# double sequence

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##### 1: 1.9 Calculus of a Complex Variable
###### DoubleSequences and Series
A set of complex numbers $\{z_{m,n}\}$ where $m$ and $n$ take all positive integer values is called a double sequence. … A double series is the limit of the double sequence … …
##### 2: Guide to Searching the DLMF
• phrase:

any double-quoted sequence of textual words and numbers.

• ##### 3: Bibliography C
• W. J. Cody (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.
• ##### 4: 3.8 Nonlinear Equations
Let $z_{1},z_{2},\dots$ be a sequence of approximations to a root, or fixed point, $\zeta$. …for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … As in the case of Table 3.8.1 the quadratic nature of convergence is clearly evident: as the zero is approached, the number of correct decimal places doubles at each iteration. … Starting this iteration in the neighborhood of one of the four zeros $\pm 1,\pm\mathrm{i}$, sequences $\{z_{n}\}$ are generated that converge to these zeros. …
##### 5: 5.17 Barnes’ $G$-Function (Double Gamma Function)
###### §5.17 Barnes’ $G$-Function (Double Gamma Function)
5.17.3 $G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$
5.17.4 $\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\,\mathrm{d}t.$
5.17.6 $A=e^{C}=1.28242\;71291\;00622\;63687\;\ldots,$
##### 6: Bibliography W
• T. Weider (1999) Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.
• E. J. Weniger (1989) Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.
• E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.
• J. Wimp (1981) Sequence Transformations and their Applications. Mathematics in Science and Engineering, Vol. 154, Academic Press Inc., New York.
• G. Wolf (1998) On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.
• ##### 7: 5.4 Special Values and Extrema
5.4.2 $n!!=\begin{cases}2^{\frac{1}{2}n}\Gamma\left(\frac{1}{2}n+1\right),&n\text{ % even},\\ \pi^{-\frac{1}{2}}2^{\frac{1}{2}n+\frac{1}{2}}\Gamma\left(\frac{1}{2}n+1\right% ),&n\text{ odd}.\end{cases}$
5.4.7 $\Gamma\left(\tfrac{1}{3}\right)=2.67893\;85347\;07747\;63365\;\dots,$
5.4.8 $\Gamma\left(\tfrac{2}{3}\right)=1.35411\;79394\;26400\;41694\;\dots,$
5.4.9 $\Gamma\left(\tfrac{1}{4}\right)=3.62560\;99082\;21908\;31193\;\dots,$
5.4.10 $\Gamma\left(\tfrac{3}{4}\right)=1.22541\;67024\;65177\;64512\;\dots.$
##### 8: 18.33 Polynomials Orthogonal on the Unit Circle
This states that for any sequence $\{\alpha_{n}\}_{n=0}^{\infty}$ with $\alpha_{n}\in\mathbb{C}$ and $|\alpha_{n}|<1$ the polynomials $\Phi_{n}(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with $\Phi_{0}(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. …
18.33.32 $\sum_{j=0}^{\infty}|\alpha_{j}|^{2}<\infty\quad\Longleftrightarrow\quad\frac{1% }{2\pi\mathrm{i}}\int_{|z|=1}\ln\left((w(z)\right)\frac{\,\mathrm{d}z}{z}>-\infty.$
##### 9: 4.2 Definitions
4.2.11 $e=2.71828\ 18284\ 59045\ 23536\dots$
4.2.17 $\operatorname{log}_{10}e=0.43429\ 44819\ 03251\ 82765\dots,$
4.2.18 $\ln 10=2.30258\ 50929\ 94045\ 68401\dots.$
4.2.25 $\exp z=\zeta\;\;\Longleftrightarrow\;\;z=\operatorname{Ln}\zeta.$
4.2.35 $z^{a}=w\;\;\Longleftrightarrow\;\;z=\exp\left(\frac{1}{a}\operatorname{Ln}w% \right).$
##### 10: Bibliography S
• M. J. Seaton (1982) Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.
• D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.
• B. L. Shea (1988) Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.
• P. N. Shivakumar and J. Xue (1999) On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.
• N. J. A. Sloane (2003) The On-Line Encyclopedia of Integer Sequences. Notices Amer. Math. Soc. 50 (8), pp. 912–915.