  By using Cauchy's formula, it is known that Bernoulli numbers and Euler
numbers can be represented by the contour integrals \begin{equation*}
\begin{aligned} B_n&=\frac{n!}{2\pi i}\oint \frac{z}{e^z-1}\frac{d
z}{z^{n+1}},\label{condefi}\\[4pt] E_n&=\frac{n!}{2\pi i}\oint
\frac{2e^z}{e^{2z}+1}\frac{d z}{z^{n+1}}, \end{aligned} \end{equation*} while
the following Witt's formula represents Euler polynomials through the fermionic
$p$-adic integrals
  $$E_n(a)=\int_{\mathbb Z_p}(x+a)^nd\mu_{-1}(x).$$
  Base on the above Witt's identity and the binomial theorem, we prove some new
identities for the Euler polynomials briefly. In particular, some symmetry
properties of Euler polynomials have been discovered, which implies many
interesting identities (known or unknown), including the Kaneko-Momiyama type
identities (shown by Wu, Sun, and Pan) and the Alzer-Kwong type identity for
Euler polynomials.
