A. Erdelyi (ed.), Higher transcendental functions, vol. 3, Mc Graw-Hill, New York, 1953.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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.

....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.

....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).

.... 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.

.... 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).

....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.

....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.

....K is a su#ciently large number. The following results state roughly that when m is small (m in (I) the distribution is Poisson; and when m is large (m in (III) it is negative binomial; the transitional behavior (m in (II) is described a parabolic cylinder function D q ( x) defined by (cf. [12, 41]) D # (x) #(#) # 1 e xt t dt (# 0) 10) Theorem 2 As # the probability distribution #m (#, #) satisfies the estimates: i) for m in (I) #m (# ; #) e # #m #) #m) 11) ii) for m in (II) m = # #) # # , q e m m ; 12) ....

....2. R m,q (#, #) 1) m q) m (# # ) where L n are Laguerre polynomials defined by n (x) z ] 1 z) # 1 exp xz R) For the intermediate range (II) we need some standard notation in the theory of special functions. Let D # (z) denote the parabolic cylinder functions (cf. [12, 41]) D # satisfies the di#erential equation D ## # (z) 4 # D # (z) 0, and the integral representation D # (z) #(# 1) e z zs s s # 1 ds ( # arg s #) which represents an entire function in the z plane. Besides the special form (10) two special cases are worthy of ....

A. Erdelyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

....large number. The following results state roughly that when m is small (m in (I) the distribution is Poisson; and when m is large (m in (III) it is negative binomial; the transitional behavior (m in (II) is described a parabolic cylinder function D Gammaq ( Gammax) defined by (cf. [12, 41]) D Gamma (x) Gamma( Gamma1 Gammaxt Gammat dt ( 0) 10) Theorem 2 As 1, the probability distribution Pi m ( satisfies the estimates: i) for m in (I) Gamma (1 Gamma ) 1 Gamma m= 11) ii) for m in (II) m = q ....

....m ( where L n are Laguerre polynomials defined by n (x) z ] 1 Gamma z) Gammaff Gamma1 exp xz (ff 2 R) For the intermediate range (II) we need some standard notation in the theory of special functions. Let D (z) denote the parabolic cylinder functions (cf. [12, 41]) D satisfies the differential equation (z) D (z) 0; and the integral representation D (z) Gamma( 1) Z (0 ) zs Gammas s ds ( Gamma arg s ) which represents an entire function in the z plane. Besides the special form (10) two special cases are worthy of ....

A. Erd'elyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

.... (6) We are left with the case # # 1 for which we apply the Mellin inversion formula (cf. 17] e w = 1 #(s)w s ds (#w 0) giving S = 1 1 U # (s)ds, 11) U # (s) 2 (k 1)s (#s 0) The function U # 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 # R, the function is analytic in the unit circle and analytically continuable to the whole z plane with the exception of a branch cut from 1 to #. Thus our U # (s) admits analytic continuation to the whole ....

....2 log 2 4#i log 2 Figure 1: The contour C. and the other from # j to # j ; and a semicircle s s = n 1 e joins these two segments; V j is the vertical segment joining j 1 and j . From expansions for Lerch s zeta function, we easily deduce that (cf. [13]) U# (s) 2 s #(# 1) s log 2 2j#i) # 1 2 s #( #) O ( s log(s log log 2 O( s 1, 12) for s 0 in the cut plane and for j Z. We observe first that n (1 2) # , since # is exponentially small at # i#: #(# it) # t e # t 2 ( t (13) for each ....

A. Erdelyi. Higher transcendental functions, volume I . Robert E. Krieger Publishing Company, Malabar, Florida, (1953).

....by absolute convergence. Since y 2s e y dy = 2 s 1 2 w s 1 2 #( s 1 2) 0 1 2) and w s (1 ds = s)#(s 1 2) #(3 2) s)#(s 1 2) 1 2 1) we have #(s)#(1 s) #( s 1 2)#(s 1 2) ds. From the functional equation of the Gamma function (cf. [1]) #(z)#(1 z) and a change of variables, it follows that c 0 = dz. By the Mellin inversion formula 1 1 u z dz (0 a 1) we deduce that log 2 = du = 1 dz and thus c 0 = 2 # 1 2 log 2, as required. In general, we have the identity dxdw dy ....

A. Erdelyi, Higher transcendental functions, volume I, Robert E. Publishing Company, Malabar, Florida, 1953.

....the asymptotic expansion of (x n ) 5: Let and M k = sup n kr k;n k. Then fn (z) a 0 a 1 Delta Delta Delta a k Gamma1 1 Gamma z Phi(z; 1; n)a 1 Delta Delta Delta Phi(z; k Gamma 1; n)a k Gamma1 ; 3) where (cf. [2]) Phi(z; s; x) l=0 (x l) l 1 Gamma(s) t s Gamma1 e Gammaxt Gammat dt (jzj 1; s 0; x 0) Applying Watson s lemma (cf. e.g. 3] to this integral, we have the asymptotic expansion Phi(z; s; x) c 0 (s; z)x c 1 (s; z)x Delta Delta Delta as x 1; 4) which ....

....2 ) 2 Gamma j=n 1 2 ) 2 S(C) Now the proof reduces to an adaptation of the proof of (5: for X = C, a 0 = a 1 = 0 and z = 1. We have that Phi(1; s; x) i(s; x) the generalized Riemann zeta function, which admits by Watson s lemma an asymptotic expansion of the following form (cf. [2]) i(s; x) d 0 (s)x 1 Gammas d 1 (s)x c 2 (s)x Delta Delta Delta as x 1: 5) Relying on 4: one can derive the asymptotic expansion of the partial products from the asymptotic expansion of the partial sums by taking logarithms. 3 Functions of the second kind The functions of ....

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A. Erd'elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions I. McGraw-Hill Boll Company, Inc., New York, 1953.

....the non negativity in the q case. For a few cases we show that the monotonicity in k follows from Stenger s theorem on quadrature. We also conjecture a cosine version in Conjecture 1, generalizing Theorem 1, in x7, and make other remarks there. We shall follow the notation and terminology in [7], 8] 2. Proof of Theorems 1 and 2. In this section we prove Theorem 1 combinatorially, the precise results are given in Theorem 2. First we review a combinatorial proof for (1.2) which is well known [10, p. 6] 11, p. 12] After finishing the proof of Theorem 2 we give an equivalent ....

....1 ; Delta Delta Delta ; PN ) where each P i has 2 steps. So the class of P is always N , and there are 2 such P , giving (1.3) Proof. Clearly a 0 0 is given by Proposition 1, so we assume that p 0. By expanding the Chebyshev polynomial T l (cos(x) cos(lx) in terms of (1 cos(x) [7], we find that the coefficient of (1 cos(x) in Theorem 2 is a p = Gammal) p (l) p p (1=2) p : 2.1) Fix p 0. We consider the same set of paths Path, but weight each path in Path l by w(l; p) jlj Gammap Since the weight is 0 for ....

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A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York.

....v] x) 0: 6.5) The transcendental equation for the BVP (6.1) 6.2) 6.5) is (s) cos(s ) s 1=2) s 1=2) 2 ln(2) 1 = 0 (6.6) where (s) s) s) with the classical gamma function and is Euler s number. For a derivation of (6. 6) the reader is referred to Erdelyi [10], Fulton [13] Kamke [15] and Titchmarsh [25] 13 0 : 0:2500 0:25 1 : 9:13773 9:13774 2 : 26:07673 26:0767 Example 2. The Bessel equation. This is the equation (x) c y(x) y(x) 0 x 1; I = 0; 1] c 2 R: 6.7) The endpoint 1 is regular and 0 is a singular endpoint for all ....

....= 3=4; I = 0; 1] y; u] 0) 0; y(1) 0: 6.11) The transcendental equation for the BVP (6.7) 6.9) 6.11) is given by (s) J (s) 0 (6.12) where J is the Bessel function of order . For a discussion of the special functions needed in the derivation of (6. 12) the reader is referred to [10]. Computed eigenvalues of the BVP (6.7) 6.9, 6.11) with 0 : 12:18714 12:1871 1 : 44:25755 44:2576 2 : 96:07161 96:0716 (ii) I = 0; 1] 3=4; y; v] 0) 0; y(1) 0: 6.13) 14 In this case the transcendental equation is (s) J (s) 0: 6.14) For a discussion of the ....

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A. Erdelyi et al. Higher transcendental functions, Volumes I and II (McGraw-Hill; New York).

....1 s # n , 6.4) where (x) n is again the Pochhammer symbol. Here we observe that continued fractions tends to be more e#ective for this example. The CF representation for the Gauss hypergeometric function in (6.3) was found by Gauss in 1812 by exploiting 8 recursions; see p. 88 of Erdelyi [17]. The CF can also be constructed by the QD scheme mentioned at the end of Section 2 from the series representation of 2 F 1 ; see p. 533 of Henrici [26] For 2 F 1 (1, p, p q; s) the elements can be taken to be b n = 1, n # 1, a 1 = 1, a 2n = p n 1) p q n 2)s (p q 2n ....

A. Erdelyi, 1953. Higher Transcendental Functions (Bateman manuscripts), vol. I, McGraw-Hill, New York.

....= 1 q (#h # (d) # h # (d # 1) for d = d(x, o) 0, 2.9) h # (1) # q 1 h # (0) This allows you to write h # (n) in terms of the Chebyshev polynomials of the rst and second kinds T n (x)andU n (x)dened by Tn (cos #) cos(n#)andUn (cos #) sin( n 1)#) sin # . See Erdelyi et al. [6], pp. 183 187, for more information on these polynomials. The nal result is (2.10) h # (n) q #n 2 2 q 1 Tn ( # 2 # q ) q # 1 q 1 Un ( # 2 # q ) Note that since # is real, so is h # (n) Figure 2 shows graphs of h # (d) as a function of # when q =2andd =0, 1, 9. We ....

A. Erdelyi et al, Higher Transcendental Functions, II, McGraw-Hill, N.Y., #953.

....(1 # p) n (#) n (q #) n , 8.2) so that f(t) # 1 0 y 1 #(1 # p; t y)b(#, q; y)dy = # 1 0 y 1 b(#, q; t y)#(1 # p; y)dy . 8.3) Proof. First express f(t) as f(t) #(# q)t # p #(#)#(1 # p) e t U(q, 2 p; t) Then apply (7) on p. 270 of Erdelyi [17] plus 15.3.5 and 13.1.29 of AS to obtain f(s) 2 F 1 (1 # p, #, q #; s) # # n=0 (1 # p) n (#) n (q #) n ( s) n n . We now give some other examples from Theorem 8.2. First, if # = 1 2, p = 1 and q = 1, then f(t) E 1 (t) 2 # #t, t # 0 . 8.4) Next, if # = 1 2, p ....

Erdelyi, A. (1953). Higher Transcendental Functions (Bateman manuscripts), Vol. I, McGraw-Hill, New York.

.... and c k = ff k Gamma 3 2 ff k c k Gamma1 ; k = 1; 2; Therefore we obtain G(z) 1 p(p Gamma 2) p 1 2 ) ff z ff 1 ff Gamma 1 2 ff 1 z (ff Gamma 1 2 ) ff 1 2 ) ff 1) ff 2) z 2 : In terms of the so called hypergeometric function (see [7]) we can write G(z) as G(z) 1 p(p Gamma 2) p 1 2 ) ff z ff F (1; ff Gamma 1 2 ; ff 1; z) 2.6) So Z R w p Gamma1 OE 0 dy = 2 Z 1 0 w p Gamma1 OE 0 dy = p 1 p Gamma 1 Z 1 0 w(1 Gamma z) Gamma 1 2 G(z)dz (by (2.2) 2.3) p 1) 2 2p(p Gamma 1) p ....

....1 2 (1 Gamma tz) Gamma1 dz = Gamma(3) Gamma( 1 2 ) Gamma( 3 2 ) Gamma( 7 2 ) Z 1 0 t ff Gamma 1 2 (1 Gamma t) 1 2 F (1; 3; 7 2 ; t)dt: Let us now compute F (1; 3; 7 2 ; t) First it is easy to see that F (1; 1; 3 2 ; sin z) 2 ) z sin z cos z : 2. 9) See Page 101 of [7]. On the other hand F (1; 1; 3 2 ; t) 1 2 3 t 8 15 t 2 8 15 t 2 (F (1; 3; 7 2 ; t) Gamma 1) 2.10) Thus F (1; 3; 7 2 ; t) 1 15 8 t Gamma2 F (1; 1; 3 2 ; t) Gamma 1 Gamma 2 3 t Gamma 8 15 t 2 : Substituting that into I, we obtain that I = ....

A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill Book, New York (1953).

.... l l = a 1 ) n1 ( a 2 ) n2 lim ## # F 2 x, n 1 , n 2 ; #, #; # a1 , # a2 , where F 2 (#, #, # # ; #, # # ; x, y) # X m=0 # X n=0 (#) m n (#) m (# # ) n (#) m (# # ) n m n x m y n , is the second of Appell s hypergeometric functions of two variables [5]. Finally we use the substitution l # j k to find the expression C a1 ,a 2 n1 ,n2 (x) a 1 ) n1 ( a 2 ) n2 n1 X k=0 k n 2 X j=k ( n 1 ) k ( n 2 ) j k ( x) j 1 a1 k k 1 a2 j k (j k) a 1 ) n 1 ( a 2 ) n 2 n 1 n 2 X j=0 min(j,n 1 ) X ....

....1 n 2 X j=0 j X k=0 ( n 1 ) k ( n 2 ) j k (#) j c 1 1 c 1 k k c 2 1 c 2 j k (j k) x) j . Here F 1 (#, #, # # ; #; x, y) # X m=0 # X n=0 (#) m n (#) m (# # ) n (#) m nm n x m y n is the first of Appell s hypergeometric functions of two variables [5]. From this explicit expression we can find the coe#cients of the recurrence relation xP n1 ,n2 (x) P n1 1,n 2 (x) b n1 ,n2 P n1 ,n2 (x) c n1 ,n2 P n1 ,n2 1 (x) d n1 ,n2 P n1 1,n 2 1 (x) with P n 1 ,n 2 (x) M #;c 1 ,c 2 n1 ,n2 (x) We denote M #;c 1 ,c 2 n1 ,n2 (x) n1 n2 ....

A. Erdelyi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, 1953.

....A 0 is skew symmetric. By the induction hypothesis, r = rank A is even, since it is the rank of the smaller skew symmetric matrix A 0 . Appendix E. Elliptic functions For reference, here are some standard notations and facts about elliptic functions used in this paper (see for example [10] [11]) Lattices. A non degenerate lattice is real if = There are two kinds of real lattices: i) rectangular: generators 1 2 R and 3 2 iR can be chosen for . ii) rhombic: generators 1 and 3 = 1 can be chosen for . For any lattice with generators 1 , 3 , let 2 = Gamma 1 Gamma 3 ....

Erdelyi, A., ed. Higher transcendental functions. New York: McGraw-Hill, 1953.

....1; 2; Delta Delta Delta ; l = 0; 1; Delta Delta Delta ; n; and m = Gammal; Gammal 1; Delta Delta Delta ; l. The function Pi ( is found from Pi ( Gamma) by replacing by Gammai and q by Gammai(n 1) 14] Alternatively, they can be written in terms of Gegenbauer polynomials [17, 18] as Pi ( nl ( sin l C l 1 n Gammal (cos ) 63) 11 z 2 4 6 8 10 0.2 0.4 0.6 0.8 1 G(z) Figure 2: Plot of the Scale Function G(z) for the Open and Flat Universes. 12 In either case, the addition theorem is found from Eq. 11.4(3) of Ref. 18] with p = 2 and = j) This reduces to ....

....in terms of Gegenbauer polynomials [17, 18] as Pi ( nl ( sin l C l 1 n Gammal (cos ) 63) 11 z 2 4 6 8 10 0.2 0.4 0.6 0.8 1 G(z) Figure 2: Plot of the Scale Function G(z) for the Open and Flat Universes. 12 In either case, the addition theorem is found from Eq. 11.4(3) of Ref. [18] (with p = 2 and = j) This reduces to X lm j Pi ( nl ( Y lm ( j 2 = n 1) 2 2 2 ; 64) from which it is easy to show that the energy density inequality, Eq. 16) becomes Delta ae Gamma 1 4 2 a 3 1 X n=0 (n 1) 2 n e Gamma2 n t 0 : 65) If we use the ....

A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. II, (MacGraw-Hill, 1953).

....and hence A 0 is skew symmetric. By the induction hypothesis, r = rank A is even, since it is the rank of the smaller skew symmetric matrix A 0 . 2 39 B Elliptic functions For reference, here are some standard notations and facts about elliptic functions used in this paper (see for example [6] [7]) Lattices. A non degenerate lattice is real if = There are two kinds of real lattices: i) rectangular: generators 1 2 R and 3 2 iR can be chosen for . ii) rhombic: generators 1 and 3 = 1 can be chosen for . For any lattice with generators 1 , 3 , let 2 = Gamma 1 Gamma 3 ....

Erdelyi, A., ed. Higher transcendental functions. New York: McGraw-Hill, 1953.

....in R n can be done using similar arguments. One just writes points on S n 1 in coordinates ( t) where 2 S n 2 and t = n 2 [ 1; 1] and then uses a function f(t)Y ( where Y ( is a spherical harmonic on S n 2 that is zero for 1 = 0 (see equation (21) on p. 239 [EMOT] for how to construct such a spherical harmonic) x3. Projections of Convex Bodies and the Cosine transform Now, we explain how the projections of a convex body give the Cosine transform of the surface area measure of the body. We review the relation between the Cosine and Funk Radon transform ....

A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, Vol. II, Krieger, Malabar, FL, 1981.

....the system of n coupled rst order di erential equations can for example be rewritten into one n th order di erential equation, which is then solved with standard methods. In some cases, the system can also be transformed into a form which is known to be solved by generalized hypergeometric series [14, 15, 16, 17]. It is clear from the above discussion, that the determination of a master integral of a certain topology with t di erent denominators requires the knowledge of all the integrals appearing in the inhomogeneous term. These integrals are subtopologies of the topology of the integral under ....

....is then retrieved by resummation of a multiple sum or by an inverse integral transformation. Both methods, when employed for arbitrary space time dimension, give rise to generalized hypergeometric functions, which can be represented as multiple sums as well as inverse Mellin Barnes integrals [15]. In the di erential equation method, one assigns momentum vectors to the loop propagators only for the sake of deriving the di erential equations and the IBP and LI identities. After applying these identities to simplify the di erential equations, one obtaines a relation between the derivative ....

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A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi (Bateman Manuscript Project) , Higher Transcendental Functions , Vol. I, McGraw Hill (New York, 1953).

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A. Erdelyi (ed.), Higher transcendental functions, vol. 3, Mc Graw-Hill, New York, 1953.

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Erdelyi, A.: Higher Transcendental Functions. McGraw Hill (1953)

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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

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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).

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A. Erdelyi (ed.), Higher transcendental functions, Vol. 1, Mc Graw--Hill, 1953.

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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.

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Erdelyi, A. et. al. (1953), Higher Transcendental Functions, McGraw Hill, New York.

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A. Erd'elyi, V. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions, McGraw-- Hill, New York, 1953.

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A. Erdelyi et al, Higher Transcendental Functions, Vol. 1, McGraw--Hill, New York, 1953.

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Erdelyi, A., and Bateman, H., (1985) Higher transcendental functions, 2nd ed., v.1,2, Robert E. Krieger Publishing Company. 10

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A. Erdelyi (Ed.), Higher Transcendental Functions, McGraw--Hill, New York, 1953-- 55.

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A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 2, McGraw-Hill, New York, 1953.

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A. Erd'elyi and H. Bateman, Higher Transcendental Functions, McGraw--Hill, New York, 1953.

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A. Erd'elyi, Higher transcendental functions, volume I, Robert E. Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erdelyi, Higher transcendental functions, volume II, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erdelyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erdelyi, Higher Transcendental Functions, Volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erdelyi, Higher Transcendental Functions, Volume I, McGraw-Hill, 1953.

No context found.

A. Erd'elyi, Higher transcendental functions, volume II, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erd'elyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erd'elyi, Higher Transcendental Functions, Volume I, McGraw-Hill, 1953.

No context found.

A. Erdelyi, Higher transcendental functions, volume I, Robert E. Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erdelyi, Higher transcendental functions, volume I, Robert E. Krieger Publishing Company, Malabar, Florida, 1953.

No context found.

A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. 1 (McGraw-Hill Book Company, New York, 1953).

No context found.

A. Erdelyi (ed.), Higher transcendental functions, Vol. 1, Mc Graw{Hill, 1953.

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