§6.15 Sums

ⓘ

6.15.1	$\displaystyle\sum_{n=1}^{\infty}\operatorname{Ci}\left(\pi n\right)$	$\displaystyle=\tfrac{1}{2}(\ln 2-\gamma),$
ⓘ 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: TeX, pMML, png See also: Annotations for §6.15 and Ch.6
6.15.2	$\displaystyle\sum_{n=1}^{\infty}\frac{\operatorname{si}\left(\pi n\right)}{n}$	$\displaystyle=\tfrac{1}{2}\pi(\ln\pi-1),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/6.15.E2 Encodings: TeX, pMML, png See also: Annotations for §6.15 and Ch.6

6.15.3	$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}\operatorname{Ci}\left(2\pi n\right)$	$\displaystyle=1-\ln 2-\tfrac{1}{2}\gamma,$
ⓘ 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: TeX, pMML, png See also: Annotations for §6.15 and Ch.6
6.15.4	$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}\frac{\operatorname{si}\left(2\pi n% \right)}{n}$	$\displaystyle=\pi(\tfrac{3}{2}\ln 2-1).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/6.15.E4 Encodings: TeX, pMML, png See also: Annotations for §6.15 and Ch.6

For further sums see Fempl (1960), Hansen (1975, pp. 423–424), Harris (2000), Prudnikov et al. (1986b, vol. 2, pp. 649–650), and Slavić (1974).