# Notations E

$\equiv$
equals by definition; Common Notations and Definitions
$e$
elementary charge; §18.39(ii)
$\mathrm{e}$
base of natural logarithm; (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; (19.2.8_2)
$e_{0}(\NVar{x})=\pi\operatorname{Hi}\left(-x\right)$
notation used by Tumarkin (1959); §9.1
$\tilde{e}_{0}(\NVar{x})=-\pi\operatorname{Gi}\left(-x\right)$
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)
$e_{\NVar{j}}$
Weierstrass lattice roots; §23.3(i)
${\rm Ec}_{\NVar{\nu}}^{\NVar{2m}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Ec}^{2m% }_{\nu}\left(z,k^{2}\right)$
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)
$\operatorname{Ei}\left(\NVar{x}\right)$
exponential integral; §6.2(i)
$\operatorname{Ein}\left(\NVar{z}\right)$
complementary exponential integral; (6.2.3)
$\operatorname{el1}\left(\NVar{x},\NVar{k_{c}}\right)$
Bulirsch’s incomplete elliptic integral of the first kind; (19.2.11_5)
$\operatorname{el2}\left(\NVar{x},\NVar{k_{c}},\NVar{a},\NVar{b}\right)$
Bulirsch’s incomplete elliptic integral of the second kind; (19.2.12)
$\operatorname{el3}\left(\NVar{x},\NVar{k_{c}},\NVar{p}\right)$
Bulirsch’s incomplete elliptic integral of the third kind; (19.2.16)
$\operatorname{envAi}\left(\NVar{x}\right)$
envelope of Airy function $\operatorname{Ai}\left(\NVar{x}\right)$; §2.8(iii)
$\operatorname{envBi}\left(\NVar{x}\right)$
envelope of Airy function $\operatorname{Bi}\left(\NVar{x}\right)$; §2.8(iii)
$\operatorname{env}\mskip-2.0muJ_{\NVar{\nu}}\left(\NVar{x}\right)$
envelope of Bessel function $J_{\NVar{\nu}}\left(\NVar{x}\right)$; §2.8(iv)
$\operatorname{env}\mskip-2.0muY_{\NVar{\nu}}\left(\NVar{x}\right)$
envelope of Bessel function $Y_{\NVar{\nu}}\left(\NVar{x}\right)$; §2.8(iv)
$\mathrm{env}\mskip-1.0muU\left(\NVar{c},\NVar{x}\right)$
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)
$\equiv$
modular equivalence; Common Notations and Definitions
$\operatorname{Erf}\NVar{z}=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z$
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)
$\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$
exponential of trace; (1.2.77)
$\exp\NVar{z}$
exponential function; (4.2.19)