On the Construction of The Laplace Transform via Gamma Function

The Laplace transform can be applied to integrable and exponential-type functions on the half-line [0,├ ∞)┤by the formula L{f}=∫_0^∞▒〖f(x) e^(-sx) dx〗. This transform reduces differential equations to algebraic equations and solves many non-homogeneous differential equations. However, the Laplace transform cannot be applied to some functions such as x^(- 9/4), because the given integral is divergent. So, the Laplace transform do not solve some differential equations with some terms such as x^(- 9/4). This transform needs a revision to include such functions to solve a wider class of differential equations. In this study, we defined the Ω-Laplace transform, which eliminates such insufficiency of the Laplace transform and is a generalization of it. We applied this new operator to previously unsolved differential equations and obtained solutions. Ω-Laplace ensform given with the help of series: f(x)=∑_(n=0)^∞▒〖c_n x^(r_n ) 〗⇒Ω{f}=∑_(n=0)^∞▒(c_n Γ(r_n+1))/s^(r_n+1) Moreover, we give the similar and different properties of this transform to the Laplace transform.