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11: 7.14 Integrals

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}+a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$,

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.29 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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12: Errata

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Sentences were added just below (10.6.5) and (10.29.3) regarding results on modified quotients of the form $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$ and $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$, respectively (suggested by Art Ballato on 2021-04-29).

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13: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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$F_{\mathrm{\ell }}(\eta ,\rho )\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$

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$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )\sim (\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }}.$

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$H_0^{\pm }(0,\rho )={\mathrm{e}}^{\pm \mathrm{i}\rho }.$

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33.5.6 $$C_{\mathrm{\ell }}\left(0\right)=\frac{2^{\mathrm{\ell }}\mathrm{\ell }!}{(2\mathrm{\ell }+1)!}=\frac{1}{(2\mathrm{\ell }+1)!!}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!!$: double factorial, $!$: factorial (as in $n!$) and $\mathrm{\ell }$: nonnegative integer

Permalink:

http://dlmf.nist.gov/33.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.5(ii), §33.5 and Ch.33

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$F_{\mathrm{\ell }}(\eta ,\rho )\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$

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14: 19.2 Definitions

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§19.2(iv) A Related Function: $R_C(x,y)$

►Let $x\in ℂ\setminus (-\mathrm{\infty },0)$ and $y\in ℂ\setminus \{0\}$. … ►When $x$ and $y$ are positive, $R_C(x,y)$ is an inverse circular function if and an inverse hyperbolic function (or logarithm) if $x>y$: …For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). ►If the line segment with endpoints $x$ and $y$ lies in $ℂ\setminus (-\mathrm{\infty },0]$, then …

15: 33.2 Definitions and Basic Properties

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33.2.3 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)2^{-\mathrm{\ell }-1}{(\mp \mathrm{i})}^{\mathrm{\ell }+1}{M}_{\pm \mathrm{i}\eta ,\mathrm{\ell }+\frac{1}{2}}\left(\pm 2\mathrm{i}\rho \right),$$

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Defines:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§13.18(vi), §33.2(ii)

Permalink:

http://dlmf.nist.gov/33.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(ii), §33.2 and Ch.33

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16: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.16 $${\int }_0^{\mathrm{\infty }}\sin \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt=-\frac{\pi A_0^{2n}(h^2)}{2{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}\left({\mathrm{ce}}_{2n}(y,h^2)\mp \frac{2}{\pi C_{2n}(h^2)}{\mathrm{fe}}_{2n}(y,h^2)\right),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, $\sin z$: sine function, $h$: parameter, $n$: integer, $y$: real variable, $A_m(q)$: Fourier coefficient and $C_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.28.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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