# normal equations

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## 11—20 of 67 matching pages

##### 11: 30.16 Methods of Computation
If $\lambda^{m}_{n}\left(\gamma^{2}\right)$ is known, then we can compute $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime}(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime}(0)=1$ if $n-m$ is odd. …
##### 12: 10.74 Methods of Computation
In the case of $J_{n}\left(x\right)$, the need for initial values can be avoided by application of Olver’s algorithm (§3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15). …
##### 13: 2.11 Remainder Terms; Stokes Phenomenon
For notational convenience assume that the original differential equation (2.7.1) is normalized so that $\lambda_{2}-\lambda_{1}=1$. …
##### 14: 14.30 Spherical and Spheroidal Harmonics
In the quantization of angular momentum the spherical harmonics $Y_{{l},{m}}\left(\theta,\phi\right)$ are normalized solutions of the eigenvalue equations
##### 15: 31.9 Orthogonality
31.9.2 $\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\mathrm{d}t=\delta_{m,k}\theta_{m}.$
The normalization constant $\theta_{m}$ is given by
31.9.3 $\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right|_{q=q_{m}},$
and the integration paths $\mathcal{L}_{1}$, $\mathcal{L}_{2}$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. For further information, including normalization constants, see Sleeman (1966a). …
##### 16: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.5 $\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ze^{\pm\pi i}\right)=% \pm\tfrac{1}{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)-iT(a,\eta),$
8.12.16 $\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ae^{\pm\pi i}\right)% \sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{k}(0)(-a)^{-k},$ $|\operatorname{ph}a|\leq\pi-\delta$,
8.12.18 $\rselection{Q\left(a,z\right)\\ P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}e^{-z}}{\Gamma\left(a\right)}{% \left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum_{k=1}^{% \infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)},$
##### 17: 28.31 Equations of Whittaker–Hill and Ince
###### §28.31(ii) Equation of Ince; Ince Polynomials
The normalization is given by … They satisfy the differential equation
##### 18: 9.13 Generalized Airy Functions
###### §9.13(i) Generalizations from the Differential Equation
Equations of the form … In Olver (1977a, 1978) a different normalization is used. … Another normalization of (9.13.17) is used in Smirnov (1960), given by …
##### 19: 3.7 Ordinary Differential Equations
The eigenvalues $\lambda_{k}$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …
##### 20: 36.10 Differential Equations
In terms of the normal form (36.2.1) the $\Psi_{K}\left(\mathbf{x}\right)$ satisfy the operator equationIn terms of the normal forms (36.2.2) and (36.2.3), the $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)$ satisfy the following operator equations