# analyticity

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##### 1: 34.9 Graphical Method
The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. Thus, any analytic expression in the theory, for example equations (34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be represented by a diagram; conversely, any diagram represents an analytic equation. …
##### 3: Simon Ruijsenaars
His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …
##### 4: 6.4 Analytic Continuation
###### §6.4 Analytic Continuation
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 $\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).$
##### 5: 28.7 Analytic Continuation of Eigenvalues
###### §28.7 Analytic Continuation of Eigenvalues
As functions of $q$, $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ can be continued analytically in the complex $q$-plane. … All the $a_{2n}\left(q\right)$, $n=0,1,2,\dots$, can be regarded as belonging to a complete analytic function (in the large). …
28.7.4 $\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2n+2)^{2}\right)=0.$
##### 6: 1.13 Differential Equations
The equation …where $z\in D$, a simply-connected domain, and $f(z)$, $g(z)$ are analytic in $D$, has an infinite number of analytic solutions in $D$. … Assume that in the equation … The inhomogeneous (or nonhomogeneous) equation …with $f(z)$, $g(z)$, and $r(z)$ analytic in $D$ has infinitely many analytic solutions in $D$. …
##### 7: 31.1 Special Notation
Sometimes the parameters are suppressed.
##### 9: William P. Reinhardt
Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics. … Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. …
##### 10: 2.4 Contour Integrals
If $q(t)$ is analytic in a sector $\alpha_{1}<\operatorname{ph}t<\alpha_{2}$ containing $\operatorname{ph}t=0$, then the region of validity may be increased by rotation of the integration paths. … is continuous in $\Re z\geq c$ and analytic in $\Re z>c$, and by inversion (§1.14(iii)) … Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion … Assume that $p(t)$ and $q(t)$ are analytic on an open domain $\mathbf{T}$ that contains $\mathscr{P}$, with the possible exceptions of $t=a$ and $t=b$. … in which $z$ is a large real or complex parameter, $p(\alpha,t)$ and $q(\alpha,t)$ are analytic functions of $t$ and continuous in $t$ and a second parameter $\alpha$. …