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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. …
2: 1 Algebraic and Analytic Methods
Chapter 1 Algebraic and Analytic Methods
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 ( z ) yields a multi-valued function with branch points at z = 0 and z = . …
6.4.4 Ci ( z e ± π i ) = ± π i + Ci ( z ) ,
6.4.7 g ( z e ± π i ) = π i e i z + g ( z ) .
5: 28.7 Analytic Continuation of Eigenvalues
§28.7 Analytic Continuation of Eigenvalues
As functions of q , a n ( q ) and b n ( q ) can be continued analytically in the complex q -plane. … All the a 2 n ( q ) , n = 0 , 1 , 2 , , can be regarded as belonging to a complete analytic function (in the large). …
28.7.4 n = 0 ( b 2 n + 2 ( q ) - ( 2 n + 2 ) 2 ) = 0 .
6: 1.13 Differential Equations
The equation …where z 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.
8: 1.10 Functions of a Complex Variable
§1.10(ii) Analytic Continuation
Schwarz Reflection Principle
Analytic Functions
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 α 1 < ph t < α 2 containing ph t = 0 , then the region of validity may be increased by rotation of the integration paths. … is continuous in z c and analytic in 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 T that contains 𝒫 , with the possible exceptions of t = a and t = b . … in which z is a large real or complex parameter, p ( α , t ) and q ( α , t ) are analytic functions of t and continuous in t and a second parameter α . …