Applications of analyticoty of functions

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(1) About mathematics, we discussed and gave the definitions of mathematics and good results in mathematics. In particular, the essentials of mathematics are stated to be relations. As an example in this sprit, we established a method connecting "analyticity of functions" and "nonlinear transforms" and derived various concrete results.(2) We found a method discussing the existence of the solutions for general linear differential equations with variable coefficients. This method gives also a contructing algorithms of the solutions, when there exist the solutions.(3) For many solutions for linear partial differential equations depending time, we found a general principle representing the solutions by means of their local deta in both space and time. We can get similar results for many elliptic linear partial differential equations. This pleasant result was named as "Principle of Telethoscope"(4) We found a general principle introducing various operators among Hilbert spaces by mens of transforms. In particular, we were able to give a general definition of the fundamental operators "convolution". As a very simple case, we derived a very simple and beautiful convolution inequality which is different from the famous Young inequality in the convolution.(5) In the viewpoint of conformal mappings, we found very nice representation formulas and their error estimates in the representations of analytic functions in terms of their local deta, using the Riemannn mapping function.(6) We continued the research for real inversion formulas of the Laplace transform from three directions ; that is, uniformly convergence formulas, error estimates and conditional stability.