New approximations to J₀ and J₁ Bessel functions

New approximate solutions to the 0th- and 1st order Bessel functions of the first kind are derived. The formulations are based upon using a new integral with no previously known solution. The new integral in the limiting case is identical to the 0th-order Bessel function integral. It is solved in closed form, and the solution is expressed as a simple even order polynomial with integer coefficients. The polynomial coefficients are all of integer value. The 1st-order Bessel function approximation can then be found through a simple derivative. Comparisons are made between the exact solution, classic solutions, and the new approximation. The new approximation proves to be much more accurate than the classic small argument approximation. It is also sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing Bessel functions