reflection formulas for arguments and orders

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1: 10.61 Definitions and Basic Properties

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§10.61(iii) Reflection Formulas for Arguments

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§10.61(iv) Reflection Formulas for Orders

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2: 11.9 Lommel Functions

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11.9.1 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=z^{\mu -1}$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.9(i), §11.9(i), §11.9(iii)

Permalink:

http://dlmf.nist.gov/11.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

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11.9.2 $$w=s_{\mu ,\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{Y}_{\nu }\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/11.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

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11.9.3 $$s_{\mu ,\nu }\left(z\right)=z^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{z^{2k}}{a_{k+1}(\mu ,\nu )},$$

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Defines:

$s_{\mu ,\nu }\left(z\right)$: Lommel function

Symbols:

$z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Permalink:

http://dlmf.nist.gov/11.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

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Reflection Formulas

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§11.9(iii) Asymptotic Expansion

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3: Bibliography F

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Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.

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Notes:

Contains Mathematica package for spheroidal wave functions of complex arguments with complex parameters.

External Links:

FullText

Referenced by:

§30.12, 1st item, 2nd item.

Encodings:

BibTeX

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Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

3rd item, 2nd item.

Encodings:

BibTeX

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Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.

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External Links:

ISSN 0036-1410, Document, FullText, MathReview (A. Reinfelds), ZentralBlatt

Referenced by:

§5.11(iii), §5.11(iii).

Encodings:

BibTeX

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4: 10.68 Modulus and Phase Functions

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§10.68(iii) Asymptotic Expansions for Large Argument

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10.68.16 $$M_{\nu }\left(x\right)=\frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}\left(1-\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}+\frac{{(\mu -1)}^2}{256}\frac{1}{x^2}-\frac{(\mu -1)({\mu }^2+14\mu -399)}{6144\sqrt{2}}\frac{1}{x^3}+O\left(\frac{1}{x^4}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.21

Permalink:

http://dlmf.nist.gov/10.68.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.17 $$\ln M_{\nu }\left(x\right)=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln \left(2\pi x\right)-\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu -1)(\mu -25)}{384\sqrt{2}}\frac{1}{x^3}-\frac{(\mu -1)(\mu -13)}{128}\frac{1}{x^4}+O\left(\frac{1}{x^5}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $\ln z$: principal branch of logarithm function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.22

Permalink:

http://dlmf.nist.gov/10.68.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.18 $${\theta }_{\nu }\left(x\right)=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu -\frac{1}{8}\right)\pi +\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu -1}{16}\frac{1}{x^2}-\frac{(\mu -1)(\mu -25)}{384\sqrt{2}}\frac{1}{x^3}+O\left(\frac{1}{x^5}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.23

Referenced by:

§10.68(i), §10.70

Permalink:

http://dlmf.nist.gov/10.68.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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5: Errata

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The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$-argument Wronskian is given by $𝒲\left\{w_1(z),\mathrm{\dots },w_n(z)\right\}=det\left[w_k^{(j-1)}(z)\right]$, where $1\le j,k\le n$. Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$-argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$th-order differential equations. A reference to Ince (1926, §5.2) was added.

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The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).

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