general%0Aelliptic%20functions
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11: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
… ►In general, does not exist when . … ►As a multivalued function of , is analytic everywhere except for possible branch points at , , and . The same properties hold for , except that as a function of , in general has poles at . … ►For example, when , , and , is a polynomial: …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
… ►can be regarded as a generalization of (11.2.7). Provided that , (11.9.1) has the general solution … ►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: 9.12 Scorer Functions
§9.12 Scorer Functions
… ►The general solution is given by … ► is a numerically satisfactory companion to the complementary functions and on the interval . is a numerically satisfactory companion to and on the interval . … ► …15: 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
…16: 5.12 Beta Function
§5.12 Beta Function
… ►In (5.12.1)–(5.12.4) it is assumed and . … ►In (5.12.8) the fractional powers have their principal values when and , and are continued via continuity. … ►When …where the contour starts from an arbitrary point in the interval , circles and then in the positive sense, circles and then in the negative sense, and returns to . …17: 5.2 Definitions
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
… ►When , is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue at . … ►18: 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). …19: 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.
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