along the real line

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1: 8.6 Integral Representations

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§8.6(i) Integrals Along the Real Line

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2: 11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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3: 10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

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10.32.11 $$K_{\nu }\left(xz\right)=\frac{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right){(2z)}^{\nu }}{{\pi }^{\frac{1}{2}}{x}^{\nu }}{\int }_0^{\mathrm{\infty }}\frac{\cos \left(xt\right)dt}{{(t^2+z^2)}^{\nu +\frac{1}{2}}},$$ $\mathrm{\Re }\nu >-\frac{1}{2}$, $x>0$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.25 (corrected)

Permalink:

http://dlmf.nist.gov/10.32.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32(i), §10.32 and Ch.10

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4: 12.5 Integral Representations

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§12.5(i) Integrals Along the Real Line

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5: 13.4 Integral Representations

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§13.4(i) Integrals Along the Real Line

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6: 13.16 Integral Representations

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§13.16(i) Integrals Along the Real Line

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7: 10.9 Integral Representations

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§10.9(i) Integrals along the Real Line

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10.9.11 $$H_{\nu }^{(2)}\left(z\right)=-\frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}z\cosh t-\nu t}dt,$$ .

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Symbols:

$H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.9(i)

Permalink:

http://dlmf.nist.gov/10.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

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10.9.16 $${\left(\frac{z+\zeta }{z-\zeta }\right)}^{\frac{1}{2}\nu }{H}_{\nu }^{(2)}\left({(z^2-{\zeta }^2)}^{\frac{1}{2}}\right)=-\frac{1}{\pi \mathrm{i}}{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}z\cosh t-\mathrm{i}\zeta \sinh t-\nu t}dt,$$ .

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Symbols:

$H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.9(i)

Permalink:

http://dlmf.nist.gov/10.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

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8: 25.5 Integral Representations

§25.5 Integral Representations

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25.5.19 $$\zeta \left(m+s\right)={(-1)}^{m-1}\frac{\mathrm{\Gamma }\left(s\right)\sin \left(\pi s\right)}{\pi \mathrm{\Gamma }\left(m+s\right)}{\int }_0^{\mathrm{\infty }}{\psi }^{(m)}\left(1+x\right)x^{-s}dx,$$ $m=1,2,3,\mathrm{\dots }$.

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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 $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, p. 178)

Referenced by:

§25.5(ii)

Permalink:

http://dlmf.nist.gov/25.5.E19

Encodings:

pMML, png, TeX

See also:

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

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9: 12.14 The Function $W(a,x)$

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§12.14(vi) Integral Representations

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10: 4.3 Graphics

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§4.3(i) Real Arguments

… ►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto rays in the $w$-plane, and lines parallel to the imaginary axis in the $z$-plane map onto circles centered at the origin in the $w$-plane. In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty })$. … ►

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). There is a branch cut along the negative real axis. Magnify 3D Help

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