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21: 9.1 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 Airy functions
and , and the Scorer functions
and (also known as inhomogeneous Airy functions).
►Other notations that have been used are as follows: and for and (Jeffreys (1928), later changed to and ); , (Fock (1945)); (Szegő (1967, §1.81)); , (Tumarkin (1959)).
nonnegative integer, except in §9.9(iii). | |
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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: 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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24: 26.10 Integer Partitions: Other Restrictions
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§26.10(ii) Generating Functions
… ►where the last right-hand side is the sum over of the generating functions for partitions into distinct parts with largest part equal to . … ►§26.10(v) Limiting Form
… ►§26.10(vi) Bessel-Function Expansion
… ►where is the modified Bessel function (§10.25(ii)), and …25: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
…26: 11.10 Anger–Weber Functions
§11.10 Anger–Weber Functions
… ►The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation … ►§11.10(vi) Relations to Other Functions
… ►For the Fresnel integrals and see §7.2(iii). …27: 7.2 Definitions
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§7.2(i) Error Functions
… ► , , and are entire functions of , as is in the next subsection. ►Values at Infinity
… ► , , and are entire functions of , as are and in the next subsection. … ►§7.2(iv) Auxiliary Functions
…28: 1.10 Functions of a Complex Variable
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►If , analytic in , equals on an arc in , or on just an infinite number of points with a limit point in , then they are equal throughout and is called an analytic continuation of .
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►An isolated singularity is always removable when exists, for example at .
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►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles.
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►A cut neighborhood is formed by deleting a ray emanating from the center.
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§1.10(xi) Generating Functions
…29: 26.12 Plane Partitions
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§26.12(ii) Generating Functions
… ►§26.12(iv) Limiting Form
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26.12.26
►where is the Riemann -function (§25.2(i)).
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30: Bibliography R
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Normal limit theorems for symmetric random matrices.
Probab. Theory Related Fields 112 (3), pp. 411–423.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Applied Bessel Functions.
Dover Publications Inc., New York.
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Precision controlled trigonometric algorithms.
Appl. Math. Comput. 2 (4), pp. 335–352.
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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media.
In Differential Operators and Related Topics, Vol. I (Odessa,
1997),
Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
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