DLMF: Notations E

Notations E

♦*♦A♦B♦C♦D♦E♦F♦G♦H♦I♦J♦K♦L♦M♦N♦O♦P♦Q♦R♦S♦T♦U♦V♦W♦X♦Y♦Z♦

$\in$: element of; Common Notations and Definitions
$\notin$: not an element of; Common Notations and Definitions
$e$: base of exponential function; (4.2.11)
$\mathop{E_{{n}}\/}\nolimits$: Euler numbers; §24.2(ii)
$\mathop{E^{{(\ell)}}_{{n}}\/}\nolimits$: generalized Euler numbers; ¶ ‣ §24.16(i)
$E(\alpha)=\mathop{E\/}\nolimits\!\left(k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind)
$\mathop{E\/}\nolimits\!\left(k\right)$: Legendre’s complete elliptic integral of the second kind; (19.2.8)
$\mathop{\eta\/}\nolimits\!\left(\tau\right)$: Dedekind’s eta function (or Dedekind modular function); (23.15.9)
$\mathop{\eta\/}\nolimits\!\left(\tau\right)$: Dedekind’s eta function (or Dedekind modular function); (27.14.12)
$E_{s}(\mathbf{z})$: elementary symmetric function; (19.19.4)
$\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$: Euler polynomials; §24.2(ii)
$\mathop{E_{1}\/}\nolimits\!\left(z\right)$: exponential integral; (6.2.1)
$\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$: generalized exponential integral; (8.19.1)
$\mathop{E_{{a,b}}\/}\nolimits\!\left(z\right)$: Mittag-Leffler function; (10.46.3)
$\mathop{E_{{q}}\/}\nolimits\!\left(x\right)$: $q$ -exponential function; (17.3.2)
$\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$: Weber function; (11.10.2)
$\mathop{e_{{q}}\/}\nolimits\!\left(x\right)$: $q$ -exponential function; (17.3.1)
$e_{0}(x)=\pi\mathop{\mathrm{Hi}\/}\nolimits\!\left(-x\right)$: notation used by Tumarkin (1959); §9.1
(with $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function))
$\tilde{e}_{0}(x)=-\pi\mathop{\mathrm{Gi}\/}\nolimits\!\left(-x\right)$: notation used by (Tumarkin, 1959); §9.1
(with $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function))
$\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)$: periodic Euler functions; §24.2(iii)
$\mathop{{E^{{\prime}}}\/}\nolimits\!\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind; (19.2.9)
$\mathop{E^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$: generalized Euler polynomials; ¶ ‣ §24.16(i)
$E(\phi\backslash\alpha)=\mathop{E\/}\nolimits\!\left(\phi,k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind)
$\mathop{E\/}\nolimits\!\left(\phi,k\right)$: Legendre’s incomplete elliptic integral of the second kind; (19.2.5)
${\rm Ec}_{\nu}^{{2m}}(z,k^{2})\propto\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$: notation used by Ince (1940b); §29.1
(with $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function)

${\rm Ec}_{\nu}^{{2m+1}}(z,k^{2})\propto\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$: notation used by Ince (1940b); §29.1
(with $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function)
$\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$: Lamé function; §29.3(iv)
$\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$: exponential integral; §6.2(i)
$\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$: complementary exponential integral; (6.2.3)
$\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)$: Bulirsch’s incomplete elliptic integral of the first kind; (19.2.15)
$\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right)$: Bulirsch’s incomplete elliptic integral of the second kind; (19.2.12)
$\mathop{\mathrm{el3}\/}\nolimits\!\left(x,k_{c},p\right)$: Bulirsch’s incomplete elliptic integral of the third kind; (19.2.16)
$\mathop{\mathrm{env}\/}\nolimits\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$: envelope of Airy function; (2.8.20)
$\mathop{\mathrm{env}\/}\nolimits\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$: envelope of Airy function; (2.8.20)
$\mathop{\mathrm{env}\/}\nolimits\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)$: envelope of Bessel function; (2.8.33)
$\mathop{\mathrm{env}\/}\nolimits\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)$: envelope of Bessel function; (2.8.33)
$\mathop{\mathrm{env}\/}\nolimits\mathop{U\/}\nolimits\!\left(-c,x\right)$: envelope of parabolic cylinder function; §14.15(v)
$\mathop{\mathrm{env}\/}\nolimits\mathop{\overline{U}\/}\nolimits\!\left(-c,x\right)$: envelope of parabolic cylinder function; §14.15(v)
$\epsilon _{{j,k,\ell}}$: Levi-Civita symbol; (1.6.14)
$\mathop{\mathcal{E}\/}\nolimits\!\left(x,k\right)$: Jacobi’s epsilon function; (22.16.14)
$\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\mathop{\mathrm{erf}\/}\nolimits z$: alternative notation for the error function; §7.1
(with $\mathop{\mathrm{erf}\/}\nolimits z$ : error function)
$\mathop{\mathrm{erf}\/}\nolimits z$: error function; (7.2.1)
$\mathop{\mathrm{erfc}\/}\nolimits z$: complementary error function; (7.2.2)
$\operatorname{Erfi}z=e^{{z^{2}}}\mathop{F\/}\nolimits\!\left(z\right)$: alternative notation for Dawson’s integral; §7.1
(with $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral and $e$ : base of exponential function)
${\rm Es}_{\nu}^{{2m+1}}(z,k^{2})\propto\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$: notation used by Ince (1940b); §29.1
(with $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function)
${\rm Es}_{\nu}^{{2m+2}}(z,k^{2})\propto\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$: notation used by Ince (1940b); §29.1
(with $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function)
$\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$: Lamé function; §29.3(iv)
$\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$: exponential of trace; §35.1
$\mathop{\exp\/}\nolimits z$: exponential function; (4.2.19)