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11: 25.12 Polylogarithms
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►The remainder of the equations in this subsection apply to principal branches.
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►The special case is the Riemann zeta function: .
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►Further properties include
…and
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►In terms of polylogarithms
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12: 20 Theta Functions
Chapter 20 Theta Functions
…13: 23.15 Definitions
§23.15 Definitions
… ►A modular function is a function of that is meromorphic in the half-plane , and has the property that for all , or for all belonging to a subgroup of SL, …(Some references refer to as the level). …If, in addition, as , then is called a cusp form. … ►14: Tom M. Apostol
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►Apostol was born on August 20, 1923.
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►He was also a coauthor of three textbooks written to accompany the physics telecourse The Mechanical Universe …and Beyond.
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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).
… Ford Award, given to recognize authors of articles of expository excellence.
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15: 27.2 Functions
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►( is defined to be 0.)
Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
…They tend to thin out among the large integers, but this thinning out is not completely regular.
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►the sum of the th powers of the positive integers that are relatively prime to
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►is the number of -tuples of integers whose greatest common divisor is relatively prime to
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16: 26.13 Permutations: Cycle Notation
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►They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►A permutation that consists of a single cycle of length can be written as the composition of two-cycles (read from right to left):
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►A permutation is even or odd according to the parity of the number of transpositions.
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►Given a permutation , the inversion number of , denoted , is the least number of adjacent transpositions required to represent .
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17: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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Special Functions, -Series and Related Topics.
Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.
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Two families of orthogonal polynomials related to Jacobi polynomials.
Rocky Mountain J. Math. 21 (1), pp. 359–375.
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Bounds for the small real and purely imaginary zeros of Bessel and related functions.
Methods Appl. Anal. 2 (1), pp. 1–21.
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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent.
Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
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18: 26.3 Lattice Paths: Binomial Coefficients
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