Geodesic
Reciprocal function method for Cauchy problems with first-order poles
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 491 (2020), pp. 102-106.

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For the numerical solution of the Cauchy problem with multiple poles, we propose a reciprocal function method. In the case of first-order poles, it makes it possible to continue the solution through the poles and to determine the solution and the pole positions with good accuracy. The method allows one to employ conventional explicit and implicit schemes, for example, explicit Runge–Kutta schemes. A test problem with multiple poles is computed as an example. The proposed method is useful for construction of software for direct computation of special functions.
Keywords: Cauchy problem, singularities, continuation through a pole. 
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     author = {A. A. Belov and N. N. Kalitkin},
     title = {Reciprocal function method for {Cauchy} problems with first-order poles},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {102--106},
     publisher = {mathdoc},
     volume = {491},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_491_a20/}
}
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A. A. Belov; N. N. Kalitkin. Reciprocal function method for Cauchy problems with first-order poles. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 491 (2020), pp. 102-106. http://geodesic.mathdoc.fr/item/DANMA_2020_491_a20/

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