analytic functions

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1: Simon Ruijsenaars
His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …
2: 2.4 Contour Integrals
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 $t=b$. …
• (a)

In a neighborhood of $a$

with $\Re\lambda>0$, $\mu>0$, $p_{0}\neq 0$, and the branches of $(t-a)^{\lambda}$ and $(t-a)^{\mu}$ continuous and constructed with $\operatorname{ph}\left(t-a\right)\to\omega$ as $t\to a$ along $\mathscr{P}$.

• 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 $\alpha$. …
3: 4.7 Derivatives and Differential Equations
For a nonvanishing analytic function $f(z)$, the general solution of the differential equation
4.7.6 $w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}.$
4: 5.2 Definitions
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\mathrm{d}t,$ $\Re z>0$.
5.2.2 $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$ $z\neq 0,-1,-2,\dots$.
5: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: …
§3.8(v) Zeros of AnalyticFunctions
Newton’s rule is the most frequently used iterative process for accurate computation of real or complex zeros of analytic functions $f(z)$. …
6: 4.14 Definitions and Periodicity
4.14.7 $\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$
7: 16.2 Definition and Analytic Properties
§16.2(iii) Case $p=q+1$
Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\infty$. …
8: 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 $\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2n+2)^{2}\right)=0.$
9: 1.13 Differential Equations
Assume that in the equation
1.13.7 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(u,z)\frac{\mathrm{d}w}{\mathrm{d% }z}+g(u,z)w=0,$
Suppose also that at (a fixed) $z_{0}\in D$, $w$ and $\ifrac{\partial w}{\partial z}$ are analytic functions of $u$. Then at each $z\in D$, $w$, $\ifrac{\partial w}{\partial z}$ and $\ifrac{{\partial}^{2}w}{{\partial z}^{2}}$ are analytic functions of $u$. … The inhomogeneous (or nonhomogeneous) equation …