multiplication theorems
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11—20 of 29 matching pages
11: 17.9 Further Transformations of Functions
12: 10.74 Methods of Computation
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βΊTo ensure that no zeros are overlooked, standard tools are the phase principle and Rouché’s theorem; see §1.10(iv).
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Multiple Zeros
…13: 5.5 Functional Relations
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§5.5(iii) Multiplication
… βΊGauss’s Multiplication Formula
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5.5.7
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§5.5(iv) Bohr–Mollerup Theorem
…14: 1.9 Calculus of a Complex Variable
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DeMoivre’s Theorem
… βΊJordan Curve Theorem
… βΊCauchy’s Theorem
… βΊLiouville’s Theorem
… βΊDominated Convergence Theorem
…15: 10.18 Modulus and Phase Functions
16: Bibliography
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Characterization theorems for orthogonal polynomials.
In Orthogonal Polynomials (Columbus, OH, 1989),
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.
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Monotonicity theorems and inequalities for the complete elliptic integrals.
J. Comput. Appl. Math. 172 (2), pp. 289–312.
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Multiple series Rogers-Ramanujan type identities.
Pacific J. Math. 114 (2), pp. 267–283.
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Theorems on generalized Dedekind sums.
Pacific J. Math. 2 (1), pp. 1–9.
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A Centennial History of the Prime Number Theorem.
In Number Theory,
Trends Math., pp. 1–14.
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17: 18.36 Miscellaneous Polynomials
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βΊThese are OP’s on the interval with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at and to the weight function for the Jacobi polynomials.
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§18.36(iii) Multiple Orthogonal Polynomials
… βΊOrthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, for as per (18.2.9_5). … βΊIn §18.39(i) it is seen that the functions, , are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …18: Bibliography D
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Multiplicative Number Theory.
3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.
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Note on the addition theorem of parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
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On multiple zeros of Bernoulli polynomials.
Acta Arith. 134 (2), pp. 149–155.
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Ramanujan’s master theorem for symmetric cones.
Pacific J. Math. 175 (2), pp. 447–490.
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On Vandermonde’s theorem, and some more general expansions.
Proc. Edinburgh Math. Soc. 25, pp. 114–132.
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19: 27.8 Dirichlet Characters
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βΊIf
is a given integer, then a function is called a Dirichlet character (mod ) if it is completely multiplicative, periodic with period , and vanishes when .
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βΊFor any character , if and only if , in which case the Euler–Fermat theorem (27.2.8) implies .
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