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31: 14.12 Integral Representations

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14.12.3 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){(\sin \theta )}^{\mu }}{2^{\mu +1}\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}\left({\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta +\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt+{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta -\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt\right),$$ , $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }\nu \pm \mu >-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

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32: 11.5 Integral Representations

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11.5.8 $${(\frac{1}{2}x)}^{-\nu -1}{𝐇}_{\nu }\left(x\right)=-\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\pi \csc \left(\pi s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}+s\right)\mathrm{\Gamma }\left(\frac{3}{2}+\nu +s\right)}{(\frac{1}{4}{x}^2)}^{s}ds,$$ $x>0$, $\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: real or complex order

Keywords:

Mellin transform

Referenced by:

§11.5(ii), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

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11.5.9 $${(\frac{1}{2}z)}^{-\nu -1}{𝐋}_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }}^{(0+)}\frac{\pi \csc \left(\pi s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}+s\right)\mathrm{\Gamma }\left(\frac{3}{2}+\nu +s\right)}{(-\frac{1}{4}{z}^2)}^{s}ds.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Keywords:

Mellin transform

Referenced by:

§11.5(ii), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

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33: 2.1 Definitions and Elementary Properties

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2.1.7 $${\mathrm{e}}^{\mathrm{i}x}=O\left(1\right),$$ $x\in ℝ$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $ℝ$: real line

Permalink:

http://dlmf.nist.gov/2.1.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(i), §2.1(i), §2.1 and Ch.2

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34: 36.5 Stokes Sets

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35: 7.13 Zeros

… ► $\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the sector , $\mathrm{erfc}z$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the first quadrant of $ℂ$ $C\left(z\right)$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. …

36: 11.4 Basic Properties

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11.4.16 $${𝐇}_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\pi \mathrm{i}(\nu +1)}{𝐇}_{\nu }\left(z\right),$$ $m\in \mathbb{Z}$,

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.1.18

Permalink:

http://dlmf.nist.gov/11.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(iii), §11.4 and Ch.11

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11.4.17 $${𝐋}_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\pi \mathrm{i}(\nu +1)}{𝐋}_{\nu }\left(z\right),$$ $m\in \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(iii), §11.4 and Ch.11

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37: 28.20 Definitions and Basic Properties

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28.20.11 $${\mathrm{M}}_{\nu }^{(1)}(z,h)=J_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({\left(\mathrm{sech}z\right)}^{3/2}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.12 $${\mathrm{M}}_{\nu }^{(2)}(z,h)=Y_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({(\mathrm{sech}z)}^{3/2}\right),$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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38: 36.12 Uniform Approximation of Integrals

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36.12.3 $$I(𝐲,k)=\frac{\exp \left(\mathrm{i}kA(𝐲)\right)}{k^{1/(K+2)}}\sum _{m=0}^K\frac{a_m(𝐲)}{k^{m/(K+2)}}\left({\delta }_{m,0}-\left(1-{\delta }_{m,0}\right)\mathrm{i}\frac{\partial }{\partial z_m}\right){\mathrm{\Psi }}_K\left(𝐳(𝐲;k)\right)\left(1+O\left(\frac{1}{k}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\delta }_{j,k}$: Kronecker delta, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: variable, $K$: codimension, $I(𝐲,k)$: oscillatory integral, $a_m(𝐲)$: function, $A(𝐲)$: function, $𝐳(𝐲,k)$: function and $z_m(𝐲,k)$: function

Permalink:

http://dlmf.nist.gov/36.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

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36.12.11 $$I(y,k)=\frac{{\mathrm{\Delta }}^{1/4}\pi \sqrt{2}}{k^{1/3}}\exp \left(\mathrm{i}k\tilde{f}\right)\left(\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime \prime }}}+\frac{g_{-}}{\sqrt{-f_{-}^{\prime \prime }}}\right)\mathrm{Ai}\left(-k^{2/3}\mathrm{\Delta }\right)\left(1+O\left(\frac{1}{k}\right)\right)-\mathrm{i}\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime \prime }}}-\frac{g_{-}}{\sqrt{-f_{-}^{\prime \prime }}}\right)\frac{{\mathrm{Ai}}^{\prime }\left(-k^{2/3}\mathrm{\Delta }\right)}{k^{1/3}{\mathrm{\Delta }}^{1/2}}\left(1+O\left(\frac{1}{k}\right)\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $k$: variable and $I(𝐲,k)$: oscillatory integral

Referenced by:

§36.13

Permalink:

http://dlmf.nist.gov/36.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(ii), §36.12 and Ch.36

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39: 28.12 Definitions and Basic Properties

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28.12.13 $${\mathrm{se}}_{\nu }(z,q)=-\frac{1}{2}\mathrm{i}\left({\mathrm{me}}_{\nu }(z,q)-{\mathrm{me}}_{\nu }(-z,q)\right).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(iii), §28.12 and Ch.28

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40: 12.11 Zeros

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12.11.1 $$z_{a,s}={\mathrm{e}}^{\frac{3}{4}\pi \mathrm{i}}\sqrt{2{\tau }_s}\left(1-\frac{\mathrm{i}a{\lambda }_s}{2{\tau }_s}+\frac{2a^2{\lambda }_s^2-8a^2{\lambda }_s+4a^2+3}{16{\tau }_s^2}+O\left({\lambda }_s^3{\tau }_s^{-3}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter, ${\tau }_s$ and ${\lambda }_s$

Permalink:

http://dlmf.nist.gov/12.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(ii), §12.11 and Ch.12

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