Anger%E2%80%93Weber%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: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
►The hypergeometric function is defined by the Gauss series … … ►On the circle of convergence, , the Gauss series: … ►§15.2(ii) Analytic Properties
…13: 5.12 Beta Function
14: 14.20 Conical (or Mehler) Functions
§14.20 Conical (or Mehler) Functions
►§14.20(i) Definitions and Wronskians
… ► … ►§14.20(ii) Graphics
… ►§14.20(x) Zeros and Integrals
…15: 11.1 Special Notation
§11.1 Special Notation
►(For other notation see Notation for the Special Functions.) … ►For the functions , , , , , and see §§10.2(ii), 10.25(ii). ►The functions treated in this chapter are the Struve functions and , the modified Struve functions and , the Lommel functions and , the Anger function , the Weber function , and the associated Anger–Weber function .16: 10.1 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 Bessel functions
, ; Hankel functions
, ; modified Bessel functions
, ; spherical Bessel functions
, , , ; modified spherical Bessel functions
, , ; Kelvin functions
, , , .
For the spherical Bessel functions and modified spherical Bessel functions the order is a nonnegative integer.
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►A common alternative notation for is .
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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).
17: 4.2 Definitions
18: 12.14 The Function
§12.14 The Function
… ►Other expansions, involving and , can be obtained from (12.4.3) to (12.4.6) by replacing by and by ; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►Bessel Functions
… ►Confluent Hypergeometric Functions
… ►§12.14(x) Modulus and Phase Functions
…19: 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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