# analyticity

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## 1—10 of 146 matching pages

##### 1: 34.9 Graphical Method

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►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.
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##### 2: 1 Algebraic and Analytic Methods

###### Chapter 1 Algebraic and Analytic Methods

…##### 3: Simon Ruijsenaars

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►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas.
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##### 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 $z=\mathrm{\infty}$. … ►
6.4.4
$$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

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6.4.7
$$\mathrm{g}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mp \pi \mathrm{i}{\mathrm{e}}^{\mp \mathrm{i}z}+\mathrm{g}\left(z\right).$$

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##### 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 ${a}_{2n}\left(q\right)$, $n=0,1,2,\mathrm{\dots}$, can be regarded as belonging to a complete analytic function (in the large). … ►
28.7.4
$$\sum _{n=0}^{\mathrm{\infty}}\left({b}_{2n+2}\left(q\right)-{(2n+2)}^{2}\right)=0.$$

##### 6: 1.13 Differential Equations

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►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 $D$.
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►Assume that in the equation
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►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

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►Sometimes the parameters are suppressed.

##### 8: 1.10 Functions of a Complex Variable

##### 9: William P. Reinhardt

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►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.
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►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.
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##### 10: 2.4 Contour Integrals

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►If $q(t)$ is analytic in a sector $$ containing $\mathrm{ph}t=0$, then the region of validity may be increased by rotation of the integration paths.
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►is continuous in $\mathrm{\Re}z\ge c$ and analytic in $\mathrm{\Re}z>c$, and by inversion (§1.14(iii))
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►Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion
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►Assume that $p(t)$ and $q(t)$ are analytic on an open domain $\mathbf{T}$ that contains $\mathcal{P}$, with the possible exceptions of $t=a$ and $t=b$.
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►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 $.
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