# analytic continuation

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##### 1: 6.4 Analytic Continuation
###### §6.4 AnalyticContinuation
Analytic continuation of the principal value of $E_{1}\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\infty$. …
6.4.7 $\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).$
##### 2: 28.7 Analytic Continuation of Eigenvalues
###### §28.7 AnalyticContinuation of Eigenvalues
28.7.4 $\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2n+2)^{2}\right)=0.$
##### 5: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions
This equation can be used as the analytic continuation for this ${{}_{1}\phi_{0}}$. … This equation can be used as the analytic continuation for this ${{}_{1}\phi_{0}}$. …
##### 7: 15.17 Mathematical Applications
By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …
##### 8: 16.15 Integral Representations and Integrals
These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large $x$, large $y$, or both. …
##### 10: 18.40 Methods of Computation
###### Stieltjes Inversion via (approximate) AnalyticContinuation
The question is then: how is this possible given only $F_{N}(z)$, rather than $F(z)$ itself? $F_{N}(z)$ often converges to smooth results for $z$ off the real axis for $\Im{z}$ at a distance greater than the pole spacing of the $x_{n}$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_{N}(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. …