# double series

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##### 1: 1.9 Calculus of a Complex Variable
###### §1.9(vii) Inversion of Limits
A double series is the limit of the double sequence …If the limit exists, then the double series is convergent; otherwise it is divergent. … If a double series is absolutely convergent, then it is also convergent and its sum is given by either of the repeated sums …
##### 2: 25.16 Mathematical Applications
25.16.1 $\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p,$
##### 3: 20.6 Power Series
In the double series the order of summation is important only when $j=1$. …
##### 4: 23.2 Definitions and Periodic Properties
The double series and double product are absolutely and uniformly convergent in compact sets in $\mathbb{C}$ that do not include lattice points. …
##### 5: 16.14 Partial Differential Equations
In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …
##### 6: 1.3 Determinants, Linear Operators, and Spectral Expansions
These have the property that the double series
##### 7: 10.53 Power Series
###### §10.53 Power Series
10.53.2 $\mathsf{y}_{n}\left(z\right)=-\frac{1}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!% (\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}\sum_{k=n+1}^{\infty}% \frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.$
10.53.4 ${\mathsf{i}^{(2)}_{n}}\left(z\right)=\frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}% \frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{% \infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.$
##### 8: Bibliography
• H. M. Antia (1993) Rational function approximations for Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 84, pp. 101–108.
• ##### 9: 10.41 Asymptotic Expansions for Large Order
For expansions in inverse factorial series see Dunster et al. (1993).
###### §10.41(iv) Double Asymptotic Properties
The series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large $|z|$. …
##### 10: Bibliography C
• L. D. Cloutman (1989) Numerical evaluation of the Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 71, pp. 677–699.