exponent%20pairs
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21—30 of 184 matching pages
21: Software Index
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20 Theta Functions | |||||||||||||||||||||||||
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28.36(ii) Exponents, Eigenvalues | ✓ | ✓ | ✓ | ✓ | ✓ | a | |||||||||||||||||||
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22: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point.
SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
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23: 2.7 Differential Equations
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►Let , denote the indices or exponents, that is, the roots of the indicial equation
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►One pair of independent solutions of the equation
…In theory either pair may be used to construct any other solution
…This kind of cancellation cannot take place with and , and for this reason, and following Miller (1950), we call and a numerically satisfactory pair of solutions.
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►This is characteristic of numerically satisfactory pairs.
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24: 31.14 General Fuchsian Equation
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►The exponents at the finite singularities are and those at are , where
…The three sets of parameters comprise the singularity parameters
, the exponent parameters
, and the free accessory parameters
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25: 31.3 Basic Solutions
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denotes the solution of (31.2.1) that corresponds to the exponent
at and assumes the value there.
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►Similarly, if , then the solution of (31.2.1) that corresponds to the exponent
at is
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►Solutions of (31.2.1) corresponding to the exponents
and at are respectively,
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►Solutions of (31.2.1) corresponding to the exponents
and at are respectively,
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►Solutions of (31.2.1) corresponding to the exponents
and at are respectively,
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26: 3.8 Nonlinear Equations
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►Let be an approximation to the real quadratic factor of that corresponds to a pair of conjugate complex zeros or to a pair of real zeros.
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3.8.15
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►Consider and .
We have and .
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3.8.16
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27: 31.8 Solutions via Quadratures
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►For half-odd-integer values of the exponent parameters:
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►The curve reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for .
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28: 26.14 Permutations: Order Notation
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►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
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►A descent of a permutation is a pair of adjacent elements for which the first is larger than the second.
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29: Bibliography C
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Numerical evaluation of the Lambert function and application to generation of generalized Gaussian noise with exponent 1/2.
IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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30: 8 Incomplete Gamma and Related
Functions
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