This article is about the Euler's formula in complex analysis. For Euler's formula in algebraic topology, see Euler characteristic. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...

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). Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler [oilər] (April 15, 1707 - September 18, 1783) was a Swiss mathematician and physicist. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, Eulers identity is the following equation: sometimes expressed as: presumably in order to use the fundamental numbers 0 and 1 (see below). ...

Euler's formula states that, for any real number x, The text or formatting below is generated by a template which has been proposed for deletion. ...

where

e is the base of the natural logarithm

i is the imaginary unit

sin and cos are trigonometric functions.

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 Wessel). Roger Cotes born July 10, 1682 in Burbage, Leicestershire, England died June 5, 1716 in Cambridge, Cambridgeshire was a mathematician. ... Events August 1 - George, elector of Hanover becomes King George I of Great Britain. ... Events April 24 - A congress assembles at Aix-la-Chapelle with the intent to conclude the struggle known as the War of Austrian Succession - at October 18 - The Treaty of Aix-la-Chapelle is signed to end the war Adam Smith begins to deliver public lectures in Edinburgh Building of... Caspar Wessel (June 8, 1745 - March 25, 1818) was a Norwegian-Danish mathematician. ...

Notes

This formula can be interpreted as saying that the function e^ix traces out the unit circle in the complex number plane as x ranges through the real numbers. Here, x is the angle 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. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... This article is about angles in geometry. ...

The proof is based on the Taylor series 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. As the degree of the taylor series rises, it approaches the correct function. ... The exponential function is one of the most important functions in mathematics. ...

The formula provides a powerful connection between analysis and trigonometry. It is used to represent complex numbers in polar coordinates and allows the definition of the logarithm for complex arguments. By using the exponential laws Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In mathematics, a logarithm of x with base b may be defined as the following: for the equation bn = x, the logarithm is a function which gives n. ...

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: In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... De Moivres formula states that for any real number x and any integer n, The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

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 or subtracting 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 mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, 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 mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, Eulers identity is the following equation: sometimes expressed as: presumably in order to use the fundamental numbers 0 and 1 (see below). ...

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. Electrical engineering is an engineering discipline that deals with the study and application of electricity and electromagnetism. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero. ...

Proofs

Using Taylor series

Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i: As the degree of the taylor series rises, it approaches the correct function. ...

and so on. The functions e^x, cos(x) and sin(x) (assuming x is real) can be written as: The text or formatting below is generated by a template which has been proposed for deletion. ...

and for complex z we define each of these function by the above series, replacing x with iz. This is possible because the radius of convergence of each series is infinite. We then find that

The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number, gives the original identity as Euler discovered it.

Q.E.D. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek oper edei deixai which was used by many early mathematicians including Euclid and Archimedes. ...

Using calculus

Define the complex number z such that

Differentiating z with respect to x:

Using the fact that i² = -1:

Separating variables and integrating both sides:

where

C is the constant of integration.

To finish the proof we have to argue that it is zero. This is easily done by substituting x = 0.

But z is just equal to:

thus

So now we just exponentiate

Q.E.D.

External links

• Euler and his beautiful and extraordinary formula (http://agutie.homestead.com/files/Eulerformula.htm) by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Euler's Formula - Puzzle: 55 pieces in a six star style of piece (http://agutie.homestead.com/files/Puzzle_EulerFormula.htm) by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.