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11: 19.35 Other Applications
§19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute π to high precision (Borwein and Borwein (1987, p. 26)). …
12: 10.44 Sums
§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
Neumann’s Addition Theorem
Graf’s and Gegenbauer’s Addition Theorems
13: 13.13 Addition and Multiplication Theorems
§13.13 Addition and Multiplication Theorems
§13.13(i) Addition Theorems for M ( a , b , z )
§13.13(ii) Addition Theorems for U ( a , b , z )
13.13.12 e y ( x + y x ) 1 - b n = 0 ( - y ) n n ! x n U ( a - n , b - n , x ) , | y | < | x | .
§13.13(iii) Multiplication Theorems for M ( a , b , z ) and U ( a , b , z )
14: 10.23 Sums
§10.23(i) Multiplication Theorem
§10.23(ii) Addition Theorems
Neumann’s Addition Theorem
Graf’s and Gegenbauer’s Addition Theorems
15: 14.28 Sums
§14.28(i) Addition Theorem
For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …
16: 13.26 Addition and Multiplication Theorems
§13.26 Addition and Multiplication Theorems
§13.26(i) Addition Theorems for M κ , μ ( z )
§13.26(ii) Addition Theorems for W κ , μ ( z )
§13.26(iii) Multiplication Theorems for M κ , μ ( z ) and W κ , μ ( z )
17: 22.18 Mathematical Applications
§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
With the identification x = sn ( z , k ) , y = d ( sn ( z , k ) ) / d z , the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …
18: Hans Volkmer
His book Multiparameter Eigenvalue Problems and Expansion Theorems was published by Springer as Lecture Notes in Mathematics No. …
19: Peter L. Walker
Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …
20: 12.13 Sums
§12.13(i) Addition Theorems