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11: 35.2 Laplace Transform

… ►where the integral is taken over all $𝐙=𝐔+\mathrm{i}𝐕$ such that $𝐔>{𝐗}_0$ and $𝐕$ ranges over $𝓢$. …

12: 1.4 Calculus of One Variable

… ►Definite integrals over the Stieltjes measure $d\alpha (x)$ could represent a sum, an integral, or a combination of the two. …

13: 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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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14: 21.1 Special Notation

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$g,h$	positive integers.
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${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

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15: 13.16 Integral Representations

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13.16.2 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\lambda }}{\mathrm{\Gamma }\left(1+2\mu -2\lambda \right)\mathrm{\Gamma }\left(2\lambda \right)}{\int }_0^1{M}_{\kappa -\lambda ,\mu -\lambda }\left(zt\right){\mathrm{e}}^{\frac{1}{2}z(t-1)}{t}^{\mu -\lambda -\frac{1}{2}}{(1-t)}^{2\lambda -1}dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\mathrm{\Re }\lambda >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.23(v)

Permalink:

http://dlmf.nist.gov/13.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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16: 9.12 Scorer Functions

… ►If $\zeta =\frac{2}{3}{z}^{3/2}$ or $\frac{2}{3}{x}^{3/2}$, and $K_{1/3}$ is the modified Bessel function (§10.25(ii)), then …

17: 3.5 Quadrature

… ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then …

18: 2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►oscillate rapidly and cancel themselves over most of the range. … ►Then … ►In the integral … ►The integral (2.3.24) transforms into …

19: 1.5 Calculus of Two or More Variables

… ►Then the double integral of $f(x,y)$ over $R$ is defined by …

20: 2.4 Contour Integrals

§2.4 Contour Integrals

… ►The result in §2.3(ii) carries over to a complex parameter $z$. … ►Then by integration by parts the integral … ►

$z$ ranges along a ray or over an annular sector ${\theta }_1\le \theta \le {\theta }_2$, $|z|\ge Z$, where $\theta =\mathrm{ph}z$, , and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

… ►Consider the integral …