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1: 28.12 Definitions and Basic Properties

… ►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. … ►

§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►For $q=0$, … ► … ►

2: 28.2 Definitions and Basic Properties

… ►converges absolutely and uniformly in compact subsets of $ℂ$. … ►In Table 28.2.1 $n=0,1,2,\mathrm{\dots }$. … ►

Change of Sign of $q$

… ►

§28.2(vi) Eigenfunctions

… ►For the connection with the basic solutions in §28.2(ii), …

3: Frank W. J. Olver

… ► 1924 in Croydon, U. …degrees in mathematics from the University of London in 1945, 1948, and 1961, respectively. … ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … ►He also spent time as a Visiting Fellow, or Professor, at the University of Lancaster, U. … ►In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olver’s 65th birthday, with Proceedings published by Marcel Dekker in 1990. …

4: 10.24 Functions of Imaginary Order

§10.24 Functions of Imaginary Order

… ►In consequence of (10.24.6), when $x$ is large ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when $x$ is small either ${\tilde{J}}_{\nu }\left(x\right)$ and $\tanh \left(\frac{1}{2}\pi \nu \right){\tilde{Y}}_{\nu }\left(x\right)$ or ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu \ne 0$ or $\nu =0$. … ►In this reference ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ are denoted respectively by $F_{\mathrm{i}\nu }(x)$ and $G_{\mathrm{i}\nu }(x)$.

5: 10.45 Functions of Imaginary Order

§10.45 Functions of Imaginary Order

… ►In consequence of (10.45.5)–(10.45.7), ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either ${\tilde{I}}_{\nu }\left(x\right)$ and $(1/\pi )\sinh \left(\pi \nu \right){\tilde{K}}_{\nu }\left(x\right)$, or ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu \ne 0$ or $\nu =0$. … ►In this reference ${\tilde{I}}_{\nu }\left(x\right)$ is denoted by $(1/\pi )\sinh \left(\pi \nu \right)L_{\mathrm{i}\nu }(x)$. …

6: 10.76 Approximations

… ►

Real Variable and Order $:$ Functions

… ►

Real Variable and Order $:$ Zeros

… ►

Real Variable and Order $:$ Integrals

… ►

Complex Variable; Real Order

… ►

Real Variable; Imaginary Order

…

7: 10.77 Software

… ►

§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

… ►

§10.77(iii) Bessel Functions–Real Order and Argument

… ►

§10.77(vi) Bessel Functions–Imaginary Order and Real Argument

… ►

§10.77(vii) Bessel Functions–Complex Order and Real Argument

… ►

§10.77(viii) Bessel Functions–Complex Order and Argument

…

8: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

…

9: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

10: 11.1 Special Notation

§11.1 Special Notation

… ► ►►►►

$x$	real variable.
…
$\nu$	real or complex order.
$n$	integer order.
…

… ►The functions treated in this chapter are the Struve functions ${𝐇}_{\nu }\left(z\right)$ and ${𝐊}_{\nu }\left(z\right)$, the modified Struve functions ${𝐋}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$, the Lommel functions $s_{\mu ,\nu }\left(z\right)$ and $S_{\mu ,\nu }\left(z\right)$, the Anger function ${𝐉}_{\nu }\left(z\right)$, the Weber function ${𝐄}_{\nu }\left(z\right)$, and the associated Anger–Weber function ${𝐀}_{\nu }\left(z\right)$.