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 transformation 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. In this thesis, 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 transform given with the help of series: f(x)=∑_(n=1)^∞▒〖c_n x^(r_n ) 〗⇒Ω{f}=∑_(n=1)^∞▒(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.