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11: 36.7 Zeros
12: Guide to Searching the DLMF
Ai^2+Bi^2
, the system modifies the query so it will find the equations containing the latter expressions.
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►To find more effectively the information you need, especially equations, you may at times wish to specify what you want with descriptive words that characterize the contents but do not occur literally.
For example, you may want equations that contain trigonometric functions, but you don’t care which trigonometric function.
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13: 18.40 Methods of Computation
14: Preface
15: 33.22 Particle Scattering and Atomic and Molecular Spectra
§33.22(i) Schrödinger Equation
… ►§33.22(iv) Klein–Gordon and Dirac Equations
… ►The Coulomb solutions of the Schrödinger and Klein–Gordon equations are almost always used in the external region, outside the range of any non-Coulomb forces or couplings. … ►§33.22(vi) Solutions Inside the Turning Point
… ►16: Bibliography D
17: Bibliography
18: 2.11 Remainder Terms; Stokes Phenomenon
§2.11(v) Exponentially-Improved Expansions (continued)
… ►For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … ►For higher-order differential equations, see Olde Daalhuis (1998a, b). … ►For nonlinear differential equations see Olde Daalhuis (2005a, b). … ►shows that this direct estimate is correct to almost 3D. …19: 14.2 Differential Equations
§14.2 Differential Equations
►§14.2(i) Legendre’s Equation
… ►§14.2(ii) Associated Legendre Equation
… ►§14.2(iii) Numerically Satisfactory Solutions
… ►When , or , and are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. …20: 23.22 Methods of Computation
In the general case, given by , we compute the roots , , , say, of the cubic equation ; see §1.11(iii). These roots are necessarily distinct and represent , , in some order.
If and are real, and the discriminant is positive, that is , then , , can be identified via (23.5.1), and , obtained from (23.6.16).
If , or and are not both real, then we label , , so that the triangle with vertices , , is positively oriented and is its longest side (chosen arbitrarily if there is more than one). In particular, if , , are collinear, then we label them so that is on the line segment . In consequence, , satisfy (with strict inequality unless , , are collinear); also , .
Finally, on taking the principal square roots of and we obtain values for and that lie in the 1st and 4th quadrants, respectively, and , are given by
where denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs (, ), corresponding to the 2 possible choices of the square root.