# value at T=0

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##### 2: 4.13 Lambert $W$-Function
βΊ $W_{0}\left(z\right)$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_{k}\left(z\right)$ are single-valued analytic functions on $\mathbb{C}\setminus(-\infty,0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … βΊAlternative notations are $\operatorname{Wp}\left(x\right)$ for $W_{0}\left(x\right)$, $\operatorname{Wm}\left(x\right)$ for $W_{-1}\left(x+0\mathrm{i}\right)$, both previously used in this section, the Wright $\omega$-function $\omega\left(z\right)=W\left({\mathrm{e}}^{z}\right)$, which is single-valued, satisfies …and has several advantages over the Lambert $W$-function (see Lawrence et al. (2012)), and the tree $T$-function $T\left(z\right)=-W\left(-z\right)$, which is a solution of … βΊIn the case of $k=0$ and real $z$ the series converges for $z\geq\mathrm{e}$. …
##### 3: 3.11 Approximation Techniques
βΊwith initial values $T_{0}\left(x\right)=1$, $T_{1}\left(x\right)=x$. … βΊFor 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). … βΊThere 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 … βΊrequires $f={\mathscr{L}}^{-1}F$ to be obtained from numerical values of $F$. … βΊGiven $n+1$ 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
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21.5.2 $\boldsymbol{{\Gamma}}\mathbf{J}_{2g}\boldsymbol{{\Gamma}}^{\mathrm{T}}=\mathbf% {J}_{2g}.$
βΊHere $\xi(\boldsymbol{{\Gamma}})$ is an eighth root of unity, that is, $(\xi(\boldsymbol{{\Gamma}}))^{8}=1$. … βΊ
21.5.5 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\boldsymbol{{0}}_{g}\\ \boldsymbol{{0}}_{g}&[\mathbf{A}^{-1}]^{\mathrm{T}}\end{bmatrix}\Rightarrow% \theta\left(\mathbf{A}\mathbf{z}\middle|\mathbf{A}\boldsymbol{{\Omega}}\mathbf% {A}^{\mathrm{T}}\right)=\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).$
βΊwhere the square root assumes its principal value. … βΊFor explicit results in the case $g=1$, see §20.7(viii).
##### 5: 3.7 Ordinary Differential Equations
βΊThe path is partitioned at $P+1$ points labeled successively $z_{0},z_{1},\dots,z_{P}$, with $z_{0}=a$, $z_{P}=b$. … βΊWrite $\tau_{j}=z_{j+1}-z_{j}$, $j=0,1,\dots,P$, expand $w(z)$ and $w^{\prime}(z)$ in Taylor series (§1.10(i)) centered at $z=z_{j}$, and apply (3.7.2). … βΊIf 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 $w^{\prime}(a)$. … βΊSimilarly, 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 $w^{\prime}(b)$. βΊ
##### 6: 1.6 Vectors and Vector-Valued Functions
βΊare 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 $\boldsymbol{{\Phi}}(u_{0},v_{0})$. … βΊA 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. … βΊ
##### 7: Mathematical Introduction
βΊWith 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. … βΊIn 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. … βΊIn 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. …
##### 8: 3.2 Linear Algebra
βΊIn 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 $\mathbf{x}=[\frac{1}{6},\frac{1}{6},\frac{1}{6}]^{\rm T}$. … βΊwhere $\rho(\mathbf{A}\mathbf{A}^{\rm T})$ is the largest of the absolute values of the eigenvalues of the matrix $\mathbf{A}\mathbf{A}^{\rm T}$; see §3.2(iv). … βΊA 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, $\mathbf{A}^{\rm T}\mathbf{y}=\lambda\mathbf{y}$. … βΊDefine 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 … βΊStart 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
βΊThe 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. … βΊLet $T$ 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: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
βΊThe operator $T$ is called self-adjoint if ${T}^{*}=T$, and referred to as symmetric if (1.18.23) holds for $v,w$ in the dense domain $\mathcal{D}(T)$ of $T$. … βΊIf $T\subset T^{**}={T}^{*}$ then $T$ is essentially self-adjoint and if $T={T}^{*}$ then $T$ is self-adjoint. … βΊSuch an operator $T$ is called injective if, for any $u,v$ in its domain, $Tu=Tv$ implies that $u=v$. … βΊ
• 1.

The point spectrum $\boldsymbol{\sigma}_{p}$. It consists of all $z\in\mathbb{C}$ for which $z-T$ is not injective, or equivalently, for which $z$ is an eigenvalue of $T$, i.e., $Tv=zv$ for some $v\in\mathcal{D}(T)\backslash\{0\}$.

• βΊIf $n_{1}=1$ then there are no nonzero boundary values at $a$; if $n_{1}=2$ then the above boundary values at $a$ form a two-dimensional class. …