Feynman path integrals

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21: 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 … ►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. 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. …However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. …

22: 31.6 Path-Multiplicative Solutions

§31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by …These solutions are called path-multiplicative. …

23: 9.13 Generalized Airy Functions

… ►

§9.13(ii) Generalizations from Integral Representations

… ►The integration paths ${ℒ}_0$, ${ℒ}_1$, ${ℒ}_2$, ${ℒ}_3$ are depicted in Figure 9.13.1. … ►and the difference equation … ►Connection formulas for the solutions of (9.13.31) include … ►

24: 6.7 Integral Representations

§6.7 Integral Representations

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§6.7(i) Exponential Integrals

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§6.7(ii) Sine and Cosine Integrals

… ►The path of integration does not cross the negative real axis or pass through the origin. …

25: 9.5 Integral Representations

§9.5 Integral Representations

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§9.5(i) Real Variable

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9.5.1 $$\mathrm{Ai}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\cos \left(\frac{1}{3}{t}^3+xt\right)dt.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Source:

Olver (1997b, p. 53)

A&S Ref:

10.4.32 (in different form)

Permalink:

http://dlmf.nist.gov/9.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(i), §9.5 and Ch.9

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9.5.3 $$\mathrm{Bi}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{1}{3}{t}^3+xt\right)dt+\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\sin \left(\frac{1}{3}{t}^3+xt\right)dt.$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b)$: half-closed interval, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b]$: half-closed interval, $\sin z$: sine function and $x$: real variable

Proof sketch:

Use (9.5.5) with the substitions: for the paths $(-\mathrm{\infty },0]$ use $t=-\tau$, for the paths $[0,\mathrm{\infty }{\mathrm{e}}^{\pm \pi \mathrm{i}/3})$ use $t={\mathrm{e}}^{\pm \pi \mathrm{i}/3}\tau$. For the resulting integrals use (4.14.1) and (4.14.2).

A&S Ref:

10.4.33

Referenced by:

(9.12.19)

Permalink:

http://dlmf.nist.gov/9.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(i), §9.5 and Ch.9

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§9.5(ii) Complex Variable

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26: 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

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Ramanujan’s Beta Integral

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27: 11.5 Integral Representations

§11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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§11.5(ii) Contour Integrals

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Mellin–Barnes Integrals

… ►In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at $s=0,1,2,\mathrm{\dots }$ from those at $s=-1,-2,-3,\mathrm{\dots }$. …

28: 31.18 Methods of Computation

… ►Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …

29: 16.17 Definition

… ►Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: …where the integration path $L$ separates the poles of the factors $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ from those of the factors $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$. … ►

$L$ goes from $-\mathrm{i}\mathrm{\infty }$ to $\mathrm{i}\mathrm{\infty }$. The integral converges if and .

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$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ($\ne 0$) if , and for if $p=q\ge 1$.

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$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all $z$ if $p>q$, and for $|z|>1$ if $p=q\ge 1$.

…

30: 7.7 Integral Representations

§7.7 Integral Representations

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§7.7(i) Error Functions and Dawson’s Integral

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§7.7(ii) Auxiliary Functions

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Mellin–Barnes Integrals

… ►In (7.7.13) and (7.7.14) the integration paths are straight lines, $\zeta =\frac{1}{16}{\pi }^2{z}^4$, and $c$ is a constant such that in (7.7.13), and in (7.7.14). …