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11: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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§26.4(i) Definitions

… ►It is also the number of $k$-dimensional lattice paths from $(0,0,\mathrm{\dots },0)$ to $(n_1,n_2,\mathrm{\dots },n_k)$. …

12: 1.6 Vectors and Vector-Valued Functions

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§1.6(iv) Path and Line Integrals

… ►A path is defined by $𝐜(t)=(x(t),y(t),z(t))$, with $t$ ranging over an interval and $x(t),y(t),z(t)$ differentiable. …then the length of a path for $a\le t\le b$ is …The path integral of a continuous function $f(x,y,z)$ is …If $h(a)=b^{\prime }$ and $h(b)=a^{\prime }$, then the reparametrization is orientation-reversing and …

13: 8.6 Integral Representations

… ► $t^{a-1}$ takes its principal value where the path intersects the positive real axis, and is continuous elsewhere on the path. …where the integration path passes above or below the pole at $t=1$, according as upper or lower signs are taken. … ►In (8.6.10)–(8.6.12), $c$ is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at $s=a$, in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots }$. …

14: 20.13 Physical Applications

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

15: 15.17 Mathematical Applications

… ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

16: 9.13 Generalized Airy Functions

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9.13.25 $$A_k(z,p)=\frac{1}{2\pi \mathrm{i}}{\int }_{{ℒ}_k}{t}^{-p}\exp \left(zt-\frac{1}{3}{t}^3\right)dt,$$ $k=1,2,3$, $p\in ℂ$,

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Defines:

$k$: index (locally) and $p$: parameter (locally)

Symbols:

$A_k(z,p)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $ℒ$: integration path

Sources:

Reid (1972, (A4), p. 362); Drazin and Reid (1981, (A4), p. 466)

Referenced by:

§9.13(ii)

Permalink:

http://dlmf.nist.gov/9.13.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(ii), §9.13 and Ch.9

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9.13.27 $$B_k(z,p)={\int }_{{ℐ}_k}{t}^{-p}\exp \left(zt-\frac{1}{3}{t}^3\right)dt,$$ $k=1,2,3$, $p=0,\pm 1,\pm 2,\mathrm{\dots }$,

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Symbols:

$B_k(z,p)$: generalized Airy function, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: complex variable, $k$: index, $p$: parameter and $ℐ$: integration path

Source:

Drazin and Reid (1981, (A20), p. 470)

Permalink:

http://dlmf.nist.gov/9.13.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(ii), §9.13 and Ch.9

… ►The integration paths ${ℒ}_0$, ${ℒ}_1$, ${ℒ}_2$, ${ℒ}_3$ are depicted in Figure 9.13.1. … ►

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Figure 9.13.1: $t$-plane. Paths ${ℒ}_0$, ${ℒ}_1$, ${ℒ}_2$, ${ℒ}_3$. Magnify

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Figure 9.13.2: $t$-plane. Paths ${ℐ}_1$, ${ℐ}_2$, ${ℐ}_3$. Magnify

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17: 16.17 Definition

… ►where the integration path $L$ separates the poles of the factors $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ from those of the factors $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$. … ►

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Figure 16.17.1: s-plane. Path $L$ for the integral representation (16.17.1) of the Meijer $G$-function.

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18: 11.13 Methods of Computation

… ►To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. …

19: 2.4 Contour Integrals

… ►If $q(t)$ is analytic in a sector containing $\mathrm{ph}t=0$, then the region of validity may be increased by rotation of the integration paths. … ►Let $𝒫$ denote the path for the contour integral … ►Cases in which $p^{\prime }(t_0)\ne 0$ are usually handled by deforming the integration path in such a way that the minimum of $\mathrm{\Re }\left(zp(t)\right)$ is attained at a saddle point or at an endpoint. Additionally, it may be advantageous to arrange that $\mathrm{\Im }\left(zp(t)\right)$ is constant on the path: this will usually lead to greater regions of validity and sharper error bounds. …However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. …

20: 4.37 Inverse Hyperbolic Functions

… ►In (4.37.1) the integration path may not pass through either of the points $t=\pm \mathrm{i}$, and the function ${(1+t^2)}^{1/2}$ assumes its principal value when $t$ is real. In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function ${(t^2-1)}^{1/2}$ assumes its principal value when $t\in (1,\mathrm{\infty })$. Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. In (4.37.3) the integration path may not intersect $\pm 1$. … ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …