EulerE

EulerE[n]

gives the Euler number .

EulerE[n,x]

gives the Euler polynomial .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Euler polynomials satisfy the generating function relation .
  • The Euler numbers are given by .
  • EulerE automatically threads over lists.

Examples

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Basic Examples  (2)

First 10 EulerE numbers:

Euler polynomials:

Scope  (4)

EulerE threads element-wise 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  (3)

Umbral calculus with Euler numbers:

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

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

Introduced in 1988
 (1.0)