?(2) b This formula served as the basis for the derivation of various precise analytical expressions for the Elsasser integral [2-4]. In [2] the solution was found in the form of a series which converges only for chz < 2. However, for large values of a the series converges poorly and, in order to calculate the integral," calculations must be made with a large number of decimal places. A much simpler analytical expression was obtained in [4] for the integral by means of the expansion of the Bessel function I0, which enters into (2) in a power series. A comparison of the results of our calculations, which were made on the B]~SM-4 computer using the formulas in [4], with table values of the integral [5] showed that for the region 0 ~ --log (z/2~r) -- + 2.4 and --1.5 - logu <- 1.5 this analytical expression gives six true places after the decimal point, except for a small number of values for the integral. Further, our calculations, using the new formulas which we introduced, showed that the indicated deviations are explained by the inaccuracy of the tabulated values of the integral as given in [5]. In this article a different, more general, method is developed for finding accurate, analytical expressions for the Elsasser integral, based on the term-by-term integration of the power series for the exponents. In fact, by expanding the exponent in a series and by changing the order of integration and summation, we write integral (1) in the form: