line integral

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21: 19.6 Special Cases

§19.6 Special Cases

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§19.6(i) Complete Elliptic Integrals

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§19.6(ii) $F(\varphi ,k)$

… ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_C(x,y)$. … ►

§19.6(v) $R_C(x,y)$

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22: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.59 $$f(x)=\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_0^{\mathrm{\infty }}\cos \left(xt\right)\cos \left(yt\right)dt\right)f(y)dy+\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_0^{\mathrm{\infty }}\sin \left(xt\right)\sin \left(yt\right)dt\right)f(y)dy,$$ $x\in ℝ$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line and $\sin z$: sine function

Source:

Titchmarsh (1962a, pp. 71–72)

Permalink:

http://dlmf.nist.gov/1.18.E59

Encodings:

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See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

… ►Surprisingly, if on any interval on the real line, even if positive elsewhere, as long as ${\int }_{X}q(x)dx\le 0$, see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding $L^2\left(X\right)$ eigenfunction. …

23: 12.14 The Function $W(a,x)$

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§12.14(vi) Integral Representations

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24: 25.11 Hurwitz Zeta Function

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§25.11(vii) Integral Representations

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25.11.25 $$\zeta (s,a)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-{\mathrm{e}}^{-x}}dx,$$ $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integral representation

Sources:

Srivastava and Choi (2001, (2), p. 89); Apostol (1976, (5), p. 251)

Referenced by:

(25.11.27), (25.11.30), (25.11.35), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E25

Encodings:

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See also:

Annotations for §25.11(vii), §25.11 and Ch.25

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25.11.26 $$\zeta (s,a)=-s{\int }_{-a}^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor -\frac{1}{2}}{{(x+a)}^{s+1}}dx,$$ , .

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

Berndt (1972, (5.3), p. 156)

Permalink:

http://dlmf.nist.gov/25.11.E26

Encodings:

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See also:

Annotations for §25.11(vii), §25.11 and Ch.25

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§25.11(viii) Further Integral Representations

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§25.11(ix) Integrals

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25: Bibliography J

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

Encodings:

BibTeX

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26: 16.17 Definition

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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 $p=q\ge 1$.

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27: 2.6 Distributional Methods

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2.6.58 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda }dt,$$ $\lambda \in ℝ$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\int$: integral and $ℝ$: real line

Permalink:

http://dlmf.nist.gov/2.6.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(iv), §2.6 and Ch.2

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28: 36.7 Zeros

… ►The zeros are lines in $𝐱=(x,y,z)$ space where $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$ is undetermined. …Away from the $z$-axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. …

29: 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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30: 7.19 Voigt Functions

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7.19.4 $$H(a,u)=\frac{a}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{{(u-t)}^2+a^2}=\frac{1}{a\sqrt{\pi }}𝖴(\frac{u}{a},\frac{1}{4a^2}).$$

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Defines:

$H(a,u)$: line-broadening function

Symbols:

$𝖴(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Permalink:

http://dlmf.nist.gov/7.19.E4

Encodings:

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See also:

Annotations for §7.19(i), §7.19 and Ch.7

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