Natural boundaries revisited through differential equations, infinite order or non-linear (Q2744099)
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scientific article; zbMATH DE number 1648123
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Natural boundaries revisited through differential equations, infinite order or non-linear |
scientific article; zbMATH DE number 1648123 |
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18 September 2001
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Chazy equation
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movable natural boundary
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0.89967316
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0.8677672
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0.8669197
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0.8654028
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0.8644538
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0.86188096
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Natural boundaries revisited through differential equations, infinite order or non-linear (English)
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The author gives some attempts to examine natural boundaries of certain types of functions by the use of differential equations. Firstly, for functions expressed by a Fourier series \(f(z)=\sum_na_n\exp(i\lambda_n z),\) the author explains the method of using linear differential operators of infinite order. Secondly, the Jacobi equation is treated, which satisfies a certain transformation of the theta-zero value \(\vartheta_3(t)=\sum_{\nu}\exp(\pi i\nu^2t)\). By a WKB-type analysis developed by Joshi and Kruskal for the Chazy equation, the author obtains an exponential expansion of a solution which indicates a movable natural boundary. Lastly, some open problems related to WKB-analysis are listed.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].
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