DLMF: Notations T

Notations T

♦*♦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♦

$T_{n}$: tangent numbers; §24.15(ii)
$\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$: Chebyshev polynomial of the first kind; Table 18.3.1
$\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$: shifted Chebyshev polynomial of the first kind; Table 18.3.1
$\mathop{\tan\/}\nolimits z$: tangent function; (4.14.4)
$\mathop{\tanh\/}\nolimits z$: hyperbolic tangent function; (4.28.4)
$\mathop{\tau\/}\nolimits\!\left(n\right)$: Ramanujan’s tau function; (27.14.18)
$\mathop{\theta\/}\nolimits\!\left(z\right)$: Airy phase function; §9.8(i)
$\mathop{\vartheta\/}\nolimits\!\left(x\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(0,x\right)$: alternative notation; (27.13.4)
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function)
$\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$: phase of Bessel functions; (10.18.3)
$\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$: theta function; §20.2(i)
$\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$: theta function; §20.2(i)
$\Theta(\boldsymbol{{\phi}}|\mathbf{B})=\mathop{\theta\/}\nolimits\!\left(\boldsymbol{{\phi}}/(2\pi i)\middle|\mathbf{B}/(2\pi i)\right)$: notation used by Belokolos et al. (1994), Dubrovin (1981); §21.1
(with $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function)

$\Theta(z|\tau)=\mathop{\theta _{{4}}\/}\nolimits\!\left(u\middle|\tau\right)$: Jacobi’s notation; ¶ ‣ §20.1
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function)
$\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$: Riemann theta function; (21.2.1)
$\Theta _{1}(z|\tau)=\mathop{\theta _{{3}}\/}\nolimits\!\left(u\middle|\tau\right)$: Jacobi’s notation; ¶ ‣ §20.1
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function)
$\vartheta _{j}(z|\tau)=\mathop{\theta _{{j}}\/}\nolimits\!\left(\pi z\middle|\tau\right)$: notation used by McKean and Moll (1999, p. 125); ¶ ‣ §20.1
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function)
$\theta _{c}(z|\tau)=\ifrac{\mathop{\theta _{{2}}\/}\nolimits\!\left(u\middle|\tau\right)}{\mathop{\theta _{{2}}\/}\nolimits\!\left(0\middle|\tau\right)}$: notation used by Neville (1951); ¶ ‣ §20.1
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function)
$\theta _{d}(z|\tau)=\ifrac{\mathop{\theta _{{3}}\/}\nolimits\!\left(u\middle|\tau\right)}{\mathop{\theta _{{3}}\/}\nolimits\!\left(0\middle|\tau\right)}$: notation used by Neville (1951); ¶ ‣ §20.1
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function)
$\theta _{n}(z|\tau)=\ifrac{\mathop{\theta _{{4}}\/}\nolimits\!\left(u\middle|\tau\right)}{\mathop{\theta _{{4}}\/}\nolimits\!\left(0\middle|\tau\right)}$: notation used by Neville (1951); ¶ ‣ §20.1
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function)
$\theta _{s}(z|\tau)=\pi{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0\middle|\tau\right)\ifrac{\mathop{\theta _{{1}}\/}\nolimits\!\left(u\middle|\tau\right)}{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0\middle|\tau\right)}$: notation used by Neville (1951); ¶ ‣ §20.1
(with $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function)
$\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$: phase of Coulomb functions; (33.2.9)
$\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$: scaled Riemann theta function; (21.2.2)
$\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$: Riemann theta function with characteristics; (21.2.5)
$\trace\mathbf{A}$: trace of matrix $\mathbf{A}$ ; Common Notations and Definitions