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Analytic continuation is the extension of an analytic function beyond its initial domain by agreeing analytic functions on overlapping domains. It can be carried out by a chain of power series centered at successive points of a path. An initial power series

with radius of convergence can be reexpanded about when . The new coefficients are

where is a nonnegative integer and is a binomial coefficient. The new expansion is at least for . The new radius of convergence can exceed this bound, so the new power series may reach points outside the original disk. Iterating this transformation along a path is the power series method of analytic continuation.

For example, has Taylor series about the origin with radius of convergence 1. Reexpansion about gives

which converges for and therefore includes points outside .

If and are analytic functions on domains and and agree on their nonempty intersection, then is an analytic continuation of , and vice versa (Flanigan 1983, p. 234). The principle of permanence makes such a direct analytic continuation to a specified connected domain unique when it exists. Along a fixed path, continuation from a specified initial function element is likewise unique. Different paths to the same point can give different branches, as described by monodromy. The monodromy theorem gives conditions under which the result is independent of the path.

Continuation need not be possible along every path. A singularity can prevent it, and a natural boundary can prevent extension across every point of a boundary. Thus, local values determine any continuation uniquely along a specified path, but do not guarantee a single-valued function on the entire complex plane.

Analytic continuation can lead to some interesting phenomena such as multivalued functions. For example, consider analytic continuation of the square root function . Although this function is not globally well-defined (since every nonzero complex number has two square roots), has a well-defined Taylor series around ,

which has radius of convergence 1 about and converges in .

The animation above shows the analytic continuation of along the path . Note that when the function goes all the way around, is the negative of the original function, so going around twice returns the function to its original value.

In the animation, the domain space (colored pink; left figures) is mapped to the image space (colored blue; right figures) by the square root function, and the light blue region indicates the negative square root. However, by continuing the function around the circle, the square root function takes values in what used to be the light blue region, so the roles of the blue and light blue region are reversed.

This can be interpreted as going from one branch of the multivalued square root function to the other. This illustrates that analytic continuation extends a function using the nearby values that provide the information on the power series.

It is possible for the function to never return to the same value. For example, increases by every time it is continued counterclockwise around zero. This gives a logarithmic branch point. The natural domain is described by a maximal chain of domains on which the continued function is a single-valued function. Its Riemann surface is the universal cover of the punctured plane for and the connected double cover for . If there is a boundary across which the function cannot be extended, it is called a natural boundary. For instance, there exists a meromorphic function in the unit disk where every point on the unit circle is a limit point of the set of poles. Then the circle is a natural boundary for .

See also

Analytic Function, Branch Cut, Branch Point, Continuity Principle, Direct Analytic Continuation, Global Analytic Continuation, Logarithmic Branch Point, Monodromy, Monodromy Theorem, Multivalued Function, Natural Boundary, Permanence of Algebraic Form, Power Series, Principle of Permanence, Riemann Surface, Schwarz Reflection Principle, Taylor Series Explore this topic in the MathWorld classroom

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 378-380, 1985.Davis, P. J. and Pollak, H. "On the Analytic Continuation of Mapping Functions." Trans. Amer. Math. Soc. 87, 198-225, 1958.Dienes, P. The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable. Oxford, England: Clarendon Press, 1931.Flanigan, F. J. Complex Variables: Harmonic and Analytic Functions. New York: Dover, 1983.Havil, J. "Analytic Continuation." §16.3 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 91-193, 2003.Henrici, P. "An Algorithm for Analytic Continuation." SIAM J. Numer. Anal. 3, 67-78, 1966. https://doi.org/10.1137/0703005.Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, 1974.Honda, K. "Notes for Math 520: Complex Analysis." https://www.math.ucla.edu/~honda/math520/notes.pdf.Jones, G. A. and Singerman, D. Complex Functions: An Algebraic and Geometric Viewpoint. Cambridge, England: Cambridge University Press, 1987.Knopp, K. "Analytic Continuation and Complete Definition of Analytic Functions." Ch. 8 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 83-111, 1996.Krantz, S. G. "Uniqueness of Analytic Continuation" and "Analytic Continuation." §3.2.3 and Ch. 10 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 38-39 and 123-141, 1999.Levinson, N. and Raymond, R. Complex Variables. New York: McGraw-Hill, pp. 398-402, 1970.McMullen, C. T. "Riemann Surfaces." Harvard University, Math 213b, Apr. 28, 2026. https://abel.math.harvard.edu/~ctm/math213b/home/course/course.pdf.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 389-390 and 392-398, 1953.Needham, T. "Analytic Continuation." §5.XI in Visual Complex Analysis. New York: Clarendon Press, pp. 247-257, 2000.Rudin, W. Real and Complex Analysis. New York: McGraw-Hill, pp. 319-327, 1987.Small, R. D. "Modified Convergence of Taylor Series." Amer. Math. Monthly 88, 439-441, 1981. https://doi.org/10.2307/2321830.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, sec. 1.6.6, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.Whittaker, E. T. and Watson, G. N. "The Process of Continuation." §5.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 96-98, 1990.

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Analytic Continuation

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Weisstein, Eric W. "Analytic Continuation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AnalyticContinuation.html

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