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In this paper, an improvement for calculating the self-inductance of circular coils of rectangular cross section is presented. An integral of elementary functions is applied to replace the Bessel and Struve functions in the integrand of the inductance expression, and the term of improper integral without the exponential factor is solved to a closed form which consists of the complete elliptic integrals of the first and second kind and generalized hypergeometric functions. Due to this method, the main difficulties of the numerical evaluations of the inductance expression are overcome. Excellent agreement with published method is confirmed by the numerical comparisons and the calculations can be accelerated to several tens to hundreds of times depending on the shape factors of the coils. Similar expressions of the special cases (thin-wall solenoids and disk coils) are also given and compared with previous methods. Furthermore, the asymptotic behaviors of the special coils which tend to be a circular ring are obtained and numerically validated. [ABSTRACT FROM AUTHOR]