# Pringsheim theorem for continued fractions

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##### 2: 13.5 Continued Fractions
###### §13.5 ContinuedFractions
This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $\mathbb{C}$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $|\operatorname{ph}{z}|<\pi$. …
##### 3: 13.17 Continued Fractions
###### §13.17 ContinuedFractions
13.17.1 $\frac{\sqrt{z}M_{\kappa,\mu}\left(z\right)}{M_{\kappa-\frac{1}{2},\mu+\frac{1}% {2}}\left(z\right)}=1+\cfrac{u_{1}z}{1+\cfrac{u_{2}z}{1+\cdots}},$
This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in\mathbb{C}$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $|\operatorname{ph}{z}|<\pi$. …
##### 4: 12.6 Continued Fraction
###### §12.6 ContinuedFraction
For a continued-fraction expansion of the ratio $\ifrac{U\left(a,x\right)}{U\left(a-1,x\right)}$ see Cuyt et al. (2008, pp. 340–341).
##### 5: 10.55 Continued Fractions
###### §10.55 ContinuedFractions
For continued fractions for $\mathsf{j}_{n+1}\left(z\right)/\mathsf{j}_{n}\left(z\right)$ and ${\mathsf{i}^{(1)}_{n+1}}\left(z\right)/{\mathsf{i}^{(1)}_{n}}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
##### 7: 28.27 Addition Theorems
###### §28.27 Addition Theorems
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
##### 8: 3.10 Continued Fractions
###### Stieltjes Fractions
A continued fraction of the form …
###### Jacobi Fractions
The continued fraction