# hyperbolic series for squares

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## 7 matching pages

##### 1: 22.11 Fourier and Hyperbolic Series
A related hyperbolic series is …
##### 2: 22.16 Related Functions
22.16.8 $\operatorname{am}\left(x,k\right)=\operatorname{gd}x-\tfrac{1}{4}{k^{\prime}}^% {2}(x-\sinh x\cosh x)\operatorname{sech}x+O\left({k^{\prime}}^{4}\right).$
###### Fourier Series
See Figure 22.16.2. …
22.16.26 $\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({\operatorname{cs}^{2}}\left(t,% k\right)-t^{-2}\right)\mathrm{d}t+x^{-1}-\operatorname{cn}\left(x,k\right)% \operatorname{ds}\left(x,k\right).$
##### 3: 4.38 Inverse Hyperbolic Functions: Further Properties
###### §4.38(ii) Derivatives
In the following equations square roots have their principal values. … All square roots have either possible value.
##### 4: Bibliography S
• F. W. Schäfke and A. Finsterer (1990) On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.
• I. J. Schwatt (1962) An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.
• T. C. Scott, G. Fee, and J. Grotendorst (2013) Asymptotic series of generalized Lambert $W$ function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.
• B. L. Shea (1988) Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.
• S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
• ##### 5: 4.45 Methods of Computation
Then we take square roots repeatedly until $|y|$ is sufficiently small, where …
###### Hyperbolic and Inverse Hyperbolic Functions
The hyperbolic functions can be computed directly from the definitions (4.28.1)–(4.28.7). …For $\operatorname{arccsch}$, $\operatorname{arcsech}$, and $\operatorname{arccoth}$ we have (4.37.7)–(4.37.9). … Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9). …
##### 6: 19.36 Methods of Computation
The reductions in §19.29(i) represent $x,y,z$ as squares, for example $x=U_{12}^{2}$ in (19.29.4). … The incomplete integrals $R_{F}\left(x,y,z\right)$ and $R_{G}\left(x,y,z\right)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_{C}$, accompanied by two quadratically convergent series in the case of $R_{G}$; compare Carlson (1965, §§5,6). … The step from $n$ to $n+1$ is an ascending Landen transformation if $\theta=1$ (leading ultimately to a hyperbolic case of $R_{C}$) or a descending Gauss transformation if $\theta=-1$ (leading to a circular case of $R_{C}$). … For series expansions of Legendre’s integrals see §19.5. Faster convergence of power series for $K\left(k\right)$ and $E\left(k\right)$ can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …
##### 7: Errata
• 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.

• Section 4.43

The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.

Let $p$ $(\neq 0)$ and $q$ be real constants and

4.43.1
$A=\left(-\tfrac{4}{3}p\right)^{1/2},$
$B=\left(\tfrac{4}{3}p\right)^{1/2}.$

The roots of

4.43.2 $z^{3}+pz+q=0$

are:

1. (a)

$A\sin a$, $A\sin\left(a+\frac{2}{3}\pi\right)$, and $A\sin\left(a+\frac{4}{3}\pi\right)$, with $\sin\left(3a\right)=4q/A^{3}$, when $4p^{3}+27q^{2}\leq 0$.

2. (b)

$A\cosh a$, $A\cosh\left(a+\frac{2}{3}\pi\mathrm{i}\right)$, and $A\cosh\left(a+\frac{4}{3}\pi\mathrm{i}\right)$, with $\cosh\left(3a\right)=-4q/A^{3}$, when $p<0$, $q<0$, and $4p^{3}+27q^{2}>0$.

3. (c)

$B\sinh a$, $B\sinh\left(a+\frac{2}{3}\pi\mathrm{i}\right)$, and $B\sinh\left(a+\frac{4}{3}\pi\mathrm{i}\right)$, with $\sinh\left(3a\right)=-4q/B^{3}$, when $p>0$.

Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

Reported 2014-10-31 by Masataka Urago.

• Table 3.5.21

The correct corner coordinates for the 9-point square, given on the last line of this table, are $(\pm\sqrt{\tfrac{3}{5}}h,\pm\sqrt{\tfrac{3}{5}}h)$. Originally they were given incorrectly as $(\pm\sqrt{\tfrac{3}{5}}h,0)$, $(\pm\sqrt{\tfrac{3}{5}}h,0)$.

Reported 2014-01-13 by Stanley Oleszczuk.

• Table 22.5.4

Originally the limiting form for $\operatorname{sc}\left(z,k\right)$ in the last line of this table was incorrect ($\cosh z$, instead of $\sinh z$).

Reported 2010-11-23.