normalized
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21—30 of 108 matching pages
21: 8.12 Uniform Asymptotic Expansions for Large Parameter
22: 33.2 Definitions and Basic Properties
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33.2.3
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33.2.5
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is a real and analytic function of on the open interval , and also an analytic function of when .
►The normalizing constant
is always positive, and has the alternative form
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33.2.6
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23: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►These are based on the Liouville normal form of (1.13.29).
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►Applying equations (1.18.29) and (1.18.30), the complete set of normalized eigenfunctions being
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►Then orthogonality and normalization relations are
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24: 3.6 Linear Difference Equations
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►It therefore remains to apply a normalizing factor .
The process is then repeated with a higher value of , and the normalized solutions compared.
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►The normalizing factor can be the true value of divided by its trial value, or can be chosen to satisfy a known property of the wanted solution of the form
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►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6).
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25: 35.4 Partitions and Zonal Polynomials
26: 8.4 Special Values
27: 22.18 Mathematical Applications
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►The special case is in Jacobian normal form.
For any two points and on this curve, their sum
, always a third point on the curve, is defined by the Jacobi–Abel addition law
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28: 30.16 Methods of Computation
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►If is known, then we can compute (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions , if is even, or , if is odd.
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►The coefficients are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).
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►Form the eigenvector of associated with the eigenvalue , , normalized according to
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