In the present work, using the Carleman function, a harmonic function and its derivatives are restored by the Cauchy data on a part of the boundary of the region. It is shown that the effective construction of the Carleman function is equivalent to the construction of a regularized solution to the Cauchy problem. We assume that a solution to the problem exists and is continuously differentiable in a closed domain with precisely given Cauchy data. For this case, we establish an explicit formula for the continuation of the solution and its derivative, as well as the regularization formula for the case when under the indicated conditions instead of the initial Cauchy data their continuous approximations are given with a given error in the uniform metric. The stability estimates for the solution to the Cauchy problem in the classical sense are obtained.

在当前工作中，使用卡尔曼函数，通过柯西数据在该区域边界的一部分上恢复了谐波函数及其导数。结果表明，Carleman函数的有效构造等同于对Cauchy问题的正则解的构造。我们假设存在问题的解决方案，并且在封闭域中使用精确给出的柯西数据可以连续地求微分。对于这种情况，我们建立了解决方案及其导数连续性的显式公式，以及当在指定条件下代替初始柯西数据时，它们的连续近似值在给定误差下给出的情况下的正则化公式。统一指标。