infinite integrals

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11: 1.9 Calculus of a Complex Variable

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1.9.28 $${\int }_{C}f(z)dz={\int }_a^{b}f(z(t))(x^{\prime }(t)+\mathrm{i}{y}^{\prime }(t))dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $C$: closed contour

Permalink:

http://dlmf.nist.gov/1.9.E28

Encodings:

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See also:

Annotations for §1.9(iii), §1.9 and Ch.1

… ►If $f(z(t_0))=\mathrm{\infty }$, $a\le t_0\le b$, then the integral is defined analogously to the infinite integrals in §1.4(v). …

12: 10.71 Integrals

… ►For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). …

13: Bibliography S

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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14: 10.43 Integrals

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§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

… ►For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

15: 2.10 Sums and Sequences

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The first infinite integral in (2.10.2) converges.

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16: 6.13 Zeros

… ► $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ each have an infinite number of positive real zeros, which are denoted by $c_k$, $s_k$, respectively, arranged in ascending order of absolute value for $k=0,1,2,\mathrm{\dots }$. …

17: 25.16 Mathematical Applications

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25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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25.16.7 $$H\left(s\right)=\frac{1}{2}\zeta \left(s+1\right)+\frac{\zeta \left(s\right)}{s-1}-\sum _{r=1}^k\left(\genfrac{}{}{0.0pt}{}{s+2r-2}{2r-1}\right)\zeta \left(1-2r\right)\zeta \left(s+2r\right)-\left(\genfrac{}{}{0.0pt}{}{s+2k}{2k+1}\right)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{s+2k+1}}dx.$$

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Symbols:

$H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (6), p. 88)

Permalink:

http://dlmf.nist.gov/25.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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18: 13.4 Integral Representations

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13.4.2 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(b-c\right)}{\int }_0^1𝐌(a,c,zt)t^{c-1}{(1-t)}^{b-c-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }c>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $c$: arbitrary

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E2

Encodings:

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See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.6 $$U(a,b,z)=\frac{{(-1)}^n{z}^{1-b-n}}{\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{𝐌(b-a,b,t){\mathrm{e}}^{-t}{t}^{b+n-1}}{t+z}dt,$$ , $n=0,1,2,\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E6

Encodings:

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See also:

Annotations for §13.4(i), §13.4 and Ch.13

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19: 10.22 Integrals

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§10.22(iii) Integrals over the Interval $(x,\mathrm{\infty })$

… ►Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

20: 2.3 Integrals of a Real Variable

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2.3.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function

Keywords:

Laplace transform

Referenced by:

§2.3(ii)

Permalink:

http://dlmf.nist.gov/2.3.E1

Encodings:

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See also:

Annotations for §2.3(i), §2.3 and Ch.2

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2.3.4 $${\int }_a^b{\mathrm{e}}^{\mathrm{i}xt}q(t)dt\sim {\mathrm{e}}^{\mathrm{i}ax}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(a){\left(\frac{\mathrm{i}}{x}\right)}^{s+1}-{\mathrm{e}}^{\mathrm{i}bx}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(b){\left(\frac{\mathrm{i}}{x}\right)}^{s+1},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $q(t)$: infinitely differentiable function, $a$: left endpoint and $b$: right endpoint

Permalink:

http://dlmf.nist.gov/2.3.E4

Encodings:

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See also:

Annotations for §2.3(i), §2.3 and Ch.2

►Alternatively, assume $b=\mathrm{\infty }$, $q(t)$ is infinitely differentiable on $[a,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges as $t\to \mathrm{\infty }$ uniformly for all sufficiently large $x$. … ►

2.3.5 $${\int }_a^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}xt}q(t)dt\sim {\mathrm{e}}^{\mathrm{i}ax}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(a){\left(\frac{\mathrm{i}}{x}\right)}^{s+1},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $q(t)$: infinitely differentiable function and $a$: left endpoint

Permalink:

http://dlmf.nist.gov/2.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(i), §2.3 and Ch.2

… ►Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges at $t=\mathrm{\infty }$, uniformly for all sufficiently large $x$. …