path integral

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1: 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). …

2: 1.6 Vectors and Vector-Valued Functions

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§1.6(iv) Path and Line Integrals

… ►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 …

3: 5.21 Methods of Computation

… ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …

4: 6.12 Asymptotic Expansions

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§6.12(i) Exponential and Logarithmic Integrals

… ►For the function $\chi$ see §9.7(i). … ►

§6.12(ii) Sine and Cosine Integrals

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5: Bibliography T

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Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

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External Links:

ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

BibTeX

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6: 11.6 Asymptotic Expansions

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11.6.3 $${\int }_0^z{𝐊}_0\left(t\right)dt-\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.32

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.4 $${\int }_0^z{𝐌}_0\left(t\right)dt+\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.8

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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7: 7.12 Asymptotic Expansions

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§7.12(ii) Fresnel Integrals

►The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and … ►They are bounded by $|\csc \left(4\mathrm{ph}z\right)|$ times the first neglected terms when . … ►

§7.12(iii) Goodwin–Staton Integral

►See Olver (1997b, p. 115) for an expansion of $G\left(z\right)$ with bounds for the remainder for real and complex values of $z$.

8: 9.17 Methods of Computation

… ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …

9: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ►where the path does not cross the negative real axis or pass through the origin. … ►

§6.2(ii) Sine and Cosine Integrals

… ►where the path does not cross the negative real axis or pass through the origin. … …

10: 16.17 Definition

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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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