This paper presents a method for translating asymptotic expansions of functions near dominant singularities into asymptotic expansions for their Taylor coefficients. The approach is based on contour integration and Cauchy's formula, and it provides an alternative to Darboux's method and Tauberian theorems. The method is particularly well-suited for combinatorial enumerations and has applications in areas such as the analysis of algorithms. The paper introduces the concept of singularity analysis of generating functions, which involves analyzing the behavior of functions near their dominant singularities to derive asymptotic information about their coefficients. The method is based on the assumption that the dominant singularity occurs at z = 1, and that the function has an asymptotic expansion of the form $ f(z) \approx (1 - z)^{\alpha} $ as $ z \to 1 $. The paper then presents a series of theorems and corollaries that demonstrate how the asymptotic behavior of the function can be translated into asymptotic behavior of its coefficients. These theorems are based on the use of contour integration and the analysis of the function's behavior near its dominant singularity. The results are applied to a variety of functions, including those with logarithmic and iterated logarithmic terms. The paper also discusses the relationship between singularity analysis and other methods such as Tauberian theorems and Darboux's method. It highlights the advantages of the singularity analysis approach, particularly in its ability to handle functions with more complex asymptotic behavior. The paper concludes with a discussion of the broader implications of the results, including their applications in combinatorial enumeration and the analysis of algorithms.This paper presents a method for translating asymptotic expansions of functions near dominant singularities into asymptotic expansions for their Taylor coefficients. The approach is based on contour integration and Cauchy's formula, and it provides an alternative to Darboux's method and Tauberian theorems. The method is particularly well-suited for combinatorial enumerations and has applications in areas such as the analysis of algorithms. The paper introduces the concept of singularity analysis of generating functions, which involves analyzing the behavior of functions near their dominant singularities to derive asymptotic information about their coefficients. The method is based on the assumption that the dominant singularity occurs at z = 1, and that the function has an asymptotic expansion of the form $ f(z) \approx (1 - z)^{\alpha} $ as $ z \to 1 $. The paper then presents a series of theorems and corollaries that demonstrate how the asymptotic behavior of the function can be translated into asymptotic behavior of its coefficients. These theorems are based on the use of contour integration and the analysis of the function's behavior near its dominant singularity. The results are applied to a variety of functions, including those with logarithmic and iterated logarithmic terms. The paper also discusses the relationship between singularity analysis and other methods such as Tauberian theorems and Darboux's method. It highlights the advantages of the singularity analysis approach, particularly in its ability to handle functions with more complex asymptotic behavior. The paper concludes with a discussion of the broader implications of the results, including their applications in combinatorial enumeration and the analysis of algorithms.