# Notations E

*ABCD♦E♦FGHIJKLMNOPQRSTUVWXYZ
$\mathrm{e}$
base of exponential function; 4.2.11
$\in$
element of; Common Notations and Definitions
$\notin$
not an element of; Common Notations and Definitions
$E_{\NVar{n}}$
Euler numbers; §24.2(ii)
$E^{(\NVar{\ell})}_{\NVar{n}}$
generalized Euler numbers; §24.16(i)
$E(\NVar{\alpha})=E\left(k\right)$
notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
$E\left(\NVar{k}\right)$
Legendre’s complete elliptic integral of the second kind; 19.2.8
$\eta\left(\NVar{\tau}\right)$
Dedekind’s eta function (or Dedekind modular function); 27.14.12
$E_{1}\left(\NVar{z}\right)$
exponential integral; 6.2.1
${E^{\prime}}\left(\NVar{k}\right)$
Legendre’s complementary complete elliptic integral of the second kind; 19.2.9
$e_{0}(\NVar{x})=\pi\mathrm{Hi}\left(-x\right)$
notation used by Tumarkin (1959); §9.1
$\tilde{e}_{0}(\NVar{x})=-\pi\mathrm{Gi}\left(-x\right)$
notation used by (Tumarkin, 1959); §9.1
$E_{\NVar{a},\NVar{b}}\left(\NVar{z}\right)$
Mittag-Leffler function; 10.46.3
$E_{\NVar{n}}\left(\NVar{x}\right)$
Euler polynomials; §24.2(ii)
$\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$
Weber function; 11.10.2
$E_{\NVar{p}}\left(\NVar{z}\right)$
generalized exponential integral; 8.19.1
$E_{\NVar{q}}\left(\NVar{x}\right)$
$q$-exponential function; 17.3.2
$e_{\NVar{q}}\left(\NVar{x}\right)$
$q$-exponential function; 17.3.1
$E_{\NVar{s}}(\NVar{\mathbf{z}})$
elementary symmetric function; 19.19.4
$E^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$
generalized Euler polynomials; §24.16(i)
$\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)$
periodic Euler functions; §24.2(iii)
$E(\NVar{\phi}\backslash\NVar{\alpha})=E\left(\phi,k\right)$
notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
$E\left(\NVar{\phi},\NVar{k}\right)$
Legendre’s incomplete elliptic integral of the second kind; 19.2.5
${\rm Ec}_{\NVar{\nu}}^{\NVar{2m}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Ec}^{2m% }_{\nu}\left(z,k^{2}\right)$
notation used by Ince (1940b); §29.1
${\rm Ec}_{\NVar{\nu}}^{\NVar{2m+1}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Es}^{% 2m+1}_{\nu}\left(z,k^{2}\right)$
notation used by Ince (1940b); §29.1
$\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$
Lamé function; §29.3(iv)
$\mathrm{Ei}\left(\NVar{x}\right)$
exponential integral; §6.2(i)
$\mathrm{Ein}\left(\NVar{z}\right)$
complementary exponential integral; 6.2.3
$\mathrm{el1}\left(\NVar{x},\NVar{k_{c}}\right)$
Bulirsch’s incomplete elliptic integral of the first kind; 19.2.15
$\mathrm{el2}\left(\NVar{x},\NVar{k_{c}},\NVar{a},\NVar{b}\right)$
Bulirsch’s incomplete elliptic integral of the second kind; 19.2.12
$\mathrm{el3}\left(\NVar{x},\NVar{k_{c}},\NVar{p}\right)$
Bulirsch’s incomplete elliptic integral of the third kind; 19.2.16
$\mathrm{envAi}\left(\NVar{x}\right)$
envelope of Airy function $\mathrm{Ai}\left(\NVar{x}\right)$; §2.8(iii)
$\mathrm{envBi}\left(\NVar{x}\right)$
envelope of Airy function $\mathrm{Bi}\left(\NVar{x}\right)$; §2.8(iii)
$\mathrm{env}\mskip-2.0mu J_{\NVar{\nu}}\left(\NVar{x}\right)$
envelope of Bessel function $J_{\NVar{\nu}}\left(\NVar{x}\right)$; §2.8(iv)
$\mathrm{env}\mskip-2.0mu Y_{\NVar{\nu}}\left(\NVar{x}\right)$
envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$; §2.8(iv)
$\mathrm{env}\mskip-1.0mu U\left(\NVar{c},\NVar{x}\right)$
envelope of parabolic cylinder function $U\left(\NVar{c},\NVar{x}\right)$; §14.15(v)
$\mathrm{env}\mskip-1.0mu \overline{U}\left(\NVar{c},\NVar{x}\right)$
envelope of parabolic cylinder function $\overline{U}\left(\NVar{c},\NVar{x}\right)$; §14.15(v)
$\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$
Levi-Civita symbol; 1.6.14
$\mathcal{E}\left(\NVar{x},\NVar{k}\right)$
Jacobi’s epsilon function; 22.16.14
$\operatorname{Erf}\NVar{z}=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z$
alternative notation for the error function; §7.1
$\operatorname{erf}\NVar{z}$
error function; 7.2.1
$\operatorname{erfc}\NVar{z}$
complementary error function; 7.2.2
$\operatorname{Erfi}\NVar{z}=e^{z^{2}}F\left(z\right)$
alternative notation for Dawson’s integral; §7.1
${\rm Es}_{\NVar{\nu}}^{\NVar{2m+1}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Ec}^{% 2m+1}_{\nu}\left(z,k^{2}\right)$
notation used by Ince (1940b); §29.1
${\rm Es}_{\NVar{\nu}}^{\NVar{2m+2}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Es}^{% 2m+2}_{\nu}\left(z,k^{2}\right)$
notation used by Ince (1940b); §29.1
$\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$
Lamé function; §29.3(iv)
$\mathrm{etr}\left(\NVar{\mathbf{X}}\right)$
exponential of trace; §35.1
$\exp\NVar{z}$
exponential function; 4.2.19