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21: 2.4 Contour Integrals

… ►The result in §2.3(ii) carries over to a complex parameter $z$. … ►Assume that $p(t)$ and $q(t)$ are analytic on an open domain $𝐓$ that contains $𝒫$, with the possible exceptions of $t=a$ and $t=b$. … ►in which … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). … ►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. …

22: Bibliography G

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Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 19(1964), no. 4, pp. 1–89

External Links:

Document, MathReview (A. Korányi), ZentralBlatt

Referenced by:

§35.3(i).

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23: Bibliography

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Arb: A C Library for Arbitrary Precision Ball Arithmetic.

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Notes:

An extended-precision C code for special functions. The library uses arbitrary-precision ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

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24: 13.9 Zeros

… ►When $a,b\in ℝ$ the number of real zeros is finite. … ►For fixed $a,b\in ℂ$ the large $z$-zeros of $M(a,b,z)$ satisfy … ►Let $P_{\alpha }$ denote the closure of the domain that is bounded by the parabola $y^2=4\alpha (x+\alpha )$ and contains the origin. … ►For fixed $a$ and $z$ in $ℂ$ the function $M(a,b,z)$ has only a finite number of $b$-zeros. … ►Inequalities for the zeros of $U(a,b,x)$ are given in Gatteschi (1990). …

25: Preface

… ► A. …A summary of the responsibilities of these groups may help in understanding the structure and results of this project. … ►The associate editors are eminent domain experts who were recruited to advise the project on strategy, execution, subject content, format, and presentation, and to help identify and recruit suitable candidate authors and validators. … ►The editors acknowledge the many other individuals who contributed to the project in a variety of ways. …Undoubtedly, the editors have overlooked some individuals who contributed, as is inevitable in a large long-lasting project. …

26: Bibliography S

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A Course in Arithmetic. Graduate Texts in Mathematics, Vol. 7, Springer-Verlag, New York.

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Notes:

Translated from the French.

External Links:

MathReview, ZentralBlatt

Referenced by:

§20.12(i), §20.7(viii), §20.9(i), §20.9(ii), §22.18(iv), §23.15(i), §23.18, §23.19, §23.20(v).

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

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Confluent hypergeometric functions on tube domains. Math. Ann. 260 (3), pp. 269–302.

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External Links:

ISSN 0025-5831, Document, MathReview (A. B. Venkov), ZentralBlatt

Referenced by:

§35.6(ii).

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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.

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External Links:

ISSN 0036-1410, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.5(ii).

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A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.

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External Links:

ISBN 0-8493-8273-4, MathReview (J. Horváth), ZentralBlatt

Referenced by:

§18.17(v).

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27: 18.18 Sums

… ►Moreover, the series (18.18.2) converges uniformly on any compact domain within $E$. … ►

Legendre

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Laguerre

… ►In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. … ►and a similar pair of equations by symmetry; compare the second row in Table 18.6.1. …

28: 10.20 Uniform Asymptotic Expansions for Large Order

… ►In the following formulas for the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $u_k$, $v_k$ are the constants defined in §9.7(i), and $U_k(p)$, $V_k(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). … ►In this way there is less usage of many-valued functions. … ►Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2. … ►The eye-shaped closed domain in the uncut $z$-plane that is bounded by $B{P}_1{E}_1$ and $B{P}_2{E}_2$ is denoted by $𝐊$; see Figure 10.20.3. … ►As $\nu \to \mathrm{\infty }$ through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for $|\mathrm{ph}z|\le \pi -\delta$, the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, being the analytic continuations of the functions defined in §10.20(i) when $\zeta$ is real. …

29: 1.14 Integral Transforms

… ►Suppose that $f(t)$ is absolutely integrable on $(-\mathrm{\infty },\mathrm{\infty })$ and of bounded variation in a neighborhood of $t=u$ (§1.4(v)). … ►If $f(t)$ is absolutely integrable on $[0,\mathrm{\infty })$ and of bounded variation (§1.4(v)) in a neighborhood of $t=u$, then … ►If $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma$ in , then the integral (1.14.32) converges and $ℳf\left(s\right)$ is an analytic function of $s$ in the vertical strip . … ►The Hilbert transform of a real-valued function $f(t)$ is defined in the following equivalent ways: … ►If the integral converges, then it converges uniformly in any compact domain in the complex $s$-plane not containing any point of the interval $(-\mathrm{\infty },0]$. …

30: Bibliography N

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Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:

ISSN 0031-9015, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

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Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, pp. 53–86.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iv).

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The Boutroux ansatz for the second Painlevé equation in the complex domain. Izv. Akad. Nauk SSSR Ser. Mat. 54 (6), pp. 1229–1251 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 37(1991), no. 3, pp. 587–609.

External Links:

ISSN 0373-2436, Document, MathReview (Alexander I. Bobenko), ZentralBlatt

Referenced by:

§32.12(ii).

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Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§36.7(iv).

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Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.

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External Links:

ISSN 1364-5021, Document, ZentralBlatt

Referenced by:

§36.7(iv).

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