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EulerE

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

EulerE[n]
    gives the Euler number E_n.
EulerE[n, x]
    gives the Euler polynomial E_n(x).
Non-negative integer n gives the exact integer E_n (odd n give 0,
E_0 = 1, E_2 = -1, E_4 = 5); an inexact integer-valued n evaluates it at
machine or arbitrary (MPFR) precision. EulerE[n, x] expands the degree-n
polynomial with exact rational coefficients, staying symbolic in x or
evaluating numerically when x is inexact; EulerE[n, 1/2] folds to
2^-n EulerE[n]. Listable.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Table[EulerE[k], {k, 0, 10}]
Out[1]= {1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521}

In[2]:= EulerE[3, z]
Out[2]= 1/4 - 3/2 z^2 + z^3

Implementation notes

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= EulerE[4]
Out[1]= 5

In[1]:= Table[EulerE[2 n], {n, 0, 6}]
Out[1]= {1, -1, 5, -61, 1385, -50521, 2702765}

In[1]:= EulerE[6, x]
Out[1]= -3 x + 5 x^3 - 3 x^5 + x^6

In[1]:= Sum[EulerE[2 k] / (2 k)! Pi^(2 k), {k, 0, 5}]
Out[1]= 1 - 1/2 Pi^2 + 5/24 Pi^4 - 61/720 Pi^6 + 277/8064 Pi^8 - 50521/3628800 Pi^10

In[1]:= N[EulerE[5, 1/3], 40]
Out[1]= -0.24897119341563786008230452674897119341565

Notes

EulerE[n] is the integer Euler number E_n (odd n vanish, E_0 = 1), and EulerE[n, x] is the degree-n Euler polynomial with exact rational coefficients. The truncated secant-style series above is the partial sum of sec(Pi/2)'s generating expansion; the polynomial form stays symbolic in x or, given an inexact argument, evaluates to arbitrary precision.