Euler numbers
Euler numbers En have the generating function 1coshx such that
1coshx=:∞∑n=0Enn!xn. |
They are integers but have no expression for calculating them. Their only are that the numbers with odd index (http://planetmath.org/IndexingSet) are all 0 and that
sgn(E2m)=(-1)m |
The Euler number have intimate relation to the Bernoulli numbers. The first Euler numbers with even index are
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One can by hand determine Euler numbers by performing the division of 1 by the Taylor series
of hyperbolic cosine
(cf. Taylor series via division and Taylor series of hyperbolic functions). Since , the division correspondingly gives only terms with plus sign, i.e. it shows the absolute values
of the Euler numbers.
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The Euler numbers may also be obtained by using the Euler polynomials
:
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If the Euler numbers are denoted as symbolic powers , then one may write the equation
which can be used as a recurrence relation for computing the values of the even-indexed Euler numbers. Cf. the Leibniz rule
for derivatives of product .
Title | Euler numbers |
---|---|
Canonical name | EulerNumbers |
Date of creation | 2014-12-02 17:43:40 |
Last modified on | 2014-12-02 17:43:40 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 11 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 11B68 |
Related topic | GudermannianFunction |
Related topic | BernoulliNumber |
Related topic | InverseGudermannianFunction |
Related topic | HermiteNumbers |