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1: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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2: 1.2 Elementary Algebra

… ►The matrix $𝐀$ has a determinant, $det(𝐀)$, explored further in §1.3, denoted, in full index form, as … ►If det($𝐀$) $\ne$ $0$, $𝐀$ has a unique inverse, ${𝐀}^{-1}$, such that …A square matrix $𝑨$ is singular if $det(𝐀)=0$, otherwise it is non-singular. … ►If $𝐀𝐁=𝐁𝐀$ the matrices $𝐀$ and $𝐁$ are said to commute. The difference between $𝐀𝐁$ and $𝐁𝐀$ is the commutator denoted as …

3: 3.2 Linear Algebra

… ►Assume that $𝐀$ can be factored as in (3.2.5), but without partial pivoting. … ►is called the characteristic polynomial of $𝐀$ and its zeros are the eigenvalues of $𝐀$. … ►Let $𝐀$ be an $n\times n$ symmetric matrix. …has the same eigenvalues as $𝐀$. … ►Many methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).

4: Bibliography H

… ►

Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

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External Links:

MathReview (C. Tanaka)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

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Notes:

A reprinting with corrections appeared in 2000.

External Links:

ISBN 3-540-56670-8, MathReview, ZentralBlatt

Referenced by:

§3.7(v), §3.7(i).

Encodings:

BibTeX

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Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

Encodings:

BibTeX

… ►

Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

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External Links:

Document

Referenced by:

§33.7.

Encodings:

BibTeX

… ►

On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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External Links:

ZentralBlatt

Referenced by:

§24.15(iii).

Encodings:

BibTeX

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5: Bibliography

… ►

Tables of $\ln \mathrm{\Gamma }(z)$ for Complex Argument. Pergamon Press, New York.

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External Links:

ZentralBlatt

Referenced by:

§5.22(iii).

Encodings:

BibTeX

… ►

Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

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Referenced by:

§9.16.

Encodings:

BibTeX

… ►

Frequency response characteristics of the multiport planar elliptic patch. IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.

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External Links:

Document

Referenced by:

6th item.

Encodings:

BibTeX

… ►

Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 588; package TOMS)

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Chiral-symmetry breaking in QCD. I. The infrared domain. Phys. Rev. D (3) 37 (8), pp. 2290–2295.

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External Links:

ISSN 0556-2821, Document, MathReview

Referenced by:

§15.18.

Encodings:

BibTeX

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6: 21.5 Modular Transformations

… ►Let $𝐀$, $𝐁$, $𝐂$, and $𝐃$ be $g\times g$ matrices with integer elements such that …Here $\xi (𝚪)$ is an eighth root of unity, that is, ${(\xi (𝚪))}^8=1$. For general $𝚪$, it is difficult to decide which root needs to be used. … ►($𝐀$ invertible with integer elements.) …For a $g\times g$ matrix $𝐀$ we define $\mathrm{diag}𝐀$, as a column vector with the diagonal entries as elements. …

7: Bibliography U

… ►

On Kelvin’s ship-wave pattern. J. Fluid Mech. 8 (3), pp. 418–431.

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External Links:

Document, MathReview (R. C. MacCamy), ZentralBlatt

Referenced by:

§36.13.

Encodings:

BibTeX

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Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.

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External Links:

ISSN 0305-0041, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§14.26.

Encodings:

BibTeX

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8: 29.7 Asymptotic Expansions

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29.7.6 $${\tau }_2=\frac{1}{2^{10}}(1+k^2){(1-k^2)}^2(5p^4+34p^2+9),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.7 $${\tau }_3=\frac{p}{2^{14}}({(1+k^2)}^4(33p^4+410p^2+405)-24k^2{(1+k^2)}^2(7p^4+90p^2+95)+16k^4(9p^4+130p^2+173)),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.8 $${\tau }_4=\frac{1}{2^{16}}({(1+k^2)}^5(63p^6+1260p^4+2943p^2+486)-8k^2{(1+k^2)}^3(49p^6+1010p^4+2493p^2+432)+16k^4(1+k^2)(35p^6+760p^4+2043p^2+378)).$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►Formulas for additional terms can be computed with the author’s Maple program LA5; see §29.22. …

9: 1.12 Continued Fractions

… ► $A_n$ and $B_n$ are called the $n$th (canonical) numerator and denominator respectively. … ►Define … ►A sequence $\{C_n\}$ in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be a sequence of convergents of the continued fraction (1.12.3) iff … ►Conversely, $C$ is called an extension of $C^{\prime }$. … ►Then the convergents $C_n$ satisfy …

10: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►For $x\ge 0$ and $p>1$ let $x=\lambda p$ and define $A_0(\lambda )=1$, …so that $A_k(\lambda )$ is a polynomial in $\lambda$ of degree $k-1$ when $k\ge 1$. … ►

$A_1(\lambda )=1,$

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$A_2(\lambda )=1-2\lambda ,$

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$A_3(\lambda )=1-8\lambda +6{\lambda }^2.$

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