In this paper a solution of the direct Cauchy problems for heat equation is founded in the Hermite polynomial series form. A wellknown classical solution of direct problem is represented in the Poisson integral form. The author shows the formulas for the solution of the inverse Cauchy problems have a symmetry with respect to the formulas for the corresponding direct problems. The obtained solution formulas for the inverse problems can serve as a basis for regularizing computational algorithms while well-known classical formula for the solution of inverse problem did not possess such properties and can't be a basis for regularizing computational algorithms.
The article considers correctness of definition for the root of the n-th degree from a real number and the degree of a real number with an arbitrary real exponent. The article analyzes the definitions of these concepts in different school textbooks and university textbooks, and discusses the differences and contradictions that arise. The article solves exponential and power-exponential equations depending on those approaches that were chosen by the authors. The solution to the problem of correctly defining these concepts lies outside the school mathematics course and goes into the theory of analytic functions. The authors of the article suggest ways that, in their opinion, can be implemented when teaching mathematics.
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