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We have found a few interesting formulas in the course of long-term study of special function theory, especially approximate formulas related to mathematical constants and special functions. Special function theory has been widely applied not only in mathematical fields, including analytical, mathematical, stochastic, but also in physics, chemistry, statistical physics, computer science and engineering. Among the special functions used by many people in mathematics and engineering, Gamma functions are the most important, and research on Gamma functions and their applications is the main topic of many papers. One of the most important in the study of special function theory is to find more accurate approximate formulas related to special functions or mathematical constants. To this end, many mathematicians and scientists are making much effort to establish more precise approximate formulas for special functions and mathematical constants, and many good results have been presented. We systematically describe previous results by comprehensive analysis and present new approximate formulas superior to previous results.