However, the phase of this contribution at any given point along the path is determined by the action along the path ( see Euler's formula ):

: This article is about Euler's formula in complex analysis.

For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic.

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,

This complex exponential function is sometimes denoted 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.

The physicist Richard Feynman called Euler's formula " our jewel " and " one of the most remarkable, almost astounding, formulas in all of mathematics.

Three-dimensional visualization of Euler's formula.

In fact, the same proof shows that Euler's formula is even valid for all complex numbers z.

Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates.

Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number.

which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities as well as de Moivre's formula.

Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:

Euler's identity is an easy consequence of Euler's formula.

In electronic 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.

Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

Here is a proof of Euler's formula using power series expansions

It follows from Euler's polyhedron formula, V − E + F = 2 ( where V, E, F are the numbers of vertices, edges, and faces ), that there are exactly 12 pentagons in a fullerene and V / 2 − 10 hexagons.

This together with Euler's formula v − e + f

One way to find that analytic continuation is to use Euler's integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,

Euler's work was presented to the St. Petersburg Academy on August 26, 1735, and published as Solutio problematis ad geometriam situs pertinentis ( The solution of a problem relating to the geometry of position ) in the journal Commentarii academiae scientiarum Petropolitanae in 1741.

The number of integers coprime to a positive integer n, between 1 and n, is given by Euler's totient function ( or Euler's phi function ) φ ( n ).

Euler's name is associated with a large number of topics.

He also introduced the modern notation for the trigonometric functions, the letter for the base of the natural logarithm ( now also known as Euler's number ), the Greek letter Σ for summations and the letter to denote the imaginary unit.

While the bel uses the decadic ( base-10 ) logarithm to compute ratios, the neper uses the natural logarithm, based on Euler's number ( e ≈ 2. 71828 ).

In number theory, Euler's totient or phi function, φ ( n ) is an arithmetic function that counts the number of positive integers less than or equal to n that are relatively prime to n. That is, if n is a positive integer, then φ ( n ) is the number of integers k in the range 1 ≤ k ≤ n for which gcd ( n, k ) = 1.

Euler's formula involves a complex power of a positive real number and this always has a defined value.

In number theory, Euler's theorem ( also known as the Fermat – Euler theorem or Euler's totient theorem ) states that if n and a are coprime positive integers, then

So if it's not known whether a number n is prime or composite, we can pick a random number a, calculate the Jacobi symbol and compare it with Euler's formula ; if they differ modulo n, then n is composite ; if they're the same modulo n for many different values of a, then n is " probably prime ".

In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime.

By Euler's formula the number of faces (' F '), vertices ( V ), and edges ( E ) of any convex polyhedron are related by the formula " F + V-E "

* Euler's totient function φ ( n ) in number theory ; also called Euler's phi function.

The order of ( i. e. the number of elements in ) Z < sub > n </ sub >< sup >×</ sup > is given by Euler's totient function Euler's theorem says that a < sup > φ ( n )</ sup > ≡ 1 ( mod n ) for every a coprime to n ; the lowest power of a which is congruent to 1 modulo n is called the multiplicative order of a modulo n. In particular, for a to be a primitive root modulo n, φ ( n ) has to be the smallest power of a which is congruent to 1 modulo n.

In addition, Euler's recognition that the key information was the number of bridges and the list of their endpoints ( rather than their exact positions ) presaged the development of topology.

Here | z | is the absolute value or modulus of the complex number z ; θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π ; and the last equality ( to | z | e < sup > iθ </ sup >) is taken from Euler's formula.

This result is known as Euler's polyhedron formula or theorem.

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees.

De Camp and Ley also claim that Sir John Leslie expanded on Euler's idea, suggesting two central suns named Pluto and Proserpine ( this was unrelated to the dwarf planet Pluto, which was discovered and named some time later ).

Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother.

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin.

A short obituary for the Russian Academy of Sciences was written by Jacob von Staehlin-Storcksburg and a more detailed eulogy was written and delivered at a memorial meeting by Russian mathematician Nicolas Fuss, one of Euler's disciples.

To commemorate the 250th anniversary of Euler's birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery.

By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent.

He also participated in the publication of Euler's selected works: he was an editor of the volumes 18 and 19.

The three other bridges remain, although only two of them are from Euler's time ( one was rebuilt in 1935 ).

The first prize was given in 2007, the 300th anniversary of Leonhard Euler's birth, to John Derbyshire for his book about Riemann and the Riemann hypothesis: Prime Obsession.

Euler's Seven Bridges of Königsberg problem was based on the rivers ' bridges in Königsberg, now Kaliningrad.

This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture.

He pursued mathematics as an amateur, his most famous achievement being his confirmation in 1901 of Leonhard Euler's conjecture that no 6 × 6 Graeco-Latin square was possible .< ref >

Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight.

Of course, Euler's original reasoning requires justification, but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series.

The earliest example of an infinitesimal transformation that may have been recognised as such was in Euler's theorem on homogeneous functions.

A forerunner of the method of variation of a celestial body's orbital elements appeared in Euler's work in 1748, while he was studying the mutual perturbations of Jupiter and Saturn.

An exhaustive analysis of the soluble generalizations of Euler's three-body problem was carried out by Adam Hiltebeitel in 1911.

Early and strong evidence was given by Euler's 1979 examination on shared features in Greek and Sanskrit nominal flection.