The Euler numbers or sometimes Euler numbers (by Leonhard Euler are) a sequence of integers, by the Taylor expansion of the hyperbolic secant hyperbolic
are defined. They are not to be confused with the two-parameter Euler numbers E (n, k).
The first Euler's numbers ≠ 0 are
All Euler's numbers with an odd index are zero, while those with an even index have alternating signs . Furthermore, with the exception of E 0, when dividing by 10 , the positive values have the remainder 5 and the negative values modulo 10 have the remainder −1 or value 9.
Some authors leave out the numbers with an odd index altogether, halve the indices, so to speak, since the values with 0 are not considered there, and define their Euler numbers as the remaining sequence. Sometimes Euler's numbers are also defined in such a way that they are all positive, i.e. correspond to ours .
The following applies to the asymptotic behavior of Euler's numbers
or more precisely
with the ~ -equivalence notation.
An easy-to-remember form of the recursion equation with the seed is
where is to be interpreted as and from what
or the explicit form through index transformation
The Euler's numbers can even be made exact
using the Hurwitz zeta function if is, represent. And using its functional equation (there with m = 1, n = 4) the elegant relationship
which identifies these numbers as scaled function values of this holomorphic function . So we also get
which creates a direct connection with the Bernoulli polynomials and thus with the Bernoulli numbers . Also applies
where the Dirichlet beta function denotes.
- Not to be confused with the Euler polynomials
Euler's polynomials are mostly characterized by their generating function
implicitly defined. The first are
But you can also add them to and then use the equation
Define inductively, where the lower limit of integration is 1/2 for odd and zero for even .
The Euler's polynomials are symmetric about , i. H.
and their functional values in the places and the relationship
suffice, whereby the Bernoulli number denotes the second kind. We also have the identity
Euler's polynomial has fewer than n real zeros for n> 5. So it has five (but two doubles, i.e. only three different ones), but only the two (trivial) zeros at 0 and at 1. Let be the set of zeros . Then
- where in the case n = 5 the number is to be assessed as 5, since the zeros must be counted with their multiplicity - and it applies
where the function actually indicates the number of elements applied to a set.
The sequence of Euler's numbers occurs, for example, in the Taylor expansion of
on. It is related to the sequence of Bernoulli numbers, which is what you can see in the representation
recognizes. From the radius of convergence of the Taylor expansion of the secant function - the cosine in the denominator becomes 0 at - from follows from the root criterion that must apply asymptotically. Of course, they also appear in the Taylor series of the higher derivatives of the hyperbolic secant or the Gudermann function .
They also appear with some improper integrals ; for example with the integral
Euler's numbers appear when counting the number of alternating permutations with an even number of elements. An alternate permutation of values a list of these values , so that these permutation no triples with containing the sorted is. In general, the number of alternating permutations of elements (which are comparable)
where the factor of two arises from the fact that each permutation can be converted into another alternating permutation by reversing the order. The following applies to
any (i.e. also odd) number
for , which gives another efficient algorithm for determining the . For odd numbers, the values are also called tangent numbers.
- JM Borwein, PB Borwein, K. Dilcher, Pi, Euler Numbers, and Asymptotic Expansions , AMM, V. 96, no. 8, (Oct. 1989), pp. 681-687
↑ M. Abramowitz and I. Stegun, Handbook of Mathematical Functions . Dover, NY 1964, p. 807