Content-Length: 51521 | pFad | https://dlmf.nist.gov/./../././.././././././.././././.././././bib/.././.././././bib/.././6.15#info

DLMF: §6.15 Sums ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals

About the Project

6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.14 Integrals 6.16 Mathematical Applications

§6.15 Sums

ⓘ

Keywords:: cosine integrals, sine integrals, sums
Notes:: See Slavić (1974).
Permalink:: http://dlmf.nist.gov/6.15
See also:: Annotations for Ch.6

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).

6.14 Integrals 6.16 Mathematical Applications

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov