Series Expansion of Bessel Function of the First Kind

Theorem
Let $n \in \Z_{\ge 0}$ be a (strictly) non-negative integer.

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.

Then:

Proof
We employ Frobenius's Method to find the solutions to the Bessel's Equation:


 * $x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {x^2 - n^2} y = 0$

for $n \ge 0$, in the form:


 * $\displaystyle \map y x = \sum_{k \mathop = 0}^\infty A_k x^{k + r}$

defined on $x > 0$, for some constants $r, A_i$, with $A_0 \neq 0$, which are to be determined.

Differentiating the expression $x$:

Substituting $y, y', y''$ into Bessel's Equation:

Comparing the constant term on both sides:

Take $r = n$. Comparing the rest of the coefficients:

From the recurrence relation above, we see that $A_k = 0$ for odd $k$, and:

Substituting this result to our original equation:

Also see

 * Bessel Function of the First Kind of Negative Integer Order for when $n \in \set {-1, -2, -3, \ldots}$