DLMF: Untitled Document

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6 Exponential, Logarithmic, Sine, and Cosine Integrals
6.21(ii) $E_{1}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Shi}\left(x\right)$ , $\operatorname{Chi}\left(x\right)$ , $x\in\mathbb{R}$	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
6.21(iii) $E_{1}\left(z\right)$ , $\operatorname{Si}\left(z\right)$ , $\operatorname{Ci}\left(z\right)$ , $\operatorname{Shi}\left(z\right)$ , $\operatorname{Chi}\left(z\right)$ , $z\in\mathbb{C}$	✓	✓				✓		✓		✓	✓	✓							✓	✓	✓
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7.25(v) $C\left(z\right)$ , $S\left(z\right)$ , $z\in\mathbb{C}$	✓									a	✓								✓	✓	✓
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20 Theta Functions
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Equation (18.28.8)

18.28.8

\frac{1}{2\pi}\int_{0}^{\pi}Q_{n}\left(\cos\theta;a,b\,|\,q\right)Q_{m}\left(% \cos\theta;a,b\,|\,q\right)\*{\left|\frac{\left({\mathrm{e}}^{2i\theta};q% \right)_{\infty}}{\left(a{\mathrm{e}}^{\mathrm{i}\theta},b{\mathrm{e}}^{% \mathrm{i}\theta};q\right)_{\infty}}\right|}^{2}\,\mathrm{d}\theta=\frac{% \delta_{n,m}}{\left(q^{n+1},abq^{n};q\right)_{\infty}},

a,b\in\mathbb{R}

or

a=\overline{b}

;

ab\neq 1

;

|a|,|b|\leq 1

The constraint which originally stated that “ $|ab|<1$ ” has been updated to be “ $ab\neq 1$ ”.

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Subsection 19.11(i)

A sentence and unnumbered equation

R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right),

were added which indicate that care must be taken with the multivalued functions in (19.11.5). See (Cayley, 1961, pp. 103-106).

Suggested by Albert Groenenboom.

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Section 1.14

There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$\mathscr{F}\left(f\right)\left(x\right)$	$\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$
Fourier Cosine	$\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Fourier Sine	$\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Laplace	$\mathscr{L}\left(f\right)\left(s\right)$	$\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{L}(f(t);s)$
Mellin	$\mathscr{M}\left(f\right)\left(s\right)$	$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{M}(f;s)$
Hilbert	$\mathcal{H}\left(f\right)\left(s\right)$	$\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{H}(f;s)$
Stieltjes	$\mathcal{S}\left(f\right)\left(s\right)$	$\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{S}(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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… ►If

f

can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►The integral (3.4.18) becomes …With the choice

r=k

(which is crucial when

k

is large because of numerical cancellation) the integrand equals

e^{k}

at the dominant points

\theta=0,2\pi

, and in combination with the factor

k^{-k}

in front of the integral sign this gives a rough approximation to

1/k!

. …As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. …

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14.30.1

Y_{{l},{m}}\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}% \right)^{1/2}e^{im\phi}\mathsf{P}^{m}_{l}\left(\cos\theta\right),

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Defines:: $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic
Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(i), §14.30(ii), §14.30(ii), §14.30(iv)
Permalink:: http://dlmf.nist.gov/14.30.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.30(i), §14.30 and Ch.14

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14.30.2

Y_{l}^{m}\left(\theta,\phi\right)=\cos\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right)\text{ or }\sin\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right).

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Defines:: $Y_{\NVar{l}}^{\NVar{m}}\left(\NVar{\theta},\NVar{\phi}\right)$ : surface harmonic of the first kind
Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(i), Erratum (V1.0.25) for Section 14.30
Permalink:: http://dlmf.nist.gov/14.30.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.30(i), §14.30 and Ch.14

… ►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►where

\mathbf{a}=\left(\tfrac{1}{2\lambda}-\tfrac{\lambda}{2},-\tfrac{\mathrm{i}}{2% \lambda}-\tfrac{\mathrm{i}\lambda}{2},1\right)

and

\mathbf{x}=\left(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta\right)

. … ►

14.30.9

\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}% \overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_% {2},\phi_{2}\right).

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §18.18(ii)
Permalink:: http://dlmf.nist.gov/14.30.E9
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the sum over $m$ has been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(iii), §14.30(iii), §14.30 and Ch.14

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§36.2 Catastrophes and Canonical Integrals

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§36.2(i) Definitions

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Canonical Integrals

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§36.2(iii) Symmetries

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See accompanying text — Figure 25.11.1: Hurwitz zeta function $\zeta\left(x,a\right)$ , $a$ = 0. …8, 1, $-20\leq x\leq 10$ . … Magnify

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§25.11(vii) Integral Representations

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§25.11(viii) Further Integral Representations

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25.11.31

\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh x}% \,\mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta% \left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right),

\Re s>0

,

\Re a>-1

.

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\Re$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Keywords:: improper integral, integral representation
Proof sketch:: The improper integral can be evaluated by using $2\cosh x={\mathrm{e}}^{x}(1+{\mathrm{e}}^{-2x})$ , changing variables $2x=t$ , and then applying (25.11.35).
Permalink:: http://dlmf.nist.gov/25.11.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

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§25.11(ix) Integrals

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cosine%20integrals

11—16 of 16 matching pages

11: Software Index

12: Errata

13: 3.4 Differentiation

14: 14.30 Spherical and Spheroidal Harmonics

15: 36.2 Catastrophes and Canonical Integrals