**NOTE: I verified the solution using the following text: Boyce, W. and DiPrima, R. ***Elementary Differential Equations. *

In this post, I shall be deriving the Bessel function of the first kind for the zeroth order Bessel differential equation. Bessel’s equation is encountered when solving differential equations in cylindrical coordinates and is of the form

where describes the order zero of Bessel’s equation. I shall be making use of the assumption

where upon taking the first and second order derivatives gives us

and

Substitution into Eq.(1) and noting the order of the equation we arrive at

Distribution and simplification of Eq.(5) yields

If we evaluate the terms in which and , we get the following

where I have introduced the dummy variable and I have shifted the indices downward by 2. Consider now the indicial equation (coefficients of ),

which upon solving gives . We may determine the recurrence relation from summation terms from which we get

To determine we let in which case the recurrence relation becomes

where . Thus we have

The only way the second term above is 0 is if . So, the successive terms are . Let , where , then the recurrence relation is again modified to

In general, for any value of , one finds the expression

Thus our solution for the Bessel function of the first kind is

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