# value at T=0

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##### 2: 18.3 Definitions
For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, the Chebyshev polynomials $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, the Legendre polynomials $P_{n}\left(x\right)$, the Laguerre polynomials $L_{n}\left(x\right)$, and the Hermite polynomials $H_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). The Jacobi polynomials are in powers of $x-1$ for $n=0,1,\dots,6$. … In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_{n}\left(x\right)$, $n=0,1,\dots,N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\dots,N+1$, of $T_{N+1}\left(x\right)$: …When $j=k=0$ the sum in (18.3.1) is $N+1$. … For another version of the discrete orthogonality property of the polynomials $T_{n}\left(x\right)$ see (3.11.9). …
##### 3: 3.11 Approximation Techniques
The Chebyshev polynomials $T_{n}$ are given by …with initial values $T_{0}\left(x\right)=1$, $T_{1}\left(x\right)=x$. … For the expansion (3.11.11), numerical values of the Chebyshev polynomials $T_{n}\left(x\right)$ can be generated by application of the recurrence relation (3.11.7). … There exists a unique solution of this minimax problem and there are at least $k+\ell+2$ values $x_{j}$, $a\leq x_{0}, such that $m_{j}=m$, where … Given $n+1$ distinct points $x_{k}$ in the real interval $[a,b]$, with ($a=$)$x_{0}($=b$), on each subinterval $[x_{k},x_{k+1}]$, $k=0,1,\ldots,n-1$, a low-degree polynomial is defined with coefficients determined by, for example, values $f_{k}$ and $f_{k}^{\prime}$ of a function $f$ and its derivative at the nodes $x_{k}$ and $x_{k+1}$. …
##### 4: 21.5 Modular Transformations
21.5.2 $\boldsymbol{{\Gamma}}\mathbf{J}_{2g}\boldsymbol{{\Gamma}}^{\mathrm{T}}=\mathbf% {J}_{2g}.$
Here $\xi(\boldsymbol{{\Gamma}})$ is an eighth root of unity, that is, $(\xi(\boldsymbol{{\Gamma}}))^{8}=1$. …
21.5.5 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\boldsymbol{{0}}_{g}\\ \boldsymbol{{0}}_{g}&[\mathbf{A}^{-1}]^{\mathrm{T}}\end{bmatrix}\Rightarrow% \theta\left(\mathbf{A}\mathbf{z}\middle|\mathbf{A}\boldsymbol{{\Omega}}\mathbf% {A}^{\mathrm{T}}\right)=\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).$
where the square root assumes its principal value. … For explicit results in the case $g=1$, see §20.7(viii).
##### 5: 3.7 Ordinary Differential Equations
The path is partitioned at $P+1$ points labeled successively $z_{0},z_{1},\dots,z_{P}$, with $z_{0}=a$, $z_{P}=b$. … Write $\tau_{j}=z_{j+1}-z_{j}$, $j=0,1,\dots,P$, expand $w(z)$ and $w^{\prime}(z)$ in Taylor series (§1.10(i)) centered at $z=z_{j}$, and apply (3.7.2). … If the solution $w(z)$ that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along $\mathscr{P}$ from $a$ to $b$, then $w(z)$ and $w^{\prime}(z)$ may be computed in a stable manner for $z=z_{0},z_{1},\dots,z_{P}$ by successive application of (3.7.5) for $j=0,1,\dots,P-1$, beginning with initial values $w(a)$ and $w^{\prime}(a)$. … Similarly, if $w(z)$ is decaying at least as fast as all other solutions along $\mathscr{P}$, then we may reverse the labeling of the $z_{j}$ along $\mathscr{P}$ and begin with initial values $w(b)$ and $w^{\prime}(b)$.
##### 6: Mathematical Introduction
With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … In referring to the numerical tables and approximations we use notation typified by $x=0(.05)1$, 8D or 8S. …05, and the corresponding function values are tabulated to 8 decimal places or 8 significant figures. … In the Handbook this information is grouped at the section level and appears under the heading Sources in the References section. In the DLMF this information is provided in pop-up windows at the subsection level. …
##### 7: 1.6 Vectors and Vector-Valued Functions
are tangent to the surface at $\boldsymbol{{\Phi}}(u_{0},v_{0})$. The surface is smooth at this point if $\mathbf{T}_{u}\times\mathbf{T}_{v}\not=0$. …The vector $\mathbf{T}_{u}\times\mathbf{T}_{v}$ at $(u_{0},v_{0})$ is normal to the surface at $\boldsymbol{{\Phi}}(u_{0},v_{0})$. … A parametrization $\boldsymbol{{\Phi}}(u,v)$ of an oriented surface $S$ is orientation preserving if $\mathbf{T}_{u}\times\mathbf{T}_{v}$ has the same direction as the chosen normal at each point of $S$, otherwise it is orientation reversing. …
##### 8: 3.2 Linear Algebra
In solving $\mathbf{A}\mathbf{x}=[1,1,1]^{\rm T}$, we obtain by forward elimination $\mathbf{y}=[1,-1,3]^{\rm T}$, and by back substitution $\mathbf{x}=[\frac{1}{6},\frac{1}{6},\frac{1}{6}]^{\rm T}$. … where $\rho(\mathbf{A}\mathbf{A}^{\rm T})$ is the largest of the absolute values of the eigenvalues of the matrix $\mathbf{A}\mathbf{A}^{\rm T}$; see §3.2(iv). … A nonzero vector $\mathbf{y}$ is called a left eigenvector of $\mathbf{A}$ corresponding to the eigenvalue $\lambda$ if $\mathbf{y}^{\rm T}\mathbf{A}=\lambda\mathbf{y}^{\rm T}$ or, equivalently, $\mathbf{A}^{\rm T}\mathbf{y}=\lambda\mathbf{y}$. … Define the Lanczos vectors $\mathbf{v}_{j}$ and coefficients $\alpha_{j}$ and $\beta_{j}$ by $\mathbf{v}_{0}=\boldsymbol{{0}}$, a normalized vector $\mathbf{v}_{1}$ (perhaps chosen randomly), $\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}$, $\beta_{1}=0$, and for $j=1,2,\ldots,n-1$ by the recursive scheme … Start with $\mathbf{v}_{0}=\boldsymbol{{0}}$, vector $\mathbf{v}_{1}$ such that $\mathbf{v}_{1}^{\rm T}\mathbf{S}\mathbf{v}_{1}=1$, $\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}$, $\beta_{1}=0$. …
##### 9: 23.20 Mathematical Applications
The curve $C$ is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element $o=(0,1,0)$ as the point at infinity, the negative of $P=(x,y)$ by $-P=(x,-y)$, and generally $P_{1}+P_{2}+P_{3}=0$ on the curve iff the points $P_{1}$, $P_{2}$, $P_{3}$ are collinear. … Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. Then $\emptyset\subseteq T\subseteq I\subseteq K\subseteq C$. …Both $T$ and $I$ are finite sets. …Values of $x$ are then found as integer solutions of $x^{3}+ax+b-y^{2}=0$ (in particular $x$ must be a divisor of $b-y^{2}$). …
##### 10: 8.12 Uniform Asymptotic Expansions for Large Parameter
where $g_{k}$, $k=0,1,2,\dots$, are the coefficients that appear in the asymptotic expansion (5.11.3) of $\Gamma\left(z\right)$. The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta=0$, and the Maclaurin series expansion of $c_{k}(\eta)$ is given by …where $d_{0,0}=-\tfrac{1}{3}$, …For numerical values of $d_{k,n}$ to 30D for $k=0(1)9$ and $n=0(1)N_{k}$, where $N_{k}=28-4\left\lfloor k/2\right\rfloor$, see DiDonato and Morris (1986). … A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by …