contour integrals

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11: 2.4 Contour Integrals

§2.4 Contour Integrals

… ►Let $𝒫$ denote the path for the contour integral ►

2.4.10 $$I(z)={\int }_a^b{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Defines:

$I(z)$: contour integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E10

Encodings:

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See also:

Annotations for §2.4(iii), §2.4 and Ch.2

… ►

2.4.14 $$I(z)={\int }_{t_0}^b{\mathrm{e}}^{-zp(t)}q(t)dt-{\int }_{t_0}^a{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $I(z)$: contour integral, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E14

Encodings:

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See also:

Annotations for §2.4(iv), §2.4 and Ch.2

… ►and assigning an appropriate value to $c$ to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12). …

12: 13.4 Integral Representations

… ►

§13.4(ii) Contour Integrals

… ►

13: 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)). …

14: 14.25 Integral Representations

… ►For corresponding contour integrals, with less restrictions on $\mu$ and $\nu$, see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).

15: 18.10 Integral Representations

… ►

§18.10(iii) Contour Integral Representations

►Table 18.10.1 gives contour integral representations of the form ►

18.10.8 $$p_n(x)=\frac{g_0(x)}{2\pi \mathrm{i}}{\int }_C{\left(g_1(z,x)\right)}^n{g}_2(z,x){(z-c)}^{-1}dz$$

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Defines:

$g_0(x)$: factors (locally), $g_1(z,x)$: factors (locally), $g_2(z,x)$: factor (locally) and $c$: center (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer, $C$: closed contour and $x$: real variable

Referenced by:

Table 18.10.1, (18.17.21_1)

Permalink:

http://dlmf.nist.gov/18.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(iii), §18.10 and Ch.18

… ►

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

► ►►

$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
…

►

…

16: 25.5 Integral Representations

… ►

§25.5(iii) Contour Integrals

►

25.5.20 $$\zeta \left(s\right)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}}^{-z}-1}dz,$$ $s\ne 1,2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

contourintegral, integral representation

Sources:

Apostol (1976, (6), p. 253); with $a=1$; Erdélyi et al. (1953a, (1.12.9), p. 32); with $z=-t$

A&S Ref:

23.2.4 (in slightly different form)

Referenced by:

§25.11(vii)

Permalink:

http://dlmf.nist.gov/25.5.E20

Encodings:

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See also:

Annotations for §25.5(iii), §25.5 and Ch.25

… ►

25.5.21 $$\zeta \left(s\right)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}(1-2^{1-s})}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}}^{-z}+1}dz,$$ $s\ne 1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: complex variable and $z$: complex variable

Keywords:

contourintegral, integral representation

Sources:

Erdélyi et al. (1953a, (1.12.10), p. 32); with $z=-t$; Magnus et al. (1966, p. 21); with $z=-t$

Permalink:

http://dlmf.nist.gov/25.5.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(iii), §25.5 and Ch.25

…

17: 10.32 Integral Representations

… ►

§10.32(ii) Contour Integrals

…

18: 10.9 Integral Representations

… ►

§10.9(ii) Contour Integrals

… ►

$H_{\nu }^{(2)}\left(z\right)=-{\displaystyle \frac{1}{\pi \mathrm{i}}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }-\pi \mathrm{i}}}{\mathrm{e}}^{z\sinh t-\nu t}dt.$

… ► …

19: 3.5 Quadrature

… ►

§3.5(viii) Complex Gauss Quadrature

… ►

§3.5(ix) Other Contour Integrals

►A frequent problem with contour integrals is heavy cancellation, which occurs especially when the value of the integral is exponentially small compared with the maximum absolute value of the integrand. … ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

20: 1.10 Functions of a Complex Variable

… ►

1.10.8 $$\frac{1}{2\pi \mathrm{i}}{\int }_{C}f(z)dz=\text{sum of the residues of }f(z)\text{ within }C.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: variable and $C$: closed contour

Referenced by:

§2.10(iii)

Permalink:

http://dlmf.nist.gov/1.10.E8

Encodings:

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See also:

Annotations for §1.10(iv), §1.10 and Ch.1

… ►

1.10.9 $$N-P=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f^{\prime }(z)}{f(z)}dz=\frac{1}{2\pi }{\mathrm{\Delta }}_C(\mathrm{ph}f(z)),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: variable, $C$: closed contour, $N$: number of zeros and $P$: number of zeros

Permalink:

http://dlmf.nist.gov/1.10.E9

Encodings:

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See also:

Annotations for §1.10(iv), §1.10(iv), §1.10 and Ch.1

… ►

1.10.10 $$\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{z{f}^{\prime }(z)}{f(z)}dz=\text{(sum of locations of zeros)}-\text{(sum of locations of poles)},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: variable and $C$: closed contour

Permalink:

http://dlmf.nist.gov/1.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(iv), §1.10(iv), §1.10 and Ch.1

… ►

§1.10(viii) Functions Defined by Contour Integrals

… ►The recurrence relation for $C_n^{(\lambda )}\left(x\right)$ in §18.9(i) follows from $(1-2xz+z^2)\frac{\partial }{\partial z}F(x,\lambda ;z)=2\lambda (x-z)F(x,\lambda ;z)$, and the contour integral representation for $C_n^{(\lambda )}\left(x\right)$ in §18.10(iii) is just (1.10.27).