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11: 5.2 Definitions
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
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5.2.1
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►It is a meromorphic function with no zeros, and with simple poles of residue at .
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5.2.2
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12: 20.2 Definitions and Periodic Properties
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§20.2(i) Fourier Series
… ►§20.2(ii) Periodicity and Quasi-Periodicity
… ►The theta functions are quasi-periodic on the lattice: … ►§20.2(iii) Translation of the Argument by Half-Periods
… ►§20.2(iv) -Zeros
…13: 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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14: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
►The hypergeometric function is defined by the Gauss series … … ►On the circle of convergence, , the Gauss series: … ►§15.2(ii) Analytic Properties
…15: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►§8.17(ii) Hypergeometric Representations
… ►§8.17(iii) Integral Representation
… ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. … ►§8.17(vi) Sums
…16: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
… ►§25.11(iii) Representations by the Euler–Maclaurin Formula
… ►§25.11(iv) Series Representations
… ►§25.11(vii) Integral Representations
… ►§25.11(x) Further Series Representations
…17: 17.17 Physical Applications
§17.17 Physical Applications
… ►They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. See Kassel (1995). …18: 17.15 Generalizations
§17.15 Generalizations
►For higher-dimensional basic hypergometric functions, see Milne (1985a, b, c, d, 1988, 1994, 1997) and Gustafson (1987).19: 10.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►For the spherical Bessel functions and modified spherical Bessel functions the order is a nonnegative integer.
For the other functions when the order is replaced by , it can be any integer.
For the Kelvin functions the order is always assumed to be real.
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►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).