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11: 14.20 Conical (or Mehler) Functions
12: 16.13 Appell Functions
§16.13 Appell Functions
►The following four functions of two real or complex variables and cannot be expressed as a product of two functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►
16.13.1
,
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16.13.4
.
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13: 11.9 Lommel Functions
§11.9 Lommel Functions
… ► … ►§11.9(ii) Expansions in Series of Bessel Functions
… ►For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … ►14: 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 . …15: 5.2 Definitions
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
… ►It is a meromorphic function with no zeros, and with simple poles of residue at . … ►
5.2.2
.
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5.2.3
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16: 31.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main functions treated in this chapter are , , , and the polynomial .
…Sometimes the parameters are suppressed.
, | real variables. |
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… |
17: 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 .
…
18: 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
…19: 25.1 Special Notation
…
►(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. |
20: William P. Reinhardt
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►Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland.
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►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions.
Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions.
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►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.