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1: 29.2 Differential Equations

§29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

… ►For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). … ►

2: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

… ►This is the hypergeometric differential equation. … ► … ►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

3: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ► … ►With $\zeta =\gamma z$ Equation (30.2.1) changes to … ►If ${\mu }^2=\frac{1}{4}$, Equation (30.2.2) reduces to the Mathieu equation; see (28.2.1). …

4: 31.2 Differential Equations

§31.2 Differential Equations

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§31.2(i) Heun’s Equation

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§31.2(v) Heun’s Equation Automorphisms

… ►There are $4!=24$ homographies $\tilde{z}(z)=(Az+B)/(Cz+D)$ that take $0,1,a,\mathrm{\infty }$ to some permutation of $0,1,a^{\prime },\mathrm{\infty }$, where $a^{\prime }$ may differ from $a$. …If $\tilde{z}=\tilde{z}(z)$ is one of the $4!-3!=18$ homographies that do not map $\mathrm{\infty }$ to $\mathrm{\infty }$, then an appropriate prefactor must be included on the right-hand side. …

5: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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6: 32.2 Differential Equations

… ►The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows: … ►The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions. … ►An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. … ►The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. … ►thus in the limit as $ϵ\to 0$, $W(\zeta )$ satisfies ${\text{P}}_{\text{I}}$ with $z=\zeta$. …

7: William P. Reinhardt

… ►Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland. … ►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

8: Gergő Nemes

… ►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. …

9: 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. …

10: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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Curtis (1964a) tabulates $P_{\mathrm{\ell }}(ϵ,r)$, $Q_{\mathrm{\ell }}(ϵ,r)$ (§33.1), and related functions for $\mathrm{\ell }=0,1,2$ and $ϵ=-2(.2)2$, with $x=0(.1)4$ for and $x=0(.1)10$ for $ϵ\ge 0$; 6D.

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