special values of the variable
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1: 22.5 Special Values
§22.5 Special Values
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2: 22.8 Addition Theorems
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§22.8(iii) Special Relations Between Arguments
…3: Bille C. Carlson
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►In his paper Lauricella’s hypergeometric function
(1963), he defined the -function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter.
If some of the parameters are equal, then the -function is symmetric in the corresponding variables.
…Also, the homogeneity of the -function has led to a new type of mean value for several variables, accompanied by various inequalities.
►The foregoing matters are discussed in Carlson’s book Special Functions of Applied Mathematics, published by Academic Press in 1977.
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4: Mathematical Introduction
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►With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature.
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►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function.
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5: 18.6 Symmetry, Special Values, and Limits to Monomials
§18.6 Symmetry, Special Values, and Limits to Monomials
►§18.6(i) Symmetry and Special Values
… ►Laguerre
… ► …6: 35.1 Special Notation
§35.1 Special Notation
►(For other notation see Notation for the Special Functions.) … ►All fractional or complex powers are principal values. ►complex variables. | |
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7: 8.19 Generalized Exponential Integral
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►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of , and unless indicated otherwise in the DLMF principal values are assumed.
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►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase.
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§8.19(iii) Special Values
…8: 1.4 Calculus of One Variable
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►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus , and well defined for all values of these variables; possible exceptions being at boundary points.
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