EulerE

EulerE[n]

gives the Euler number TemplateBox[{n}, EulerE].

EulerE[n,x]

gives the Euler polynomial TemplateBox[{n, x}, EulerE2].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Euler polynomials satisfy the generating function relation 2e^(xt)/(e^t+1)=sum_(n=0)^(infty)TemplateBox[{n, x}, EulerE2](t^n/n!).
  • The Euler numbers are given by TemplateBox[{n}, EulerE]=2^nTemplateBox[{n, {1, /, 2}}, EulerE2].
  • EulerE automatically threads over lists.

Examples

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Basic Examples  (2)

First 10 EulerE numbers:

Euler polynomials:

Scope  (4)

EulerE threads elementwise over lists:

Plot Euler polynomials:

Simple exact values are generated automatically:

TraditionalForm formatting:

Applications  (2)

Implement the Boole summation formula:

First a sequence of approximations to sum_(k=0)^n(-1)^kk^3:

The sequence converges to the exact answer:

Plot roots of Euler polynomials in the complex plane:

Properties & Relations  (5)

Find Euler numbers from their generating function:

Find Euler polynomials from their generating function:

EulerE can be represented as a DifferenceRoot:

FindSequenceFunction can recognize the EulerE sequence:

The exponential generating function for EulerE:

Possible Issues  (1)

Algorithmically produced results are often expressed using Zeta instead of EulerE:

Neat Examples  (4)

Umbral calculus with Euler numbers:

Histogram of digits of 10000^(th) Euler number:

The sequence of Euler numbers modulo a fixed number is periodic:

Define a Hankel matrix whose entries are the Euler numbers:

Its determinant can be expressed in terms of the Barnes G-function:

Wolfram Research (1988), EulerE, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerE.html.

Text

Wolfram Research (1988), EulerE, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerE.html.

CMS

Wolfram Language. 1988. "EulerE." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/EulerE.html.

APA

Wolfram Language. (1988). EulerE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EulerE.html

BibTeX

@misc{reference.wolfram_2025_eulere, author="Wolfram Research", title="{EulerE}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/EulerE.html}", note=[Accessed: 21-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_eulere, organization={Wolfram Research}, title={EulerE}, year={1988}, url={https://reference.wolfram.com/language/ref/EulerE.html}, note=[Accessed: 21-February-2025 ]}