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11: 16.7 Relations to Other Functions
§16.7 Relations to Other Functions
… ►Further representations of special functions in terms of functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).12: 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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13: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
… ►The Riemann zeta function is a special case: … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►When , (25.11.35) reduces to (25.2.3). … ►uniformly with respect to bounded nonnegative values of . …14: 26.3 Lattice Paths: Binomial Coefficients
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►
§26.3(i) Definitions
► is the number of ways of choosing objects from a collection of distinct objects without regard to order. is the number of lattice paths from to . …The number of lattice paths from to , , that stay on or above the line is … ►§26.3(iii) Recurrence Relations
…15: 12.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values.
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►These notations are due to Miller (1952, 1955).
An older notation, due to Whittaker (1902), for is .
The notations are related by .
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16: 6.11 Relations to Other Functions
§6.11 Relations to Other Functions
… ►Incomplete Gamma Function
… ►Confluent Hypergeometric Function
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6.11.2
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6.11.3
17: 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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18: 26.5 Lattice Paths: Catalan Numbers
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§26.5(i) Definitions
… ►It counts the number of lattice paths from to that stay on or above the line . … ►§26.5(ii) Generating Function
… ►§26.5(iii) Recurrence Relations
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26.5.6
,
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19: 11.10 Anger–Weber Functions
§11.10 Anger–Weber Functions
… ►§11.10(vi) Relations to Other Functions
… ► … ►where the prime on the second summation symbols means that the first term is to be halved. ►§11.10(ix) Recurrence Relations and Derivatives
…20: 25.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main function treated in this chapter is the Riemann zeta function
.
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►The main related functions are the Hurwitz zeta function
, the dilogarithm , the polylogarithm (also known as Jonquière’s function
), Lerch’s transcendent , and the Dirichlet -functions
.
nonnegative integers. | |
… | |
primes | on function symbols: derivatives with respect to argument. |