limits%20of%20functions
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1: 8.17 Incomplete Beta Functions
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8.17.24
positive integers; .
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2: 18.40 Methods of Computation
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►It is now necessary to take the limit
of , and the imaginary part is the required Stieltjes–Perron inversion:
…Results of low ( to decimal digits) precision for are easily obtained for to .
Gautschi (2004, p. 119–120) has explored the
limit via the Wynn -algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in , depending smoothly on , for , for an example involving first numerator Legendre OP’s.
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being the Heaviside step-function, see (1.16.13).
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►The example chosen is inversion from the for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50).
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3: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Coulomb Functions for Large Values of the Parameter
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Technical report
Atomic Energy of Canada Limited, Chalk
River, Ontario.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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Hypergeometric functions.
Acta Math. 94, pp. 289–349.
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4: 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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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter.
SIAM J. Math. Anal. 21 (4), pp. 995–1018.
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5: 26.12 Plane Partitions
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§26.12(ii) Generating Functions
… ►§26.12(iv) Limiting Form
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26.12.26
►where is the Riemann -function (§25.2(i)).
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6: Bibliography C
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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial as the index and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials.
Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
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Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée.
Mem. Ac. Sc. St. Pétersbourg 6, pp. 141–157.
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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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7: 6.16 Mathematical Applications
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►These limits are not approached uniformly, however.
…Hence if and , then the limiting value of overshoots by approximately 18%.
Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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►It occurs with Fourier-series expansions of all piecewise continuous functions.
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8: 26.5 Lattice Paths: Catalan Numbers
9: Bibliography R
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Normal limit theorems for symmetric random matrices.
Probab. Theory Related Fields 112 (3), pp. 411–423.
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Elliptic Functions, Theta Functions, and Riemann Surfaces.
The Williams & Wilkins Co., Baltimore, MD.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Applied Bessel Functions.
Dover Publications Inc., New York.
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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function.
Mathematics 9 (16).
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