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
(with $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind)
$E\left(\NVar{k}\right)$: 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
(with $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function) and $\pi$ : the ratio of the circumference of a circle to its diameter)
$\tilde{e}_{0}(\NVar{x})=-\pi\operatorname{Gi}\left(-x\right)$: notation used by (Tumarkin, 1959); §9.1
(with $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function) and $\pi$ : the ratio of the circumference of a circle to its diameter)
$E_{\NVar{a},\NVar{b}}\left(\NVar{z}\right)$: 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
(with $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind)
$E\left(\NVar{\phi},\NVar{k}\right)$: 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
(with $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function)
${\rm Ec}_{\NVar{\nu}}^{\NVar{2m+1}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Es}^{% 2m+1}_{\nu}\left(z,k^{2}\right)$: notation used by Ince (1940b); §29.1
(with $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function)
$\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: 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
(with $\pi$ : the ratio of the circumference of a circle to its diameter and $\operatorname{erf}\NVar{z}$ : error function)
$\operatorname{erf}\NVar{z}$: 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
(with $F\left(\NVar{z}\right)$ : Dawson’s integral and $\mathrm{e}$ : base of natural logarithm)
${\rm Es}_{\NVar{\nu}}^{\NVar{2m+1}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Ec}^{% 2m+1}_{\nu}\left(z,k^{2}\right)$: notation used by Ince (1940b); §29.1
(with $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function)
${\rm Es}_{\NVar{\nu}}^{\NVar{2m+2}}(\NVar{z},\NVar{k^{2}})\propto\mathit{Es}^{% 2m+2}_{\nu}\left(z,k^{2}\right)$: notation used by Ince (1940b); §29.1
(with $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function)
$\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: 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)

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