Weierstrass%0Aelliptic%20functions
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21: 12.14 The Function
§12.14 The Function
… ►§12.14(ii) Values at and Wronskian
… ►the branch of being zero when and defined by continuity elsewhere. … ►When … ►where is defined in (12.14.5), and ( 0), , ( 0), and are real. …22: 11.10 Anger–Weber Functions
§11.10 Anger–Weber Functions
… ►(11.10.4) also applies when and . … ►§11.10(vi) Relations to Other Functions
… ► … ►§11.10(viii) Expansions in Series of Products of Bessel Functions
…23: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
… ►Other Inverse Functions
… ►Care needs to be taken on the cuts, for example, if then . … ►§4.23(viii) Gudermannian Function
… ►The inverse Gudermannian function is given by …24: 30.1 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 the eigenvalues and the spheroidal wave functions
, , , , and , .
…Meixner and Schäfke (1954) use , , , for , , , , respectively.
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Other Notations
…25: 16.2 Definition and Analytic Properties
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§16.2(i) Generalized Hypergeometric Series
… ► … ►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at , and . … ►On the circle the series (16.2.1) is absolutely convergent if , convergent except at if , and divergent if , where … ►§16.2(v) Behavior with Respect to Parameters
…26: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
►§4.37(i) General Definitions
… ►the upper or lower sign being taken according as ; compare Figure 4.37.1(ii). … ►Other Inverse Functions
… ►§4.37(vi) Interrelations
…27: 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.
►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: , , , and .
…An older notation, due to Whittaker (1902), for is .
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28: 17.1 Special Notation
§17.1 Special Notation
►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the basic hypergeometric (or -hypergeometric) function , the bilateral basic hypergeometric (or bilateral -hypergeometric) function , and the -analogs of the Appell functions , , , and . ►Another function notation used is the “idem” function: …29: 14.1 Special Notation
§14.1 Special Notation
►(For other notation see Notation for the Special Functions.) … ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. ►The main functions treated in this chapter are the Legendre functions , , , ; Ferrers functions , (also known as the Legendre functions on the cut); associated Legendre functions , , ; conical functions , , , , (also known as Mehler functions). …30: 35.1 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 the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric or ; generalized hypergeometric or .
►An alternative notation for the multivariate gamma function is (Herz (1955, p. 480)).
Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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