Applications of analyticoty of functions

(1)关于数学，我们讨论并给出了数学的定义和数学中的好成绩。特别是，数学的本质被认为是关系。作为例子，我们建立了“函数的解析性”和“非线性变换”相结合的方法，并得到了各种具体的结果。(2)找到了一种讨论一般变系数线性微分方程解的存在性的方法。对于许多依赖时间的线性偏微分方程解，我们找到了用它们在空间和时间上的局部DETA来表示解的一般原理。对于许多椭圆型线性偏微分方程组，我们都能得到类似的结果。这一令人满意的结果被命名为“望远镜原理”。(4)通过变换，我们发现了在Hilbert空间中引入各种算子的一般原理。特别是，我们能够给出基本运算符“卷积”的一般定义。作为一个非常简单的情形，我们导出了一个非常简单而美丽的卷积不等式，它不同于著名的Young卷积不等式。(5)从共形映射的角度，利用黎曼映射函数，我们在解析函数的局部Deta表示中找到了非常好的表示公式及其误差估计。(6)从一致收敛公式、误差估计和条件稳定性三个方面继续研究了拉普拉斯变换的实逆公式。

(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.