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Circle of Convergence

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For complex-valued functions, the circle of convergence (also called the region of convergence) is the set of values for which a power series is valid [1]. The circle of convergence is basically the radius of convergence extended to the complex setting.


Formally, the circle of convergence CR(z0 is where the power series
power series

converges at each point inside the circle.

If a function has a power series expansion around a point, then the circle will extend outward to the closest point where the function isn’t analytic; In other words, a function is analytic, but then fails to be analytic for at least one point (or perhaps all points) on the circle of convergence. This circle could therefore represent all complex numbers (ℂ), but also could represent a specific region (the complex analysis equivalent of an interval in real analysis).


Graph of the Circle of Convergence

circle of convergence

The above image shows the greatest circle centered at z0.

The radius of the circle of convergence could be infinite [2], which doesn’t work well on paper! If you’re having trouble in imagining an infinitely wide circle, think of how an arrow on the x-axis or y-axis of a Cartesian plane could also imply infinity.

uniform convergence 2

Example of convergence in real analysis. The arrows on the x and y axis imply “to infinity”.


Absolute and Uniform Convergence

If the infinite complex power series an(z – z0)n has circle of convergence |z – z0| = R, then for any positive valued r, the series is uniformly convergent on the closed disc |z – z0| ≤ r. In addition, it is absolutely convergent for each point in |z – z0| ≤ r [3].

References

[1] Manogue, C. & Dray, C. 7.6 Convergence of Power Series. Retrieved April 24, 2021 from: https://books.physics.oregonstate.edu/LinAlg/convergence.html
[2] Section 5.63. Absolute and Uniform Convergence of Power Series. Retrieved April 24, 2021 from: https://faculty.etsu.edu/gardnerr/5337/notes/Chapter5-63.pdf
[3] Uniqueness of Taylor Series. Retrieved April 24, 2021 from: http://web.math.ucsb.edu/~helena/teaching/math122b/taylor_series.pdf

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Stephanie Glen. "Circle of Convergence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/circle-of-convergence/
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