Jacobian%0Aelliptic%20functions
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11: 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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12: 5.15 Polygamma Functions
§5.15 Polygamma Functions
►The functions , , are called the polygamma functions. In particular, is the trigamma function; , , are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►For see §24.2(i). …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: 20.2 Definitions and Periodic Properties
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§20.2(i) Fourier Series
… ►§20.2(ii) Periodicity and Quasi-Periodicity
… ►For fixed , each of , , , and is an analytic function of for , with a natural boundary , and correspondingly, an analytic function of for with a natural boundary . ►The four points are the vertices of the fundamental parallelogram in the -plane; see Figure 20.2.1. … ►§20.2(iv) -Zeros
…15: 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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16: 4.2 Definitions
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►This is a multivalued function of with branch point at .
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is a single-valued analytic function on and real-valued when ranges over the positive real numbers.
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§4.2(iii) The Exponential Function
… ►§4.2(iv) Powers
… ►In all other cases, is a multivalued function with branch point at . …17: 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 Kelvin functions the order is always assumed to be real.
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►Abramowitz and Stegun (1964): , , , , for , , , , respectively, when .
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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).
18: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►Throughout §§8.17 and 8.18 we assume that , , and . … ►§8.17(ii) Hypergeometric Representations
… ►With , , and , … ►§8.17(vi) Sums
…19: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
… ►As a function of , with () fixed, is analytic in the half-plane . … ►Most references treat real with . … ►Throughout this subsection . … ►As with fixed, …20: 1.10 Functions of a Complex Variable
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►An analytic function
has a zero of order (or multiplicity) () at if the first nonzero coefficient in its Taylor series at is that of .
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►Lastly, if for infinitely many negative , then is an isolated essential singularity.
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►where , , and the series converges in a neighborhood of .
…Let .
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