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Euler's identity therefore states that the limit, as n approaches infinity, of (+) is equal to −1. This limit is illustrated in the animation to the right. Euler's formula for a general angle. Euler's identity is a special case of Euler's formula, which states that for any real number x,
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.
This mathematical term forms part of an identity, a special case of Euler's formula, written = + (). Setting x {\displaystyle x} to a value of π {\displaystyle \pi } , as with the above term, Euler's formula reduces to a famous equation relating seven important mathematical symbols (and none that are unimportant!), namely e i π + 1 ...
The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula χ = V − E + F {\displaystyle \chi =V-E+F} where V , E , and F are respectively the numbers of v ertices (corners), e dges and f aces in the given polyhedron.
Euler's theorem Euler's theorem states that if n and a are coprime positive integers, then a φ(n) is congruent to 1 mod n. Euler's theorem generalizes Fermat's little theorem. Euler's totient function For a positive integer n, Euler's totient function of n, denoted φ(n), is the number of integers coprime to n between 1 and n inclusive.
Euler's identity is a special case of this: + =. This identity is particularly remarkable as it involves e, , i, 1, and 0, arguably the five most important constants in mathematics, as well as the four fundamental arithmetic operators: addition, multiplication, exponentiation, and equality.
The Euler function may be expressed as a q-Pochhammer symbol: = (;). The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q = 0, yielding
Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple yet ambiguous names such as Euler's function , Euler's equation , and Euler's formula .