relation to Fuchsian equation
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11: 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
12: 9 Airy and Related Functions
Chapter 9 Airy and Related Functions
…13: 25 Zeta and Related Functions
Chapter 25 Zeta and Related Functions
…14: Ranjan Roy
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►Roy has published many papers on differential equations, fluid mechanics, special functions, Fuchsian groups, and the history of mathematics.
…He also authored another two advanced mathematics books: Sources in the development of mathematics (Roy, 2011), Elliptic and modular functions from Gauss to Dedekind to Hecke (Roy, 2017).
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15: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
…16: 19.10 Relations to Other Functions
§19.10 Relations to Other Functions
►§19.10(i) Theta and Elliptic Functions
►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►§19.10(ii) Elementary Functions
… ►For relations to the Gudermannian function and its inverse (§4.23(viii)), see (19.6.8) and …17: 25.17 Physical Applications
§25.17 Physical Applications
… ►This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. … ►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).18: 25.13 Periodic Zeta Function
§25.13 Periodic Zeta Function
►The notation is used for the polylogarithm with real: ►
25.13.1
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►Also,
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►
25.13.3
if ; if .