# ascending

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##### 2: 6.13 Zeros
$\mathrm{Ci}\left(x\right)$ 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 $k=0,1,2,\dots$. …
##### 3: 22.17 Moduli Outside the Interval [0,1]
###### Ascending Landen Transformation
We consider only the descending Gauss transformation because its (ascending) inverse moves $F\left(\phi,k\right)$ closer to the singularity at $k=\sin\phi=1$. …
If $x,y,z$ are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when $x,y (implying $a), and descending Gauss transformations when $z (implying $z_{+}). …Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. …
if the expansion of its $n$th convergent $C_{n}$ in ascending powers of $z$ agrees with (3.10.7) up to and including the term in $z^{n-1}$, $n=1,2,3,\dots$. … We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_{n}$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\dots$. …
The function $\ln x$ can always be computed from its ascending power series after preliminary scaling. … The function $\operatorname{arctan}x$ can always be computed from its ascending power series after preliminary transformations to reduce the size of $x$. …
and real eigenvalues $\alpha_{1,d}$, $\alpha_{2,d}$, $\dots$, $\alpha_{d,d}$, arranged in ascending order of magnitude. …
They are denoted by $a_{k}$, $a^{\prime}_{k}$, $b_{k}$, $b^{\prime}_{k}$, respectively, arranged in ascending order of absolute value for $k=1,2,\ldots.$They lie in the sectors $\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi$ and $-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi$, and are denoted by $\beta_{k}$, $\beta^{\prime}_{k}$, respectively, in the former sector, and by $\overline{\beta_{k}}$, $\overline{\beta^{\prime}_{k}}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\ldots.$ See §9.3(ii) for visualizations. …