for 3F2 hypergeometric functions of matrix argument
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21: 14.20 Conical (or Mehler) Functions
§14.20 Conical (or Mehler) Functions
… ► … ► exists except when and ; compare §14.3(i). … ►For extensions to complex arguments (including the range ), asymptotic expansions, and explicit error bounds, see Dunster (1991). For the case of purely imaginary order and argument see Dunster (2013). …22: 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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23: 5.12 Beta Function
24: 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).
25: 4.2 Definitions
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§4.2(iii) The Exponential Function
… ►§4.2(iv) Powers
… ►In particular, , and if , then … …26: 12.14 The Function
§12.14 The Function
… ►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument and parameter . … ►Bessel Functions
… ►Confluent Hypergeometric Functions
… ►§12.14(x) Modulus and Phase Functions
…27: 23.2 Definitions and Periodic Properties
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►If and are nonzero real or complex numbers such that , then the set of points , with , constitutes a lattice
with and
lattice generators.
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§23.2(ii) Weierstrass Elliptic Functions
… ►The function is quasi-periodic: for , … ►For , the function satisfies …More generally, if , , , and , then …28: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►§8.17(ii) Hypergeometric Representations
… ►For the hypergeometric function see §15.2(i). … ►The and convergents are less than , and the and convergents are greater than . … ►§8.17(vi) Sums
…29: 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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