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11: 4.2 Definitions
This is a multivalued function of z with branch point at z = 0 . … ln z is a single-valued analytic function on ( - , 0 ] and real-valued when z ranges over the positive real numbers. …
§4.2(iii) The Exponential Function
4.2.27 z a = z z z n  times = 1 / z - a .
This is an analytic function of z on ( - , 0 ] , and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless a . …
12: 22.17 Moduli Outside the Interval [0,1]
§22.17(ii) Complex Moduli
13: 1.10 Functions of a Complex Variable
In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in with at most one exception. …
Analytic Functions
Suppose that a n ( z ) are analytic functions in D . …
14: 32.2 Differential Equations
be a nonlinear second-order differential equation in which F is a rational function of w and d w / d z , and is locally analytic in z , that is, analytic except for isolated singularities in . … in which a ( z ) , b ( z ) , c ( z ) , d ( z ) , and ϕ ( z ) are locally analytic functions. …
15: 1.9 Calculus of a Complex Variable
Analyticity
A function analytic at every point of is said to be entire. … Inside the circle the sum of the series is an analytic function f ( z ) . …
16: 14.21 Definitions and Basic Properties
17: 14.24 Analytic Continuation
§14.24 Analytic Continuation
18: 10.34 Analytic Continuation
§10.34 Analytic Continuation
19: 10.11 Analytic Continuation
§10.11 Analytic Continuation
20: 2.7 Differential Equations
2.7.1 d 2 w d z 2 + f ( z ) d w d z + g ( z ) w = 0
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … In a finite or infinite interval ( a 1 , a 2 ) let f ( x ) be real, positive, and twice-continuously differentiable, and g ( x ) be continuous. …