Notations Notations D Notations F

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

$E_n$

Euler numbers; §24.2(ii)

$E_n^{(\mathrm{\ell })}$

generalized Euler numbers; §24.16(i)

$E(\alpha )=E\left(k\right)$

notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind)

$E\left(k\right)$

Legendre’s complete elliptic integral of the second kind; (19.2.8)

$\eta \left(\tau \right)$

Dedekind’s eta function (or Dedekind modular function); (27.14.12)

$E_1\left(z\right)$

exponential integral; (6.2.1)

$E^{\prime }\left(k\right)$

Legendre’s complementary complete elliptic integral of the second kind; (19.2.8_2)

$e_0(x)=\pi \mathrm{Hi}\left(-x\right)$

notation used by Tumarkin (1959); §9.1
(with $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function) and $\pi$: the ratio of the circumference of a circle to its diameter)

${\tilde{e}}_0(x)=-\pi \mathrm{Gi}\left(-x\right)$

notation used by (Tumarkin, 1959); §9.1
(with $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function) and $\pi$: the ratio of the circumference of a circle to its diameter)

$E_{a,b}\left(z\right)$

Mittag-Leffler function; (10.46.3)

$E_n\left(x\right)$

Euler polynomials; §24.2(ii)

${𝐄}_{\nu }\left(z\right)$

Weber function; (11.10.2)

$E_p\left(z\right)$

generalized exponential integral; (8.19.1)

$E_q\left(x\right)$

$q$-exponential function; (17.3.2)

$e_q\left(x\right)$

$q$-exponential function; (17.3.1)

$E_s(𝐳)$

elementary symmetric function; (19.19.4)

$E_n^{(\mathrm{\ell })}\left(x\right)$

generalized Euler polynomials; §24.16(i)

${\tilde{E}}_n\left(x\right)$

periodic Euler functions; §24.2(iii)

$E(\varphi \backslash \alpha )=E(\varphi ,k)$

notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind)

$E(\varphi ,k)$

Legendre’s incomplete elliptic integral of the second kind; (19.2.5)

$e_j$

Weierstrass lattice roots; §23.3(i)

${\mathrm{Ec}}_{\nu }^{2m}(z,k^2)\propto {\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$

notation used by Ince (1940b); §29.1
(with ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function)

${\mathrm{Ec}}_{\nu }^{2m+1}(z,k^2)\propto {\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$

notation used by Ince (1940b); §29.1
(with ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function)

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$

Lamé function; §29.3(iv)

$\mathrm{Ei}\left(x\right)$

exponential integral; §6.2(i)

$\mathrm{Ein}\left(z\right)$

complementary exponential integral; (6.2.3)

$\mathrm{el1}(x,k_c)$

Bulirsch’s incomplete elliptic integral of the first kind; (19.2.11_5)

$\mathrm{el2}(x,k_c,a,b)$

Bulirsch’s incomplete elliptic integral of the second kind; (19.2.12)

$\mathrm{el3}(x,k_c,p)$

Bulirsch’s incomplete elliptic integral of the third kind; (19.2.16)

$\mathrm{envAi}\left(x\right)$

envelope of Airy function $\mathrm{Ai}\left(x\right)$; §2.8(iii)

$\mathrm{envBi}\left(x\right)$

envelope of Airy function $\mathrm{Bi}\left(x\right)$; §2.8(iii)

$\mathrm{env}{J}_{\nu }\left(x\right)$

envelope of Bessel function $J_{\nu }\left(x\right)$; §2.8(iv)

$\mathrm{env}{Y}_{\nu }\left(x\right)$

envelope of Bessel function $Y_{\nu }\left(x\right)$; §2.8(iv)

$\mathrm{env}U(c,x)$

envelope of parabolic cylinder function $U(c,x)$; §14.15(v)

$\mathrm{env}\overline{U}(c,x)$

envelope of parabolic cylinder function $\overline{U}(c,x)$; §14.15(v)

${ϵ}_{jk\mathrm{\ell }}$

Levi-Civita symbol; (1.6.14)

$ℰ(x,k)$

Jacobi’s epsilon function; (22.16.14)

$\equiv$

modular equivalence; Common Notations and Definitions

$\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$

alternative notation for the error function; §7.1
(with $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{erf}z$: error function)

$\mathrm{erf}z$

error function; (7.2.1)

$\mathrm{erfc}z$

complementary error function; (7.2.2)

$\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$

alternative notation for Dawson’s integral; §7.1
(with $F\left(z\right)$: Dawson’s integral and $\mathrm{e}$: base of natural logarithm)

${\mathrm{Es}}_{\nu }^{2m+1}(z,k^2)\propto {\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$

notation used by Ince (1940b); §29.1
(with ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function)

${\mathrm{Es}}_{\nu }^{2m+2}(z,k^2)\propto {\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$

notation used by Ince (1940b); §29.1
(with ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function)

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$

Lamé function; §29.3(iv)

$\mathrm{etr}\left(𝐀\right)$

exponential of trace; (1.2.77)

$\exp z$

exponential function; (4.2.19)

Notations D Notations F

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