A. Erdelyi (ed.), Higher transcendental functions, vol. 3, Mc Graw-Hill, New York, 1953. Home/Search Document Not in Database Summary Related Articles Check |
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Superintegrable Systems in Darboux spaces - Kalnins Department Of (Correct)
....2) arctanh 1) c (E c 2) E(a 2 cos 2u) c cv . The corresponding Schrodinger equation is which has the solution # = 2 F 1 2 ) 1 2 (# # # ) # 1 1 2 1 2)E , 4. 3) and 2 F 1 is a Gaussian hypergeometric function [11]. If we choose new coordinates 2 2 ( i#) 4.4) then the Hamiltonian takes the rational form . In this case the corresponding choice of coordinates has already been given, and the quadratic constant in these coordinates is X 2 = 25 The ....
A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi. Higher Transcendental Functions, Vol. I. McGraw-Hill, New York 1953.
Superintegrable Systems in Darboux spaces - Kalnins Department Of (Correct)
....b # E 1# # 2 . # Eb , # = cos #,# = cosh #. The corresponding Schrodinger equation has solutions of the form cos # cosh # S i sinh #, P s sin #, j = 1, 2 where S n (z, #) and P s n (t, #) are spheroidal functions [8] and E = m . Here we use polar coordinates: u = r cos # , v = r sin # . 2.6) The classical Hamiltonian has the form # p . The Hamilton Jacobi equation in these coordinates is with solution S(r, #) # Er # arctanh Er # # sin # # E # ....
....1 E arccosh (E #) cos #) cos . The corresponding Schrodinger equation is 2 # sin # # sin # ## and has solutions of the form # = # r sin # C # P # (cos #) E = m where C # (z) is a Bessel function and P # (cos #) is an associated Legendre polynomial [8]. A suitable choice of coordinates is u = ##, v = 1 2 ) 2.7) The classical Hamiltonian in these coordinates has the form . The corresponding quadratic constant is . which has solution #) ....
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A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi. Higher Transcendental Functions, Vol. II. McGraw-Hill, New York 1953.
On Random Variate Generation When Only Moments Or Fourier.. - Devroye (1989) (Correct)
....there is a simple recurslye formula that allows us to incrementally compute each ; at O(1) cost (see, e.g. 29, p. 1781) j 1)v (x) 2j 1)x2(x) j (x) 0, with 0 = I and = x. It should be noted that the Legendre series is but a special case of the Gegenbauer, Ferrer and Jacobi series [16,29,32]. 2.3. Example (Gram Charlier series of type A) The Legendre series of Example 2.2 is only applicable to functions with compact support. If f has support on the real line, one can use a special form of the Herte series, the Gram Charlier series of type A. The strategy followed in this paper ....
A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, Vol. II (McGraw-Hill, New York, 1953).
A Constant Arising From the Analysis of Algorithms for Determining .. - Hwang (1997) (1 citation) (Correct)
.... 1=2) 0 s 1=2) and ds = Gamma(1 Gamma s) Gamma(s 1=2) Gamma(3=2) Gamma(1 Gamma s) Gamma(s 1=2) Gamma1=2 s 1) we have 3=2 Gamma(s) Gamma(1 Gamma s) Gamma( Gammas 1=2) Gamma(s 1=2) ds: From the functional equation of the Gamma function (cf. [1]) Gamma(z) Gamma(1 Gamma z) and a change of variables, it follows that c 0 = dz: By the Mellin inversion formula 1 1 u Gammaz dz (0 a 1) we deduce that log 2 = du = 1 dz and thus c 0 = 2 log 2, as required. In general, we have the ....
A. Erd'elyi, Higher transcendental functions, volume I, Robert E. Publishing Company, Malabar, Florida, 1953.
Measures of Distinctness for Random Partitions and Compositions .. - Hwang, Yeh (1997) (4 citations) (Correct)
.... with the case ff Gamma1 for which we apply the Mellin inversion formula (cf. 17] Gammaw Z Gamma 2 Gammai1 Gamma(s)w ds ( w 0) giving S = 1 Gammai1 U ff (s)ds; 11) Gamma(k 1)s The function U ff is essentially a special case of the Lerch zeta function (cf. [13]) From known properties of the Lerch zeta function, it follows that, for fixed ff 2 R, the function k1 k is analytic in the unit circle and analytically continuable to the whole z plane with the exception of a branch cut from 1 to 1. Thus our U ff (s) admits analytic continuation to the ....
....2 Figure 1: The contour C. and the other from j to Gamma j ; and a semicircle fs : js Gamma j j = n : Gamma joins these two segments; V j is the vertical segment joining C j Gamma1 and C j . From expansions for Lerch s zeta function, we easily deduce that (cf. [13]) Gamma(ff 1) s log 2 Gamma 2ji) 2 i( Gammaff) O (js Gamma j j) if ff Gamma1; Gamma log(s Gamma j ) Gamma log log 2 O(js Gamma j j) if ff = Gamma1; 12) for s 0 in the cut plane and for j 2 Z. We observe first that V j U ff (s)ds n since Gamma is ....
A. Erd'elyi. Higher transcendental functions, volume I . Robert E. Krieger Publishing Company, Malabar, Florida, (1953).
Limit Theorems for the Number of Summands in Integer Partitions - Hwang (1997) (2 citations) (Correct)
....no longer useful when juj 1. In particular, Y (1; s) 1 Gamma 2 )i(s 1) Gamma(s) so that = AY (1; ff) Now by integration by parts, we see that Y (u; s) is related to the Lerch zeta function Phi(z; s; v) by: Y (u; s) u Gamma(s) Phi( Gammau; s 1; 1) with Phi defined by (cf. [6]) Phi(z; s; v) k v) for jzj 1; s 2 C; and v 6= 0; Gamma1; Gamma2; Analytic properties of Y (u; s) are summarized in the following lemma. Lemma 1 For each fixed u lying in the cut plane Cn ( Gamma1; Gamma1] the Mellin transform Y (u; s) can be meromorphically continued ....
....s = 0; Gamma1; Gamma2; Moreover, Y (u; s) satisfies the estimate Y (u; oe it)j e Gamma( 2 Gamma )jtj for any 0 as jtj 1; 10) uniformly for finite oe and u in the cut plane. Proof. For completeness, we sketch here a self contained proof. For further properties, see Erd elyi [6]. First of all, by integration by parts, we have the right hand side providing a meromorphic continuation of Y to the half plane e s Gamma1. The first assertion of the lemma follows from repeating the same process. As to (10) since log(1 ue is an analytic function of x in the ....
A. Erd'elyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.
Limit Theorems for the Number of Summands in Integer Partitions - Hwang (2000) (2 citations) (Correct)
....s 1 , 10) a representation no longer useful when u 1. In particular, Y (1, s) 1 2 s )#(s 1)#(s) so that # = AY (1, #) Now by integration by parts, we see that Y (u, s) is related to the Lerch zeta function #(z, s, v) by: Y (u, s) u#(s)#( u, s 1, 1) with # defined by (see [6]) k v) 1, s C, and v 0, Analytic properties of Y (u, s) are summarized in the following lemma. Lemma 1. For each fixed u lying in the cut plane C ( #, 1] the Mellin transform Y (u, s) can be meromorphically continued into the whole s plane with simple poles at s = 0, ....
....whole s plane with simple poles at s = 0, Moreover, Y (u, s) satisfies the estimate (u, # it) e (# 2 #) t for any # 0 as #, 11) uniformly for finite # and u in the cut plane. Proof. For completeness, we sketch here a self contained proof. For further properties, see Erdelyi [6]. First, by integration by parts, we have 7 the right hand side providing a meromorphic continuation of Y to the half plane s 1. The first assertion of the lemma follows from repeating the same process. As to (11) since log(1 ue x ) is an analytic function of x in the ....
A. Erdelyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.
Mathematics Of Computation - Volume Number Xxxx (Correct)
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A. Erdelyi (ed.), Higher transcendental functions, vol. 3, Mc Graw-Hill, New York, 1953.
Analysis of a Queueing Fairness Measure - Sandmann (2006) (Correct)
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Erdelyi, A.: Higher Transcendental Functions. McGraw Hill (1953)
Algorithmic Support for Commodity-Based Parallel Computing.. - Leung, Phillips, al. (2003) (Correct)
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A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume 1. McGraw-Hill, New York, 1953. 87
Ornstein-Uhlenbeck Process - Steven Finch May (2004) (Correct)
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A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, v. II, based on notes left by H. Bateman, McGraw-Hill, 1953, pp. 116---126; MR0058756 (15,419i).
Random Young Tableaux and Combinatorial Identities - Olshanski, Regev (Correct)
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A. Erdelyi (ed.), Higher transcendental functions, Vol. 1, Mc Graw--Hill, 1953.
On the Stability of the J Transformation - Homeier (Correct)
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A. Erd'elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, volume I, McGraw-Hill, New York, 1953.
Pricing Continuous Asian Options: A Comparison of Monte Carlo .. - Fu, Madan, Wang (1998) (4 citations) (Correct)
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Erdelyi, A. et. al. (1953), Higher Transcendental Functions, McGraw Hill, New York.
The Number of Rhombus Tilings of a Symmetric Hexagon Which .. - Fulmek, Krattenthaler (2000) (15 citations) (Correct)
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A. Erd'elyi, V. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions, McGraw-- Hill, New York, 1953.
Unknown - (Correct)
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A. Erdelyi et al, Higher Transcendental Functions, Vol. 1, McGraw--Hill, New York, 1953.
The Priority Queueing With Nite Bu er Size and.. - Konstantin.. (Correct)
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Erdelyi, A., and Bateman, H., (1985) Higher transcendental functions, 2nd ed., v.1,2, Robert E. Krieger Publishing Company. 10
Algebraic Solutions of the Lam e Equation, Revisited - Robert Maier Depts (Correct)
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A. Erdelyi (Ed.), Higher Transcendental Functions, McGraw--Hill, New York, 1953-- 55.
Orthogonal Polynomials For A Family Of Product Weight Functions On.. - Xu (1997) (4 citations) (Correct)
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A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 2, McGraw-Hill, New York, 1953.
Difference Equations for Multiple Charlier and Meixner Polynomials - Van Assche (Correct)
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A. Erdelyi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, 1953.
Quasi-Exact Solvability - González-López, Kamran, Olver (Correct)
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A. Erd'elyi and H. Bateman, Higher Transcendental Functions, McGraw--Hill, New York, 1953.
Normal Approximations of the Number of Records in.. - Zhi-Dong Bai Department (1998) (2 citations) (Correct)
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A. Erd'elyi, Higher transcendental functions, volume I, Robert E. Publishing Company, Malabar, Florida, 1953.
Asymptotic Estimates of Elementary Probability Distributions - Hwang (2 citations) (Correct)
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A. Erdelyi, Higher transcendental functions, volume II, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.
Asymptotic Estimates of Elementary Probability Distributions - Hwang (2 citations) (Correct)
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A. Erdelyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.
Distribution of the Number of Consecutive Records - Chern, Hwang, Yeh (2000) (Correct)
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A. Erdelyi, Higher Transcendental Functions, Volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.
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