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1: 28.12 Definitions and Basic Properties

… ►Without loss of generality, from now on we replace $\nu +2n$ by $\nu$. … ►

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

… ►For $q=0$, … ► … ►

2: 28.2 Definitions and Basic Properties

… ►(28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with … ►A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. … ►The Fourier series of a Floquet solution …leads to a Floquet solution. … ►

§28.2(vi) Eigenfunctions

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3: Michael V. Berry

… ►is Melville Wills Professor (emertitus) (homepage) in the Physics Department of the University of Bristol, U. … ►He was elected a Fellow of the Royal Society of London in 1982, and of the Royal Society of Edinburgh in 2005. …He was elected a Foreign Member of the Royal Society of Sciences, Uppsala, Sweden, in 1986, of the Indian Academy of Sciences in 1990, of the U. … ►Berry was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. … ►Berry was named a Senior Associate Editor of the DLMF in November 2015.

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 $\nu =0$. … ►

5: 10.45 Functions of Imaginary Order

§10.45 Functions of Imaginary Order

… ►The corresponding result for ${\tilde{K}}_{\nu }\left(x\right)$ is given by … ►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 $\nu =0$. … ►

6: 10.76 Approximations

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Real Variable and Order $:$ Functions

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Real Variable and Order $:$ Zeros

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Real Variable and Order $:$ Integrals

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Complex Variable; Real Order

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Real Variable; Imaginary Order

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7: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

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8: 10.77 Software

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

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

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

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

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

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9: 10.3 Graphics

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§10.3(i) Real Order and Variable

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§10.3(ii) Real Order, Complex Variable

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§10.3(iii) Imaginary Order, Real Variable

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Figure 10.3.18: ${\tilde{J}}_1\left(x\right)$, ${\tilde{Y}}_1\left(x\right)$, $0.01\le x\le 10$. Magnify

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Figure 10.3.19: ${\tilde{J}}_5\left(x\right)$, ${\tilde{Y}}_5\left(x\right)$, $0.01\le x\le 10$. Magnify

10: 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. …