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Mar 16, 2022 at 22:49 comment added Alexandre Eremenko Your answer and intuitive justification are essentially correct for LINEAR equations (the number and the nature of singularities are important, and there is no "finite list"). For non-linear equations it is completely hopeless. Linear equations can be classified in principle, into a countable number of finite-parametric families.
Mar 16, 2022 at 19:01 history asked Diego Santos CC BY-SA 4.0