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21: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►where ${\alpha }_n(a)$ and ${\beta }_n(a)$ satisfy the recursion relations … ►

§12.14(vii) Relations to Other Functions

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Bessel Functions

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Confluent Hypergeometric Functions

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22: Gergő Nemes

… ►Nemes has research interests in asymptotic analysis, Écalle theory, exact WKB analysis, and special functions. ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. ►In March 2022, Nemes was named Contributing Developer of the NIST Digital Library of Mathematical Functions.

23: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. ►In July 2023, Groenevelt was named Contributing Developer of the NIST Digital Library of Mathematical Functions.

24: 1.10 Functions of a Complex Variable

… ►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. … ►each location again being counted with multiplicity equal to that of the corresponding zero or pole. … ►

§1.10(x) Infinite Partial Fractions

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§1.10(xi) Generating Functions

… ►The recurrence relation for $C_n^{(\lambda )}\left(x\right)$ in §18.9(i) follows from $(1-2xz+z^2)\frac{\partial }{\partial z}F(x,\lambda ;z)=2\lambda (x-z)F(x,\lambda ;z)$, and the contour integral representation for $C_n^{(\lambda )}\left(x\right)$ in §18.10(iii) is just (1.10.27).

25: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

… ►the upper/lower signs corresponding to the upper/lower sides. … ►

§4.23(viii) Gudermannian Function

►The Gudermannian $\mathrm{gd}\left(x\right)$ is defined by … ►The inverse Gudermannian function is given by …

26: 4.2 Definitions

… ►Most texts extend the definition of the principal value to include the branch cut … ►In contrast to (4.2.5) the closed definition is symmetric. … ►

§4.2(ii) Logarithms to a General Base $a$

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§4.2(iii) The Exponential Function

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§4.2(iv) Powers

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27: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iv) Relations to Other Functions

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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28: 22.16 Related Functions

§22.16 Related Functions

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Relation to Elliptic Integrals

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Relation to Theta Functions

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Relation to the Elliptic Integral $E(\varphi ,k)$

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Definition

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29: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

30: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

… ►When $z\notin 𝕃$ the functions are related by … ►When it is important to display the lattice with the functions they are denoted by $\mathrm{\wp }\left(z|𝕃\right)$, $\zeta \left(z|𝕃\right)$, and $\sigma \left(z|𝕃\right)$, respectively. …