Notations Notations S Notations U

Notations T

$T_n$

tangent numbers; §24.15(ii)

$T\left(z\right)$

Tree $T$-function; §4.13

$T_n^{\ast }\left(x\right)$

shifted Chebyshev polynomial of the first kind; Table 18.3.1

$\tan z$

tangent function; (4.14.4)

$\tanh z$

hyperbolic tangent function; (4.28.4)

$\tau \left(n\right)$

Ramanujan’s tau function; (27.14.18)

$\vartheta \left(x\right)={\theta }_3(0,x)$

alternative notation; (27.13.4)
(with ${\theta }_j(z,q)$: theta function)

$\theta \left(z\right)$

Airy phase function; (9.8.4)

${\theta }_{\nu }\left(x\right)$

phase of Bessel functions; (10.18.3)

$\mathrm{\Theta }(\mathit{\varphi }|𝐁)=\theta \left(\mathit{\varphi }/(2\pi \mathrm{i})\right|𝐁/(2\pi \mathrm{i}))$

notation used by Belokolos et al. (1994), Dubrovin (1981); §21.1
(with $\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{i}$: imaginary unit)

$\theta \left(𝐳\right|𝛀)$

Riemann theta function; (21.2.1)

$\mathrm{\Theta }(z|\tau )={\theta }_4\left(u\right|\tau )$

Jacobi’s notation; §20.1
(with ${\theta }_j\left(z\right|\tau )$: theta function)

${\mathrm{\Theta }}_1(z|\tau )={\theta }_3\left(u\right|\tau )$

Jacobi’s notation; §20.1
(with ${\theta }_j\left(z\right|\tau )$: theta function)

${\theta }_c(z|\tau )={\theta }_2\left(u\right|\tau )/{\theta }_2\left(0\right|\tau )$

notation used by Neville (1951); §20.1
(with ${\theta }_j\left(z\right|\tau )$: theta function)

${\theta }_d(z|\tau )={\theta }_3\left(u\right|\tau )/{\theta }_3\left(0\right|\tau )$

notation used by Neville (1951); §20.1
(with ${\theta }_j\left(z\right|\tau )$: theta function)

$\widehat{\theta }\left(𝐳\right|𝛀)$

scaled Riemann theta function; (21.2.2)

${\theta }_j(z,q)$

theta function; §20.2(i)

${\theta }_j\left(z\right|\tau )$

theta function; §20.2(i)

${\vartheta }_j(z|\tau )={\theta }_j\left(\pi z\right|\tau )$

notation used by McKean and Moll (1999, p. 125); §20.1
(with ${\theta }_j\left(z\right|\tau )$: theta function and $\pi$: the ratio of the circumference of a circle to its diameter)

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$

phase of Coulomb functions; (33.2.9)

${\theta }_n(z|\tau )={\theta }_4\left(u\right|\tau )/{\theta }_4\left(0\right|\tau )$

notation used by Neville (1951); §20.1
(with ${\theta }_j\left(z\right|\tau )$: theta function)

${\theta }_s(z|\tau )=\pi {\theta }_3^2\left(0\right|\tau ){\theta }_1\left(u\right|\tau )/{\theta }_1^{\prime }\left(0\right|\tau )$

notation used by Neville (1951); §20.1
(with ${\theta }_j\left(z\right|\tau )$: theta function and $\pi$: the ratio of the circumference of a circle to its diameter)

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$

Riemann theta function with characteristics; (21.2.5)

$\mathrm{tr}𝐀$

trace of matrix; (1.2.62)

Notations S Notations U

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