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1: 1.13 Differential Equations
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§1.13(vii) Closed-Form Solutions
… ►§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
►A standard form for second order ordinary differential equations with , and with a real parameter , and real valued functions and , with and positive, is …Assuming that satisfies un-mixed boundary conditions of the form … ►Transformation to Liouville normal Form
…2: Gergő Nemes
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►Nemes has research interests in asymptotic analysis, Écalle theory, exact WKB analysis, and special functions.
►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions.
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3: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
►For asymptotic approximations for the accessory parameter eigenvalues , see Fedoryuk (1991) and Slavyanov (1996). ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).4: 20 Theta Functions
Chapter 20 Theta Functions
…5: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►Gauss and Legendre conjectured that is asymptotic to as :
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►An equivalent form states that the th prime (when the primes are listed in increasing order) is asymptotic to as :
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27.2.4
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6: 28.16 Asymptotic Expansions for Large
7: 17.1 Special Notation
§17.1 Special Notation
… ►The main functions treated in this chapter are the basic hypergeometric (or -hypergeometric) function , the bilateral basic hypergeometric (or bilateral -hypergeometric) function , and the -analogs of the Appell functions , , , and . ►Another function notation used is the “idem” function: … ►Fine (1988) uses for a particular specialization of a function.8: 26.3 Lattice Paths: Binomial Coefficients
9: 25.12 Polylogarithms
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►For a uniform asymptotic approximation for see Temme and Olde Daalhuis (1990).
25.12.2
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