cosine integrals

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11: 6.15 Sums

§6.15 Sums

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6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: 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 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: 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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12: 6.13 Zeros

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6.13.1 $$x_0=\mathrm{0.37250\hspace{0.33em}74107\hspace{0.33em}81366\hspace{0.33em}63446\hspace{0.33em}19918\hspace{0.33em}66580}\mathrm{\dots }.$$

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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

► $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ 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 $k=0,1,2,\mathrm{\dots }$. Values of $c_1$ and $c_2$ to 30D are given by MacLeod (1996b). … ►

6.13.2 $$c_k,s_k\sim \alpha +\frac{1}{\alpha }-\frac{16}{3}\frac{1}{{\alpha }^3}+\frac{1673}{15}\frac{1}{{\alpha }^5}-\frac{\mathrm{5\hspace{0.33em}07746}}{105}\frac{1}{{\alpha }^7}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $c_k$: zeros of $\mathrm{Ci}\left(x\right)$ and $s_k$: zeros of $\mathrm{Si}\left(x\right)$

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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13: 6.17 Physical Applications

§6.17 Physical Applications

… ►Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.

14: 6.3 Graphics

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Figure 6.3.2: The sine and cosine integrals $\mathrm{Si}\left(x\right),\mathrm{Ci}\left(x\right)$, $0\le x\le 15$. Magnify

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15: 7.4 Symmetry

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$C\left(-z\right)=-C\left(z\right),$

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$C\left(\mathrm{i}z\right)=\mathrm{i}C\left(z\right),$

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16: 6.7 Integral Representations

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§6.7(ii) Sine and Cosine Integrals

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§6.7(iii) Auxiliary Functions

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6.7.12 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)={\mathrm{e}}^{-\mathrm{i}z}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}t}}{t}dt,$$ $|\mathrm{ph}z|\le \pi .$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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6.7.13 $$\mathrm{f}\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{t+z}dt={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^2+1}dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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6.7.15 $$\mathrm{f}\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_0\left(2\sqrt{zt}\right)\cos tdt,$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{f}\left(z\right)$: 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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17: 6.18 Methods of Computation

§6.18 Methods of Computation

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§6.18(ii) Auxiliary Functions

►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. …Then $\mathrm{f}\left(z\right)=B_0$, $\mathrm{g}\left(z\right)=A_0$, and … ►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

18: 8.1 Special Notation

… ►Unless otherwise indicated, primes denote derivatives with respect to the argument. ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

19: 6.12 Asymptotic Expansions

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§6.12(ii) Sine and Cosine Integrals

►The asymptotic expansions of $\mathrm{Si}\left(z\right)$ and $\mathrm{Ci}\left(z\right)$ are given by (6.2.19), (6.2.20), together with … … ►

6.12.5 $$\mathrm{f}\left(z\right)=\frac{1}{z}\sum _{m=0}^{n-1}{(-1)}^m\frac{(2m)!}{z^{2m}}+R_n^{(\mathrm{f})}(z),$$

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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20: 7.25 Software

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§7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$

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§7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$

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