Euler numbers: Difference between revisions
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In the field of [[mathematics]], Euler numbers, named after the Swiss mathematician [[Leonhard Euler]], are a sequence of integers that play a key role in the analysis of series and the computation of certain types of integrals. They are closely related to, but distinct from, the Eulerian numbers and the Euler's totient function. | In the field of [[mathematics]], Euler numbers, named after the Swiss mathematician [[Leonhard Euler]], are a sequence of integers that play a key role in the analysis of series and the computation of certain types of integrals. They are closely related to, but distinct from, the Eulerian numbers and the Euler's totient function. | ||
[[Image:Detail-144863.jpg|thumb|center|A sequence of numbers on a blackboard, representing the Euler numbers]] | |||
== Definition == | == Definition == | ||
Revision as of 03:54, 29 October 2025
Introduction
In the field of mathematics, Euler numbers, named after the Swiss mathematician Leonhard Euler, are a sequence of integers that play a key role in the analysis of series and the computation of certain types of integrals. They are closely related to, but distinct from, the Eulerian numbers and the Euler's totient function.
Definition
Euler numbers are defined through the generating function:
- E(x) = sec(x) + tan(x) = ∑ E_n * x^n / n!
where E_n denotes the nth Euler number, sec(x) is the secant function, tan(x) is the tangent function, and the sum extends over all nonnegative integers n.
Properties
Euler numbers have several interesting properties. For instance, they are all integers, and they alternate in sign. That is, E_0 is positive, E_1 is negative, E_2 is positive, and so on. The sequence of Euler numbers begins 1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, and so on.
Another property of Euler numbers is that they appear in the Maclaurin series expansions of the secant and tangent functions. This is a direct consequence of their definition through the generating function.
Applications
Euler numbers have numerous applications in mathematics. They appear in the computation of definite integrals, in the analysis of series, and in the study of combinatorial objects known as alternating permutations.
In particular, the nth Euler number gives the number of alternating permutations of the set {1, 2, ..., n}, where a permutation is said to be alternating if it starts with a rise and then alternates between falls and rises. This connection between Euler numbers and alternating permutations is a special case of a more general phenomenon in combinatorics, where sequences of numbers are often associated with certain types of combinatorial structures.
Relation to Other Mathematical Concepts
Euler numbers are related to several other mathematical concepts. For instance, they are connected to the Bernoulli numbers through the Euler–Maclaurin formula, which is a formula for approximating integrals in terms of finite sums.
Euler numbers are also related to the Stirling numbers of the second kind, which count the number of ways to partition a set into a given number of nonempty subsets. The relationship between Euler numbers and Stirling numbers is given by a certain combinatorial identity.