# analyticity

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##### 11: David M. Bressoud
His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …
##### 13: Peter L. Walker
Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. …
##### 14: 5.2 Definitions
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\mathrm{d}t,$ $\Re z>0$.
When $\Re z\leq 0$, $\Gamma\left(z\right)$ is defined by analytic continuation. …
5.2.2 $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$ $z\neq 0,-1,-2,\dots$.
##### 15: 4.7 Derivatives and Differential Equations
For a nonvanishing analytic function $f(z)$, the general solution of the differential equation
4.7.6 $w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}.$
##### 16: 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. …
##### 17: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point $z_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\tfrac{1}{3}$9.6(i)). … The number $m$ can also be replaced by any real constant $\lambda$ $(>-2)$ in the sense that $(z-z_{0})^{-\lambda}$ $f(z)$ is analytic and nonvanishing at $z_{0}$; moreover, $g(z)$ is permitted to have a single or double pole at $z_{0}$. … In regions in which the function $f(z)$ has a simple pole at $z=z_{0}$ and $(z-z_{0})^{2}g(z)$ is analytic at $z=z_{0}$ (the case $\lambda=-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm\sqrt{1+4\rho}$, where $\rho$ is the limiting value of $(z-z_{0})^{2}g(z)$ as $z\to z_{0}$. … Assume that whether or not $\alpha=a$, $z^{2}g(z,\alpha)$ is analytic at $z=0$. …
##### 18: 31.4 Solutions Analytic at Two Singularities: Heun Functions
###### §31.4 Solutions Analytic at Two Singularities: Heun Functions
For an infinite set of discrete values $q_{m}$, $m=0,1,2,\dots$, of the accessory parameter $q$, the function $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic at $z=1$, and hence also throughout the disk $|z|. … with $(s_{1},s_{2})\in\{0,1,a,\infty\}$, denotes a set of solutions of (31.2.1), each of which is analytic at $s_{1}$ and $s_{2}$. …
##### 19: 15.2 Definitions and Analytical Properties
###### §15.2 Definitions and Analytical Properties
on the disk $|z|<1$, and by analytic continuation elsewhere. … again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. …
###### §15.2(ii) Analytic Properties
The right-hand side can be seen as an analytical continuation for the left-hand side when $a$ approaches $-m$. …