# Clausen integral

(0.001 seconds)

## 7 matching pages

##### 1: 25.19 Tables
• Abramowitz and Stegun (1964) tabulates: $\zeta\left(n\right)$, $n=2,3,4,\dots$, 20D (p. 811); $\operatorname{Li}_{2}\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta)$, $\theta=15^{\circ}(1^{\circ})30^{\circ}(2^{\circ})90^{\circ}(5^{\circ})180^{\circ}$, $f(\theta)+\theta\ln\theta$, $\theta=0(1^{\circ})15^{\circ}$, 6D (p. 1006). Here $f(\theta)$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

• Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

• Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

##### 3: 25.12 Polylogarithms
The right-hand side is called Clausen’s integral. …
##### 4: Software Index
 NetNUMPAC Open Source With Book Commercial … 25.21(vi) Clausen’s Integral ✓ ✓ a ✓ ✓ …
##### 5: Bibliography K
• J. Kamimoto (1998) On an integral of Hardy and Littlewood. Kyushu J. Math. 52 (1), pp. 249–263.
• E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.
• N. M. Katz (1975) The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216 (1), pp. 1–4.
• A. D. Kerr (1978) An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.
• E. Kreyszig (1957) On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.
• ##### 6: Bibliography C
• L. Carlitz (1961b) The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.
• B. C. Carlson (1965) On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.
• B. C. Carlson (1970) Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.
• Th. Clausen (1828) Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha}{1}\cdot\frac{\beta}{\gamma}x+\frac{\alpha\cdot\alpha+1}{1% \cdot 2}\cdot\frac{\beta\cdot\beta+1}{\gamma\cdot\gamma+1}x^{2}+$ etc. ein Quadrat von der Form $z=1+\frac{\alpha^{\prime}}{1}\cdot\frac{\beta^{\prime}}{\gamma^{\prime}}\cdot% \frac{\delta^{\prime}}{\epsilon^{\prime}}x+\frac{\alpha^{\prime}\cdot\alpha^{% \prime}+1}{1\cdot 2}\cdot\frac{\beta^{\prime}\cdot\beta^{\prime}+1}{\gamma^{% \prime}\cdot\gamma^{\prime}+1}\cdot\frac{\delta^{\prime}\cdot\delta^{\prime}+1% }{\epsilon^{\prime}\cdot\epsilon^{\prime}+1}x^{2}+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.
• D. Cvijović and J. Klinowski (1999) Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.
• ##### 7: Errata
• Subsection 17.9(iii)

The title of the paragraph which was previously “Gasper’s $q$-Analog of Clausen’s Formula” has been changed to “Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)”.

• Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12

Scales were corrected in all figures. The interval $-8.4\leq\frac{x-y}{\sqrt{2}}\leq 8.4$ was replaced by $-12.0\leq\frac{x-y}{\sqrt{2}}\leq 12.0$ and $-12.7\leq\frac{x+y}{\sqrt{2}}\leq 4.2$ replaced by $-18.0\leq\frac{x+y}{\sqrt{2}}\leq 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

Reported 2016-09-12 by Dan Piponi.

• Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21

The scaling error reported on 2016-09-12 by Dan Piponi also applied to contour and density plots for the phase of the hyperbolic umbilic canonical integrals. Scales were corrected in all figures. The interval $-8.4\leq\frac{x-y}{\sqrt{2}}\leq 8.4$ was replaced by $-12.0\leq\frac{x-y}{\sqrt{2}}\leq 12.0$ and $-12.7\leq\frac{x+y}{\sqrt{2}}\leq 4.2$ replaced by $-18.0\leq\frac{x+y}{\sqrt{2}}\leq 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

Reported 2016-09-28.

• Equation (36.10.14)
36.10.14 $3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2\mathrm{i}z\frac{% \partial\Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0$

Originally this equation appeared with $\frac{\partial\Psi^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial\Psi^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

• Notations

The definition of $R_{C}\left(x,y\right)$ was revised in Notations.