path integral

SearchAdvancedHelp

(0.003 seconds)

11—20 of 51 matching pages

11: 5.9 Integral Representations

§5.9 Integral Representations

… ►

Hankel’s Loop Integral

… ►where the path is the real axis. … ► … ► …

12: 5.13 Integrals

§5.13 Integrals

►In (5.13.1) the integration path is a straight line parallel to the imaginary axis. … ►

Barnes’ Beta Integral

… ►

Ramanujan’s Beta Integral

… ►

13: 2.4 Contour Integrals

… ►Let $𝒫$ denote the path for the contour integral … ►Paths on which $\mathrm{\Im }\left(zp(t)\right)$ is constant are also the ones on which $|\exp \left(-zp(t)\right)|$ decreases most rapidly. …

14: 8.6 Integral Representations

§8.6 Integral Representations

… ►

§8.6(ii) Contour Integrals

… ► $t^{a-1}$ 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 $s=0,1,2,\mathrm{\dots }$. …

15: 8.19 Generalized Exponential Integral

… ►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_p\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed. … ►The general function $E_p\left(z\right)$ is attained by extending the path in (8.19.2) across the negative real axis. …

16: 3.5 Quadrature

… ►

§3.5(ix) Other Contour Integrals

…

17: 9.13 Generalized Airy Functions

… ►

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

ⓘ

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

►

9.13.26 $$B_0(z,p)=\frac{1}{2\pi \mathrm{i}}{\int }_{{ℒ}_0}{t}^{-p}\exp \left(zt-\frac{1}{3}{t}^3\right)dt,$$ $p=0,\pm 1,\pm 2,\mathrm{\dots }$,

ⓘ

Symbols:

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

Source:

Reid (1972, (A16), p. 364)

Permalink:

http://dlmf.nist.gov/9.13.E26

Encodings:

pMML, png, TeX

See also:

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

►

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

ⓘ

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

…

18: 19.2 Definitions

… ►The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. …

19: 23.6 Relations to Other Functions

… ►where the integral is taken along any path from $z$ to $\mathrm{\infty }$ that does not pass through any of $e_1,e_2,e_3$. …

20: 31.9 Orthogonality

… ►

31.9.5 $${\int }_{{ℒ}_1}{\int }_{{ℒ}_2}\rho (s,t)w_1(s)w_1(t)w_2(s)w_2(t)dsdt=0,$$ $|n_1-n_2|+|m_1-m_2|\ne 0$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $\rho (s,t)$: weight function and ${ℒ}_1$, ${ℒ}_2$: integration paths

Permalink:

http://dlmf.nist.gov/31.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(ii), §31.9 and Ch.31

…