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- W2897838887 abstract "In calculus, we use Taylor series to represent functions on intervals centered at fixed points with a radius of convergence that could be positive, infinite, or zero, depending on the remainder associated with the function. For example, $$cos x$$ , $$e^x$$ , $$frac{1}{1+x^2}$$ , and the function defined by $$e^{-1/x^2}$$ for $$xne 0$$ and 0 if $$x=0$$ are all infinitely differentiable for all real x. The radius of convergence of the Taylor series representation around zero is $$infty $$ for the first two, 1 for the third one, and 0 for the last one. However in complex analysis, Taylor series are much nicer, in the sense that the remainder will play no role in determining their convergence. If a function is analytic on a disk of radius R centered at $$z_0$$ , then it has a Taylor series representation centered at $$z_0$$ with radius at least R. For example the function $$ frac{1}{1+z^2}$$ is analytic on the disk $$|z|<1$$ but, as it is not differentiable at $$z=pm i$$ , we do not expect the series to have a radius of convergence larger than 1." @default.
- W2897838887 created "2018-10-26" @default.
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- W2897838887 date "2018-01-01" @default.
- W2897838887 modified "2023-09-23" @default.
- W2897838887 title "Series of Analytic Functions and Singularities" @default.
- W2897838887 doi "https://doi.org/10.1007/978-3-319-94063-2_4" @default.
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