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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

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§28.2(vi) Eigenfunctions

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3: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

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4: 10.24 Functions of Imaginary Order

§10.24 Functions of Imaginary Order

… ►and ${\tilde{J}}_{\nu }\left(x\right)$, ${\tilde{Y}}_{\nu }\left(x\right)$ are linearly independent solutions of (10.24.1): … ►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). … … ►

5: 10.45 Functions of Imaginary Order

§10.45 Functions of Imaginary Order

… ►and ${\tilde{I}}_{\nu }\left(x\right)$, ${\tilde{K}}_{\nu }\left(x\right)$ are real and linearly independent solutions of (10.45.1): … ►The corresponding result for ${\tilde{K}}_{\nu }\left(x\right)$ is given by … ► … ►

6: 10.26 Graphics

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§10.26(i) Real Order and Variable

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§10.26(ii) Real Order, Complex Variable

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§10.26(iii) Imaginary Order, Real Variable

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Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.8: ${\tilde{I}}_1\left(x\right)$, ${\tilde{K}}_1\left(x\right)$, $0.01\le x\le 3$. Magnify

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7: 2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols

… ►As $x\to c$ in $𝐗$ … ►

§2.1(ii) Integration and Differentiation

►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … ►Differentiation requires extra conditions. …

8: 11.7 Integrals and Sums

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11.7.1 $$\int z^{\nu }{𝐇}_{\nu -1}\left(z\right)dz=z^{\nu }{𝐇}_{\nu }\left(z\right),$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E1

Encodings:

pMML, png, TeX

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Annotations for §11.7(i), §11.7 and Ch.11

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11.7.5 $$f_{\nu }(z)={\int }_0^z{t}^{\nu }{𝐇}_{\nu }\left(t\right)dt,$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

… ►The following Laplace transforms of ${𝐇}_{\nu }\left(t\right)$ require $\mathrm{\Re }a>0$ for convergence, while those of ${𝐋}_{\nu }\left(t\right)$ require $\mathrm{\Re }a>1$. … ►

§11.7(iv) Integrals with Respect to Order

►For integrals of ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ with respect to the order $\nu$, see Apelblat (1989). …

9: 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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10: 18.36 Miscellaneous Polynomials

… ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … ►This infinite set of polynomials of order $n\ge k$, the smallest power of $x$ being $x^k$ in each polynomial, is a complete orthogonal set with respect to this measure. … ►This lays the foundation for consideration of exceptional OP’s wherein a finite number of (possibly non-sequential) polynomial orders are missing, from what is a complete set even in their absence. … ►Exceptional type I $X_m$-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order $m$, or, said another way, the first $m$ polynomial orders, $0,1,\mathrm{\dots },m-1$ are missing. The exceptional type III $X_m$-EOP’s are missing orders $1,\mathrm{\dots },m$. …