Debye functions
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41: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
►§35.8(i) Definition
… ►Convergence Properties
… ►§35.8(ii) Relations to Other Functions
… ►Confluence
…42: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
… ►§7.18(iv) Relations to Other Functions
… ►Confluent Hypergeometric Functions
… ►Parabolic Cylinder Functions
… ►Probability Functions
…43: 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
…44: 28.20 Definitions and Basic Properties
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§28.20(ii) Solutions , , , ,
… ►§28.20(iv) Radial Mathieu Functions ,
… ►§28.20(vi) Wronskians
… ►§28.20(vii) Shift of Variable
… ►And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).45: 19.16 Definitions
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§19.16(ii)
… ►The -function is often used to make a unified statement of a property of several elliptic integrals. …where is the beta function (§5.12) and … ►For generalizations and further information, especially representation of the -function as a Dirichlet average, see Carlson (1977b). …46: 28.2 Definitions and Basic Properties
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►Since (28.2.1) has no finite singularities its solutions are entire functions of .
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§28.2(vi) Eigenfunctions
… ► … ► ►The functions are orthogonal, that is, …47: 4.13 Lambert -Function
§4.13 Lambert -Function
►The Lambert -function is the solution of the equation … ►and has several advantages over the Lambert -function (see Lawrence et al. (2012)), and the tree -function , which is a solution of … ►Properties include: … ►For these and other integral representations of the Lambert -function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020). …48: 22.16 Related Functions
§22.16 Related Functions
►§22.16(i) Jacobi’s Amplitude () Function
… ►§22.16(ii) Jacobi’s Epsilon Function
… ►Relation to Theta Functions
… ►§22.16(iii) Jacobi’s Zeta Function
…49: 19.2 Definitions
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►Let be a cubic or quartic polynomial in with simple zeros, and let be a rational function of and containing at least one odd power of .
…where is a polynomial in while and are rational functions of .
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§19.2(iv) A Related Function:
… ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if : …50: 8.22 Mathematical Applications
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