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11: 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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12: 29.17 Other Solutions

… ►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …

13: 36.4 Bifurcation Sets

… ►Swallowtail self-intersection line: … ►Swallowtail cusp lines (ribs): … ►Elliptic umbilic cusp lines (ribs): … ►Hyperbolic umbilic cusp line (rib): …

14: 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. There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by …

15: Bibliography I

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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16: Mathematical Introduction

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$ℂ$	complex plane (excluding infinity).
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$(a,b)$	open interval in $ℝ$, or open straight-line segment joining $a$ and $b$ in $ℂ$.
$[a,b]$	closed interval in $ℝ$, or closed straight-line segment joining $a$ and $b$ in $ℂ$.

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$(a,b]$ or $[a,b)$	half-closed intervals.
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$ℝ$	real line (excluding infinity).
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… ►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. …

17: 26.6 Other Lattice Path Numbers

… ► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ► $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … ► $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … ► $r(n)$ is the number of paths from $(0,0)$ to $(n,n)$ that stay on or above the diagonal $y=x$ and are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. …

18: 6.7 Integral Representations

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6.7.5 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{a^2+t^2}dt=-\frac{1}{2a\mathrm{i}}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)-{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$,

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.43 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.6 $${\int }_x^{\mathrm{\infty }}\frac{t{\mathrm{e}}^{-t}}{a^2+t^2}dt=\frac{1}{2}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)+{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.44 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.7 $${\int }_0^1\frac{{\mathrm{e}}^{-at}\sin \left(bt\right)}{t}dt=\mathrm{\Im }\mathrm{Ein}\left(a+\mathrm{i}b\right),$$ $a,b\in ℝ$,

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $\sin z$: sine function

A&S Ref:

5.1.36 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.8 $${\int }_0^1\frac{{\mathrm{e}}^{-at}(1-\cos \left(bt\right))}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Ein}\left(a\right),$$ $a,b\in ℝ$.

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $ℝ$: real line

A&S Ref:

5.1.38 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.11 $${\int }_0^1\frac{(1-{\mathrm{e}}^{-at})\cos \left(bt\right)}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Cin}\left(b\right),$$ $a,b\in ℝ$.

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $ℝ$: real line

A&S Ref:

5.2.33 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(ii), §6.7 and Ch.6

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19: 4.15 Graphics

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Figure 4.15.2: $\mathrm{Arcsin}x$ and $\mathrm{Arccos}x$. Principal values are shown with thickened lines. Magnify

… ►Lines parallel to the real axis in the $z$-plane map onto ellipses in the $w$-plane with foci at $w=\pm 1$, and lines parallel to the imaginary axis in the $z$-plane map onto rectangular hyperbolas confocal with the ellipses. …

20: 13.16 Integral Representations

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§13.16(i) Integrals Along the Real Line

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