Ritt theorem
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11: 13.26 Addition and Multiplication Theorems
§13.26 Addition and Multiplication Theorems
►§13.26(i) Addition Theorems for
… ►§13.26(ii) Addition Theorems for
… ►§13.26(iii) Multiplication Theorems for and
…12: 22.18 Mathematical Applications
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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
… ►With the identification , , 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). …13: Hans Volkmer
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►His book Multiparameter Eigenvalue Problems and Expansion Theorems was published by Springer as Lecture Notes in Mathematics No.
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14: Peter L. Walker
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►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.
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15: 12.13 Sums
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§12.13(i) Addition Theorems
…16: 27.2 Functions
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§27.2(i) Definitions
… ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. …This is the number of positive integers that are relatively prime to ; is Euler’s totient. ►If , then the Euler–Fermat theorem states that …17: 5.5 Functional Relations
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§5.5(iv) Bohr–Mollerup Theorem
…18: 19.15 Advantages of Symmetry
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►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration.
…These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)).
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19: 23.23 Tables
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►05, and in the case of the user may deduce values for complex by application of the addition theorem (23.10.1).
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20: 27.11 Asymptotic Formulas: Partial Sums
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►where , .
►Letting in (27.11.9) or in (27.11.11) we see that there are infinitely many primes if are coprime; this is Dirichlet’s theorem
on primes in arithmetic progressions.
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27.11.15
►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3).
The prime number theorem for
arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if , then the number of primes with is asymptotic to as .