function e^ix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. Euler's identity is an easy consequence of Euler's formula.

In electrical engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier analysis), and these are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula.