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1: 27.15 Chinese Remainder Theorem
§27.15 Chinese Remainder Theorem
… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod ), (mod ), (mod ), and (mod ), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits. …2: 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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3: 28.27 Addition Theorems
§28.27 Addition Theorems
►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …4: 20 Theta Functions
Chapter 20 Theta Functions
…5: 27.2 Functions
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§27.2(i) Definitions
… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(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. … ►If , then the Euler–Fermat theorem states that …6: Tom M. Apostol
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►Apostol was born on August 20, 1923.
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►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem).
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7: Bibliography I
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Ordinary Differential Equations.
Longmans, Green and Co., London.
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Tables of the elliptic cylinder functions.
Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.
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The periodic Lamé functions.
Proc. Roy. Soc. Edinburgh 60, pp. 47–63.
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Further investigations into the periodic Lamé functions.
Proc. Roy. Soc. Edinburgh 60, pp. 83–99.
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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8: 3.8 Nonlinear Equations
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►Initial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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3.8.15
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►Consider and .
We have and .
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9: 8 Incomplete Gamma and Related
Functions
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10: 28 Mathieu Functions and Hill’s Equation
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