normal equations

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21: 33.22 Particle Scattering and Atomic and Molecular Spectra
§33.22(vi) Solutions Inside the Turning Point
The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity $\rho/({F_{\ell}}^{2}\left(\eta,\rho\right)+{G_{\ell}}^{2}\left(\eta,\rho\right))$ (Mott and Massey (1956, pp. 63–65)). …
• Solution of relativistic Coulomb equations. See for example Cooper et al. (1979) and Barnett (1981b).

• 22: 28.2 Definitions and Basic Properties
For simple roots $q$ of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations
23: 30.8 Expansions in Series of Ferrers Functions
Then the set of coefficients $a^{m}_{n,k}(\gamma^{2})$, $k=-R,-R+1,-R+2,\dots$ is the solution of the difference equation
30.8.4 $A_{k}f_{k-1}+\left(B_{k}-\lambda^{m}_{n}\left(\gamma^{2}\right)\right)f_{k}+C_% {k}f_{k+1}=0,$
(note that $A_{-R}=0$) that satisfies the normalizing condition …
30.8.8 $\frac{\lambda^{m}_{n}\left(\gamma^{2}\right)-B_{k}}{A_{k}}\frac{a^{m}_{n,k}(% \gamma^{2})}{a^{m}_{n,k-1}(\gamma^{2})}=1+O\left(\frac{1}{k^{4}}\right).$
The set of coefficients ${a^{\prime}}^{m}_{n,k}(\gamma^{2})$, $k=-N-1,-N-2,\dots$, is the recessive solution of (30.8.4) as $k\to-\infty$ that is normalized by …
24: 32.14 Combinatorics
The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). … See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.
25: 28.15 Expansions for Small $q$
§28.15(i) Eigenvalues $\lambda_{\nu}\left(q\right)$
28.15.1 $\lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.$
Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a=\lambda_{\nu}\left(q\right)$:
28.15.2 $a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.$
28.15.3 $\mathrm{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1% }{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}\right)+% \frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+\frac{1}{% (\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)^{2}}e^{% \mathrm{i}\nu z}\right)+\cdots;$
26: Bibliography M
• R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
• R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.
• H. R. McFarland and D. St. P. Richards (2001) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.
• H. R. McFarland and D. St. P. Richards (2002) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.
• J. P. Mills (1926) Table of the ratio: Area to bounding ordinate, for any portion of normal curve. Biometrika 18, pp. 395–400.
• 27: 31.11 Expansions in Series of Hypergeometric Functions
Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. …The coefficients $c_{j}$ satisfy the equationsEvery Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. … In this case the accessory parameter $q$ is a root of the continued-fraction equation
28: Bibliography L
• G. Labahn and M. Mutrie (1997) Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.
• C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
• E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.
• D. W. Lozier and J. M. Smith (1981) Algorithm 567: Extended-range arithmetic and normalized Legendre polynomials [A1], [C1]. ACM Trans. Math. Software 7 (1), pp. 141–146.
• N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).
• 29: 22.18 Mathematical Applications
§22.18(iii) Uniformization and Other Parametrizations
The special case $y^{2}=(1-x^{2})(1-k^{2}x^{2})$ is in Jacobian normal form. For any two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on this curve, their sum $(x_{3},y_{3})$, always a third point on the curve, is defined by the Jacobi–Abel addition law …
30: 3.2 Linear Algebra
With $\mathbf{y}=[y_{1},y_{2},\dots,y_{n}]^{\rm T}$ the process of solution can then be regarded as first solving the equation $\mathbf{L}\mathbf{y}=\mathbf{b}$ for $\mathbf{y}$ (forward elimination), followed by the solution of $\mathbf{U}\mathbf{x}=\mathbf{y}$ for $\mathbf{x}$ (back substitution). … A normalized eigenvector has Euclidean norm 1; compare (3.2.13) with $p=2$. … where $\mathbf{x}$ and $\mathbf{y}$ are the normalized right and left eigenvectors of $\mathbf{A}$ corresponding to the eigenvalue $\lambda$. … 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 …