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21: 1.9 Calculus of a Complex Variable

… ►If $f(z)$ is continuous within and on a simple closed contour $C$ and analytic within $C$, then … ►If $f(z)$ is continuous within and on a simple closed contour $C$ and analytic within $C$, and if $z_0$ is a point within $C$, then … ►If $C$ is a closed contour, and $z_0\notin C$, then … ►Suppose $f(z)$ is analytic in a domain $D$ and $C_1,C_2$ are two arcs in $D$ passing through $z_0$. … ►for any finite contour $C$ in $D$. …

22: Bibliography D

… ►

The principal frequencies of vibrating systems with elliptic boundaries. Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.

ⓘ

External Links:

Document, MathReview (M. A. Hyman), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

BibTeX

… ►

Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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External Links:

ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

Encodings:

BibTeX

… ►

Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

ⓘ

Notes:

(electronic only, article id. JHEP01(2000)019, 8 pp.).

External Links:

ISSN 1029-8479, Document, MathReview, ZentralBlatt

Referenced by:

§22.11.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

ⓘ

External Links:

ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

…

23: 19.37 Tables

… ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 10D by Fettis and Caslin (1964). ►Tabulated for $\varphi =0(1^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 7S by Beli͡akov et al. (1962). … ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $k=0(.01)1$ to 10D by Fettis and Caslin (1964). ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971), for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(2^{\circ }){90}^{\circ }$ and $5^{\circ }({10}^{\circ }){85}^{\circ }$ to 8D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({10}^{\circ }){90}^{\circ }$, $\arcsin k=0(5^{\circ }){90}^{\circ }$ to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated for $\varphi =5^{\circ }(5^{\circ }){80}^{\circ }({2.5}^{\circ }){90}^{\circ }$, ${\alpha }^2=-1(.1)-0.1,0.1(.1)1$, $k^2=0(.05)0.9(.02)1$ to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). …

24: 1.12 Continued Fractions

… ► $C_n$ is called the $n$th approximant or convergent to $C$ . … ►Define … ►A contraction of a continued fraction $C$ is a continued fraction $C^{\prime }$ whose convergents $\{C_n^{\prime }\}$ form a subsequence of the convergents $\{C_n\}$ of $C$. Conversely, $C$ is called an extension of $C^{\prime }$. … ►Then the convergents $C_n$ satisfy …

25: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ►

26.6.12 $$C\left(n\right)=\sum _{k=1}^{n}N(n,k),$$

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Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer, $n$: nonnegative integer and $N(n,k)$: Narayana number

Permalink:

http://dlmf.nist.gov/26.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.13 $$M(n)=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)C\left(n+1-k\right),$$

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Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.14 $$C\left(n\right)=\sum _{k=0}^{2n}{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{2n}{k}\right)M(2n-k).$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

26: Frank W. J. Olver

… ►

9 Airy and Related Functions

… ►He continued his editing work until the time of his death on April 22, 2013 at age 88.

27: 18.17 Integrals

… ►For the beta function $\mathrm{B}(a,b)$ see §5.12, and for the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. … ►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. … ►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). … ►For the generalized hypergeometric function ${}_2{}^{}F_2^{}$ see (16.2.1). … ►For further integrals, see Apelblat (1983, pp. 189–204), Erdélyi et al. (1954a, pp. 38–39, 94–95, 170–176, 259–261, 324), Erdélyi et al. (1954b, pp. 42–44, 271–294), Gradshteyn and Ryzhik (2000, pp. 788–806), Gröbner and Hofreiter (1950, pp. 23–30), Marichev (1983, pp. 216–247), Oberhettinger (1972, pp. 64–67), Oberhettinger (1974, pp. 83–92), Oberhettinger (1990, pp. 44–47 and 152–154), Oberhettinger and Badii (1973, pp. 103–112), Prudnikov et al. (1986b, pp. 420–617), Prudnikov et al. (1992a, pp. 419–476), and Prudnikov et al. (1992b, pp. 280–308).

28: 7.14 Integrals

… ►

7.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}at}\mathrm{erfc}\left(bt\right)dt=\frac{1}{a\sqrt{\pi }}F\left(\frac{a}{b}\right)+\frac{\mathrm{i}}{2a}\left(1-{\mathrm{e}}^{-{(a/b)}^2}\right),$$ $a\in ℂ$, .

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{ph}$: phase

A&S Ref:

7.4.18 (in different form)

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

… ►

7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

… ►

7.14.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}+a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.29 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

… ►For collections of integrals see Apelblat (1983, pp. 131–146), Erdélyi et al. (1954a, vol. 1, pp. 40, 96, 176–177), Geller and Ng (1971), Gradshteyn and Ryzhik (2000, §§5.4 and 6.28–6.32), Marichev (1983, pp. 184–189), Ng and Geller (1969), Oberhettinger (1974, pp. 138–139, 142–143), Oberhettinger (1990, pp. 48–52, 155–158), Oberhettinger and Badii (1973, pp. 171–172, 179–181), Prudnikov et al. (1986b, vol. 2, pp. 30–36, 93–143), Prudnikov et al. (1992a, §§3.7–3.8), and Prudnikov et al. (1992b, §§3.7–3.8). …

29: 12.14 The Function $W(a,x)$

… ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►Here ${𝒜}_s(t)$ is as in §12.10(ii), $\sigma$ is defined by … ►uniformly for $t\in [-1+\delta ,1-\delta ]$, with $\eta$ given by (12.10.23) and ${\widetilde{𝒜}}_s(t)$ given by (12.10.24). … ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, and $B_s(\zeta )$ as in §12.10(vii). … ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …

30: 19.36 Methods of Computation

… ►If (19.36.1) is used instead of its first five terms, then the factor ${(3r)}^{-1/6}$ in Carlson (1995, (2.2)) is changed to ${(3r)}^{-1/8}$. ►For both $R_D$ and $R_J$ the factor ${(r/4)}^{-1/6}$ in Carlson (1995, (2.18)) is changed to ${(r/5)}^{-1/8}$ when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: … ►All cases of $R_F$, $R_C$, $R_J$, and $R_D$ are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). … ►The incomplete integrals $R_F(x,y,z)$ and $R_G(x,y,z)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_C$, accompanied by two quadratically convergent series in the case of $R_G$; compare Carlson (1965, §§5,6). … ►Here $R_C$ is computed either by the duplication algorithm in Carlson (1995) or via (19.2.19). …