# normal equations

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## 1—10 of 98 matching pages

##### 1: 31.14 General Fuchsian Equation

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###### Normal Form

…##### 2: 30.2 Differential Equations

##### 3: 31.2 Differential Equations

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###### §31.2(ii) Normal Form of Heun’s Equation

…##### 4: 3.11 Approximation Techniques

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►From the equations
$\partial S/\partial {a}_{k}=0$, $k=0,1,\mathrm{\dots},n$, we derive the

*normal equations*…##### 5: 29.20 Methods of Computation

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►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5).
►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8.
…Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6.
(Equation (29.6.3) serves as a check.)
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►§29.15(i) includes formulas for normalizing the eigenvectors.
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##### 6: 28.7 Analytic Continuation of Eigenvalues

##### 7: 28.5 Second Solutions ${\mathrm{fe}}_{n}$, ${\mathrm{ge}}_{n}$

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28.5.5
$${({C}_{n}(q))}^{2}{\int}_{0}^{2\pi}{({f}_{n}(x,q))}^{2}dx={({S}_{n}(q))}^{2}{\int}_{0}^{2\pi}{({g}_{n}(x,q))}^{2}dx=\pi .$$

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##### 8: 3.6 Linear Difference Equations

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►Let us assume the normalizing condition is of the form ${w}_{0}=\lambda $, where $\lambda $ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns ${w}_{1}^{(N)},{w}_{2}^{(N)},\mathrm{\dots},{w}_{N-1}^{(N)}$; see §3.2(ii).
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##### 9: 18.2 General Orthogonal Polynomials

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►It is assumed throughout this chapter that for each polynomial ${p}_{n}(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$

*unless indicated otherwise.*(However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … ►
18.2.11_2
$${a}_{n-1}{c}_{n}>0,$$
$n\ge 1$.

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18.2.11_6
$${\beta}_{n}>0,$$
$n\ge 1$.

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18.2.11_7
$${q}_{n}(x)={p}_{n}(x)/\sqrt{{h}_{n}},$$
$n\ge 0$.

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►Equations (18.14.3_5) and (18.14.8), both for $\alpha =0$, can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.
##### 10: 28.12 Definitions and Basic Properties

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28.12.2
$${\lambda}_{\nu}\left(-q\right)={\lambda}_{\nu}\left(q\right)={\lambda}_{-\nu}\left(q\right).$$

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►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization
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28.12.5
$${\int}_{0}^{\pi}{\mathrm{me}}_{\nu}(x,q){\mathrm{me}}_{\nu}(-x,q)dx=\pi .$$

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