The application of residue theorem in Mellin transform integrals

Fengqin Dai; Tian Qiu

doi:10.1117/12.2648800

27 September 2022 The application of residue theorem in Mellin transform integrals

Fengqin Dai, Tian Qiu

Proceedings Volume 12345, International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2022); 123450C (2022) https://doi.org/10.1117/12.2648800
Event: 2022 International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2022), 2022, Qingdao, China

Abstract

The integral was introduced by Newton a hundred years ago. After that, Riemann gave a strict mathematical definition for integration. Integration has been widely applied in many fields. However, the Riemann integration has some limitations. Some integrals can be solved but they are not Riemann integrable. The most famous example is the Dirichlet function which is not Riemann integrable but Lebesgue integrable. We noticed that the Mellin transforms are closely connected to the theory of the Dirichlet series. Meanwhile, the Mellin transforms of some functions are not Riemann integrable, and they are difficult to be solved by some traditional methods but still can be solved in some other ways. Here, we will report a sample of solving one complicated integral of Mellin transform that has the form ∫∞0 t^s−1 f(t)dt by complex analysis. The result shows that some integrals of Mellin transform that are not Riemann integrable could be easily solved by the residue theorem.

Citation Download Citation

Fengqin Dai and Tian Qiu "The application of residue theorem in Mellin transform integrals", Proc. SPIE 12345, International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2022), 123450C (27 September 2022); https://doi.org/10.1117/12.2648800

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