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21: 6.19 Tables

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§6.19(ii) Real Variables

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•

Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$ , $-x^{-2}\operatorname{Cin}\left(x\right)$ , $x^{-1}\operatorname{Ein}\left(x\right)$ , $-x^{-1}\operatorname{Ein}\left(-x\right)$ , $x=0(.01)0.5$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=0.5(.01)2$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x=2(.1)10$ ; $x\mathrm{f}\left(x\right)$ , $x^{2}\mathrm{g}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x^{-1}=0(.005)0.1$ ; $\operatorname{Si}\left(\pi x\right)$ , $\operatorname{Cin}\left(\pi x\right)$ , $x=0(.1)10$ . Accuracy varies but is within the range 8S–11S.

►

•

Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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22: 14.18 Sums

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14.18.1

\mathsf{P}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{P}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right),

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : 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, $m$ : nonnegative integer, $\nu$ : general degree, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(ii), §14.18 and Ch.14

►

14.18.2

\mathsf{P}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\phi\right)=\sum_{m=-n}^{n}(-1)^{m}\mathsf{P}^{-m}_{n}\left(\cos\theta_{1}% \right)\mathsf{P}^{m}_{n}\left(\cos\theta_{2}\right)\cos\left(m\phi\right).

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(ii), §14.18 and Ch.14

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14.18.3

\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right).

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), Ch.14
Permalink:: http://dlmf.nist.gov/14.18.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(ii), §14.18 and Ch.14

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14.18.4

P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=% P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{% \infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)P^{m}_{\nu}\left(\cosh\xi_% {2}\right)\cos\left(m\phi\right),

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, $m$ : nonnegative integer, $\nu$ : general degree, $\phi$ : real and $\xi_{1}$ , $\xi_{2}$ : positive
Referenced by:: §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(ii), §14.18 and Ch.14

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14.18.5

Q_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=% P_{\nu}\left(\cosh\xi_{1}\right)Q_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{% \infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)Q^{m}_{\nu}\left(\cosh\xi_% {2}\right)\cos\left(m\phi\right).

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $m$ : nonnegative integer, $\nu$ : general degree, $\phi$ : real and $\xi_{1}$ , $\xi_{2}$ : positive
Referenced by:: §14.18(ii)
Permalink:: http://dlmf.nist.gov/14.18.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(ii), §14.18 and Ch.14

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23: 6.15 Sums

§6.15 Sums

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6.15.1

\sum_{n=1}^{\infty}\operatorname{Ci}\left(\pi n\right)=\tfrac{1}{2}(\ln 2-% \gamma),

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\ln\NVar{z}$ : principal branch of logarithm function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/6.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.15 and Ch.6

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6.15.3

\sum_{n=1}^{\infty}(-1)^{n}\operatorname{Ci}\left(2\pi n\right)=1-\ln 2-\tfrac% {1}{2}\gamma,

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\ln\NVar{z}$ : principal branch of logarithm function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/6.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.15 and Ch.6

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24: 6.21 Software

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§6.21(ii) $E_{1}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Shi}\left(x\right)$ , $\operatorname{Chi}\left(x\right)$ , $x\in\mathbb{R}$

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§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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25: 6.7 Integral Representations

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§6.7(ii) Sine and Cosine Integrals

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6.7.9

\operatorname{si}\left(z\right)=-\int_{0}^{\pi/2}e^{-z\cos t}\cos\left(z\sin t% \right)\,\mathrm{d}t,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 5.2.10
Permalink:: http://dlmf.nist.gov/6.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §6.7(ii), §6.7 and Ch.6

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6.7.10

\operatorname{Ein}\left(z\right)-\operatorname{Cin}\left(z\right)=\int_{0}^{% \pi/2}e^{-z\cos t}\sin\left(z\sin t\right)\,\mathrm{d}t,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\cos\NVar{z}$ : cosine function, $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 5.2.11 (modified form of)
Permalink:: http://dlmf.nist.gov/6.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §6.7(ii), §6.7 and Ch.6

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§6.7(iii) Auxiliary Functions

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6.7.15

\mathrm{f}\left(z\right)=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\cos t% \,\mathrm{d}t,

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.7(iii)
Permalink:: http://dlmf.nist.gov/6.7.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §6.7(iii), §6.7 and Ch.6

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26: 14.19 Toroidal (or Ring) Functions

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x=\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta},

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y=\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta},

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z=\frac{c\sin\theta}{\cosh\eta-\cos\theta},

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14.19.5

\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+% \frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}\right)\Gamma\left(n-m+\frac{1% }{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(mt\right)}{(\cosh\xi+\cosh t% \sinh\xi)^{n+(1/2)}}\,\mathrm{d}t,

m<n+\tfrac{1}{2}

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable
A&S Ref:: 8.11.4 (modified and corrected)
Permalink:: http://dlmf.nist.gov/14.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §14.19(iii), §14.19 and Ch.14

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14.19.6

\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)+2\sum_{n=1}^{\infty}% \frac{\Gamma\left(\mu+n+\tfrac{1}{2}\right)}{\Gamma\left(\mu+\tfrac{1}{2}% \right)}\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right)\cos\left(n% \phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu% }}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}},

\Re\mu>-\tfrac{1}{2}

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\Re$ : real part, $n$ : nonnegative integer, $\mu$ : general order and $\xi>0$ : variable
Permalink:: http://dlmf.nist.gov/14.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §14.19(iv), §14.19 and Ch.14

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27: 28.26 Asymptotic Approximations for Large $q$

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28.26.1

{\operatorname{Mc}^{(3)}_{m}}\left(z,h\right)=\dfrac{e^{\mathrm{i}\phi}}{(\pi h% \cosh z)^{\ifrac{1}{2}}}\*\left(\mathrm{Fc}_{m}\left(z,h\right)-\mathrm{i}% \mathrm{Gc}_{m}\left(z,h\right)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $\phi$
A&S Ref:: 20.9.7 (in slightly different notation)
Permalink:: http://dlmf.nist.gov/28.26.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

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28.26.2

\mathrm{i}{\operatorname{Ms}^{(3)}_{m+1}}\left(z,h\right)=\dfrac{e^{\mathrm{i}% \phi}}{(\pi h\cosh z)^{\ifrac{1}{2}}}\*{\left(\mathrm{Fs}_{m}\left(z,h\right)-% \mathrm{i}\mathrm{Gs}_{m}\left(z,h\right)\right)},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $\phi$
A&S Ref:: 20.9.7 (in slightly different notation)
Permalink:: http://dlmf.nist.gov/28.26.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

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28.26.4

\mathrm{Fc}_{m}\left(z,h\right)\sim 1+\dfrac{s}{8h{\cosh}^{2}z}+\dfrac{1}{2^{1% 1}h^{2}}\left(\dfrac{s^{4}+86s^{2}+105}{{\cosh}^{4}z}-\dfrac{s^{4}+22s^{2}+57}% {{\cosh}^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(-\dfrac{s^{5}+14s^{3}+33s}{{% \cosh}^{2}z}-\dfrac{2s^{5}+124s^{3}+1122s}{{\cosh}^{4}z}+\dfrac{3s^{5}+290s^{3% }+1627s}{{\cosh}^{6}z}\right)+\cdots,

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cosh\NVar{z}$ : hyperbolic cosine function, $m$ : integer, $h$ : parameter and $z$ : complex variable
A&S Ref:: 20.9.9 (in slightly different notation)
Referenced by:: §28.26(i)
Permalink:: http://dlmf.nist.gov/28.26.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.5

\mathrm{Gc}_{m}\left(z,h\right)\sim\dfrac{\sinh z}{{\cosh}^{2}z}\left(\dfrac{s% ^{2}+3}{2^{5}h}+\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cosh}% ^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(5s^{4}+34s^{2}+9-\dfrac{s^{6}-47s^{4% }+667s^{2}+2835}{12{\cosh}^{2}z}+\dfrac{s^{6}+505s^{4}+12139s^{2}+10395}{12{% \cosh}^{4}z}\right)\right)+\cdots.

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $m$ : integer, $h$ : parameter and $z$ : complex variable
A&S Ref:: 20.9.10 (in slightly different notation)
Referenced by:: §28.26(i)
Permalink:: http://dlmf.nist.gov/28.26.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

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28: 6.3 Graphics

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See accompanying text — Figure 6.3.2: The sine and cosine integrals $\operatorname{Si}\left(x\right),\operatorname{Ci}\left(x\right)$ , $0\leq x\leq 15$ . Magnify

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29: 6.13 Zeros

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6.13.1

x_{0}=0.37250\;74107\;81366\;63446\;19918\;66580\dots.

ⓘ

Symbols:: $x$ : real variable
Notes:: For more digits see OEIS Sequence A091723; see also Sloane (2003).
Referenced by:: §6.13
Permalink:: http://dlmf.nist.gov/6.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.13 and Ch.6

►

\operatorname{Ci}\left(x\right)

and

\operatorname{si}\left(x\right)

each have an infinite number of positive real zeros, which are denoted by

c_{k}

,

s_{k}

, respectively, arranged in ascending order of absolute value for

k=0,1,2,\dots

. Values of

c_{1}

and

c_{2}

to 30D are given by MacLeod (1996b). … ►

6.13.2

c_{k},s_{k}\sim\alpha+\frac{1}{\alpha}-\frac{16}{3}\frac{1}{\alpha^{3}}+\frac{% 1673}{15}\frac{1}{\alpha^{5}}-\frac{5\;07746}{105}\frac{1}{\alpha^{7}}+\cdots,

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $c_{k}$ : zeros of $\operatorname{Ci}\left(x\right)$ and $s_{k}$ : zeros of $\operatorname{Si}\left(x\right)$
Referenced by:: §6.13, §6.18(iii)
Permalink:: http://dlmf.nist.gov/6.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.13 and Ch.6

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30: 24.7 Integral Representations

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24.7.7

B_{2n}\left(x\right)=(-1)^{n+1}2n\*\int_{0}^{\infty}\frac{\cos\left(2\pi x% \right)-e^{-2\pi t}}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n-1}% \,\mathrm{d}t,

n=1,2,\dots

,

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.7(ii), §24.7 and Ch.24

►

24.7.8

B_{2n+1}\left(x\right)=(-1)^{n+1}(2n+1)\*\int_{0}^{\infty}\frac{\sin\left(2\pi x% \right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n}\,\mathrm{d}t.

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.7(ii), §24.7 and Ch.24

►

24.7.9

E_{2n}\left(x\right)=(-1)^{n}4\int_{0}^{\infty}\frac{\sin\left(\pi x\right)% \cosh\left(\pi t\right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2% n}\,\mathrm{d}t,

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.7(ii), §24.7 and Ch.24

►

24.7.10

E_{2n+1}\left(x\right)=(-1)^{n+1}4\*\int_{0}^{\infty}\frac{\cos\left(\pi x% \right)\sinh\left(\pi t\right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x% \right)}t^{2n+1}\,\mathrm{d}t.

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.7(ii), §24.7 and Ch.24

…

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