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11: 1.6 Vectors and Vector-Valued Functions
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►where is the unit vector normal to and whose direction is determined by the right-hand rule; see Figure 1.6.1.
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►where is the surface element with an attached normal direction .
►A surface is orientable if a continuously varying normal can be defined at all points of the surface.
An orientable surface is oriented if suitable normals have been chosen.
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►Suppose is an oriented surface with boundary which is oriented so that its direction is clockwise relative to the normals of .
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12: 28.19 Expansions in Series of Functions
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►Let be a normal value (§28.12(i)) with respect to , and be a function that is analytic on a doubly-infinite open strip that contains the real axis.
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13: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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Theta relations and projective normality of Abelian varieties.
Amer. J. Math. 98 (4), pp. 865–889.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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14: 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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15: 28.15 Expansions for Small
16: 18.40 Methods of Computation
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►There are many ways to implement these first two steps, noting that the expressions for and of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010).
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18.40.6
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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►Achieving precisions at this level shown above requires higher than normal computational precision, see Gautschi (2009).
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17: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Normalization integrals of orthogonal Heun functions.
J. Math. Phys. 38 (7), pp. 3692–3699.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Tables of Normalized Associated Legendre Polynomials.
Pergamon Press, The Macmillan Co., Oxford-New York.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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18: 33.7 Integral Representations
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33.7.2
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33.7.3
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33.7.4
►Noninteger powers in (33.7.1)–(33.7.4) and the arctangent assume their principal values (§§4.2(i), 4.2(iv), 4.23(ii)).