# relation to infinite partial fractions

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##### 1: 1.9 Calculus of a Complex Variable
If $f(z(t_{0}))=\infty$, $a\leq t_{0}\leq b$, then the integral is defined analogously to the infinite integrals in §1.4(v). …
###### §1.9(v) Infinite Sequences and Series
This sequence converges pointwise to a function $f(z)$ if …
##### 2: 1.10 Functions of a Complex Variable
###### §1.10(ix) Infinite Products
The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. …
##### 4: 3.10 Continued Fractions
###### §3.10(ii) Relationsto Power Series
can be converted into a continued fraction $C$ of type (3.10.1), and with the property that the $n$th convergent $C_{n}=A_{n}/B_{n}$ to $C$ is equal to the $n$th partial sum of the series in (3.10.3), that is, …
###### Stieltjes Fractions
The $A_{n}$ and $B_{n}$ of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). … This forward algorithm achieves efficiency and stability in the computation of the convergents $C_{n}=A_{n}/B_{n}$, and is related to the forward series recurrence algorithm. …
##### 5: 6.11 Relations to Other Functions
###### Confluent Hypergeometric Function
6.11.3 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$
##### 7: 4.36 Infinite Products and Partial Fractions
###### §4.36 Infinite Products and PartialFractions
4.36.4 ${\operatorname{csch}^{2}}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},$
##### 8: Bibliography R
• E. M. Rains (1998) Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.
• Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.
• M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
• R. R. Rosales (1978) The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.
• J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
• ##### 9: 15.19 Methods of Computation
However, by appropriate choice of the constant $z_{0}$ in (15.15.1) we can obtain an infinite series that converges on a disk containing $z=e^{\pm\pi\mathrm{i}/3}$. Moreover, it is also possible to accelerate convergence by appropriate choice of $z_{0}$. …
###### §15.19(iv) Recurrence Relations
The relations in §15.5(ii) can be used to compute $F\left(a,b;c;z\right)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …
##### 10: 25.13 Periodic Zeta Function
###### §25.13 Periodic Zeta Function
The notation $F\left(x,s\right)$ is used for the polylogarithm $\mathrm{Li}_{s}\left(e^{2\pi ix}\right)$ with $x$ real: Also,
25.13.2 $F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$ $0, $\Re s>1$,