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Analytic Function
Acomplex function is said to be analytic on a region
if it iscomplex differentiable at every point in
. The termsholomorphic function, differentiable function, and complex differentiable function are sometimes used interchangeably with "analytic function" (Krantz 1999, p. 16). Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic function" (Krantz 1999, p. 16), while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts (e.g., Morse and Feshbach 1953, pp. 356-374; Knopp 1996, pp. 83-111; Whittaker and Watson 1990, p. 83).
If acomplex function is analytic on a region
, it is infinitelydifferentiable in
. A complex function may fail to be analytic at one or more points through the presence ofsingularities, or along lines or line segments through the presence ofbranch cuts.
Acomplex function that is analytic at all finite points of thecomplex plane is said to beentire. Asingle-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities goes to infinity like a polynomial (i.e., these exceptional points must bepoles and notessential singularities), is called ameromorphic function.
See also
Anti-Analytic Function,Bergman Space,Cauchy-Riemann Equations,Complex Differentiable,Complex Function,Complex Plane,Differentiable,Entire Function,Holomorphic Function,Meromorphic Function,Pseudoanalytic Function,Real Analytic Function,Regular Function,Semianalytic,Singularity,SubanalyticExplore this topic in the MathWorld classroomExplore with Wolfram|Alpha
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References
Knopp, K. "Analytic Continuation and Complete Definition of Analytic Functions." Ch. 8 inTheory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 83-111, 1996.Krantz, S. G. "Alternative Terminology for Holomorphic Functions." §1.3.6 inHandbook of Complex Variables. Boston, MA: Birkhäuser, p. 16, 1999.Morse, P. M. and Feshbach, H. "Analytic Functions." §4.2 inMethods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 356-374, 1953.Whittaker, E. T. and Watson, G. N.A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Referenced on Wolfram|Alpha
Analytic FunctionCite this as:
Weisstein, Eric W. "Analytic Function."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/AnalyticFunction.html