In this paper, it will be shown a generalization of the series of the exponential function using non-integer and negative exponents for the correspondent polynomial such as:e^x=⋯-Γ(7/2)/π x^(-7/2)+Γ(5/2)/π x^(-5/2)-Γ(3/2)/π x^(-3/2)+Γ(1/2)/π x^(-1/2)+1/Γ(3/2) x^(1/2)+1/Γ(5/2) x^(3/2)+1/Γ(7/2) x^(5/2)+⋯Being Γ(z) the gamma function. The series with negative exponents seems to diverge. But it will be demonstrated that they can be calculated using the following integral. This integral converges and therefore, it can be solved, leading for the exact result for the exponential function.e^x=x^(-1/2)/π ∫_0^∞〖(t^(-1/2) e^(-t))/(1+t/x) dt〗+1/Γ(3/2) x^(1/2)+1/Γ(5/2) x^(3/2)+1/Γ(7/2) x^(5/2)+⋯A generalization with complex exponents for the series will be shown in the paper.The Newton binomial theorem is also generalized this way (being w a free complex parameter):(m+1)^s=∑_(k=-∞@k=integer)^∞〖m^(k+w) Γ(s+1)/Γ(k+w+1)Γ(s-k-w+1) ==⋯+m^(-2+w) Γ(s+1)/Γ(-1+w)Γ(s+3-w) +m^(-1+w) Γ(s+1)/Γ(w)Γ(s+2-w) 〗+m^w Γ(s+1)/Γ(w+1)Γ(s+1-w) +m^(1+w) Γ(s+1)/Γ(w+2)Γ(s-w) +m^(2+w) Γ(s+1)/Γ(w+3)Γ(s-1-w) +⋯.It will be also shown that there are infinite solutions for the fractional derivative of a constant. But the only one that at the same time keeps the derivative of the exponential function being itself again, is the following:(d^(1/2) C)/(dx^(1/2) )=C/π ∫_0^∞〖(t^(-1/2) e^(-t) x^(-1/2))/(1+t/x) dt〗Again, a generalization for fractional derivatives of zth grade (being z a general complex number) of a constant will be shown.