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1: 22.7 Landen Transformations

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§22.7(ii) Ascending Landen Transformation

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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 $k=0,1,2,\mathrm{\dots }$. …

3: 22.17 Moduli Outside the Interval [0,1]

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§22.17(ii) Complex Moduli

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4: 19.8 Quadratic Transformations

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Ascending Landen Transformation

… ►We consider only the descending Gauss transformation because its (ascending) inverse moves $F(\varphi ,k)$ closer to the singularity at $k=\sin \varphi =1$. …

5: 19.22 Quadratic Transformations

… ►If $x,y,z$ are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when (implying ), and descending Gauss transformations when (implying ). …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. …

6: 3.10 Continued Fractions

… ►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 $z^{n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►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 $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. …

7: 4.45 Methods of Computation

… ►The function $\ln x$ can always be computed from its ascending power series after preliminary scaling. … ►The function $\arctan x$ can always be computed from its ascending power series after preliminary transformations to reduce the size of $x$. …

8: 22.20 Methods of Computation

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§22.20(iii) Landen Transformations

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9: 30.16 Methods of Computation

… ►and real eigenvalues ${\alpha }_{1,d}$, ${\alpha }_{2,d}$, $\mathrm{\dots }$, ${\alpha }_{d,d}$, arranged in ascending order of magnitude. …

10: 9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►They lie in the sectors and , and are denoted by ${\beta }_k$, ${\beta }_k^{\prime }$, respectively, in the former sector, and by $\overline{{\beta }_k}$, $\overline{{\beta }_k^{\prime }}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\mathrm{\dots }.$ See §9.3(ii) for visualizations. …