# ascending

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## 1—10 of 13 matching pages

##### 1: 22.7 Landen Transformations

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###### §22.7(ii) Ascending Landen Transformation

…##### 2: 6.13 Zeros

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$\mathrm{Ci}\left(x\right)$ 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 $k=0,1,2,\mathrm{\dots}$.
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##### 3: 22.17 Moduli Outside the Interval [0,1]

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###### §22.17(ii) Complex Moduli

…##### 4: 4.45 Methods of Computation

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►The function $\mathrm{ln}x$ can always be computed from its ascending power series after preliminary scaling.
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►The function $\mathrm{arctan}x$ can always be computed from its ascending power series after preliminary transformations to reduce the size of $x$.
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##### 5: 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 $k=\mathrm{sin}\varphi =1$. …##### 6: 19.22 Quadratic Transformations

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►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.
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►The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations.
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##### 7: 3.10 Continued Fractions

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►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 ${z}^{n-1}$, $n=1,2,3,\mathrm{\dots}$.
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►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 ${z}^{2n-1}$, $n=1,2,3,\mathrm{\dots}$. …##### 8: 22.20 Methods of Computation

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###### §22.20(iii) Landen Transformations

…##### 9: 30.16 Methods of Computation

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►and real eigenvalues ${\alpha}_{1,d}$, ${\alpha}_{2,d}$, $\mathrm{\dots}$, ${\alpha}_{d,d}$, arranged in ascending order of magnitude.
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##### 10: 9.9 Zeros

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►They are denoted by ${a}_{k}$, ${a}_{k}^{\prime}$, ${b}_{k}$, ${b}_{k}^{\prime}$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots}.$
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►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.
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