ERRORS OF COMPLEX VARIABLE FUNCTION NUMERICAL INTEGRATION METHOD

Abstract

The purpose of work of this work is to obtain and analyze the relations for errors that occur during the numerical integration of functions of a complex variable. The research methods has been used are: mathematical analysis, the theory of approximate calculations and the theory of functions of a complex variable. As a research results, we obtain analytical relations for calculating the errors of complex variable function numerical integration: the modulus of local error at each integration step, as well as the true, approximate and upper limit values of the modulus of errors on the integration contour. Verification of the obtained formulas of absolute errors of complex variable function numerical integration was carried out on the example of calculation of an integral which has an analytical solution. The dependences of the errors on the module of the integration step has been constructed and analyzed. For small values of the step modulus, the true value of the error modulus corresponds well to the approximation. For the maximum value of the modulus of the step, which corresponds to the calculation of the integral for one step, the approximate value of the modulus of error coincides with the value of the estimate of its upper threshold value. The graph of the dependence of the logarithm of the module of the upper limit value of the error on the logarithm of the module of the integration step is linear. The errors of the numerical method for calculating the transient function of an automatic control system with a PD-PID-controller has been calculated. A comparison of the results obtained by analytical and numerical methods shows a good correspodnence of the numerical solution with the analytical within the calculated errors. The scientific novelty of the work is to obtain analytical relations for calculating the errors of complex variable function numerical integration: the modulus of local error at each integration step, as well as the approximate and upper limit of the modulus of errors in the integration contour. The practical significance of work results is ability to obtain predetermined errors of complex variable function numerical integration by choosing the integration step, which can be used to improve the quality of transients analysis in electrotechnical and automatic control systems.