Deriving the Bessel Function of the First Kind for Zeroth Order

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

\displaystyle x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+(x^{2}-\nu^{2})y(x)=0, (1)

where \nu = 0 describes the order zero of Bessel’s equation. I shall be making use of the assumption

\displaystyle y(x)=\sum_{j=0}^{\infty}a_{j}x^{j+r}, (2)

where upon taking the first and second order derivatives gives us

\displaystyle \frac{dy}{dx}=\sum_{j=0}^{\infty}(j+r)a_{j}x^{j+r-1}, (3)

and

\displaystyle \frac{d^{2}y}{dx^{2}}=\sum_{j=0}^{\infty}(j+r)(j+r-1)a_{j}x^{j+r-2}. (4)

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

\displaystyle x^{2}\sum_{j=0}^{\infty}(j+r)(j+r-1)a_{j}x^{j+r-2}+x\sum_{j=0}^{\infty}(j+r)a_{j}x^{j+r-1}+x^{2}\sum_{j=0}^{\infty}a_{j}x^{j+r}=0. (5)

Distribution and simplification of Eq.(5) yields

\displaystyle \sum_{j=0}^{\infty}\bigg\{(j+r)(j+r-1)+(j+r)\bigg\}a_{j}x^{j+r}+\sum_{j=0}^{\infty}a_{j}x^{j+r+2}=0. (6)

If we evaluate the terms in which j=0 and j=1, we get the following

\displaystyle a_{0}\bigg\{r(r-1)+r\bigg\}x^{r}+a_{1}\bigg\{(1+r)r+(1+r)\bigg\}x^{r+1}+\sum_{j=2}^{\infty}\bigg\{[(j+r)(j+r-1)+(j+r)]a_{j}+a_{j-2}\bigg\}x^{j+r}=0, (7)

where I have introduced the dummy variable m=(j+r)-2 and I have shifted the indices downward by 2. Consider now the indicial equation (coefficients of a_{0}x^{r}),

\displaystyle r(r-1)+r=0, (8)

which upon solving gives r=r_{1}=r_{2}=0. We may determine the recurrence relation from summation terms from which we get

\displaystyle a_{j}(r)=\frac{-a_{j-2}(r)}{[(j+r)(j+r-1)+(j+r)]}=\frac{-a_{j-2}(r)}{(j+r)^{2}}. (9)

To determine J_{0}(x) we let r=0 in which case the recurrence relation becomes

\displaystyle a_{j}=\frac{-a_{j-2}}{j^{2}}, (10)

where j=2,4,6,.... Thus we have

\displaystyle J_{0}(x)=a_{0}x^{0}+a_{1}x+...  (11)

The only way the second term above is 0 is if a_{1}=0. So, the successive terms are a_{3},a_{5},a_{7},..., = 0. Let j=2k, where k\in \mathbb{Z}^{+}, then the recurrence relation is again modified to

\displaystyle a_{2k}=\frac{-a_{2k-2}}{(2k)^{2}}. (12)

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

\displaystyle ... \frac{(-1)^{k}a_{0}x^{2k}}{2^{2k}(k!)^{2}}. (13)

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

\displaystyle J_{0}(x)=a_{0}\bigg\{1+\sum_{k=1}^{\infty}\frac{(-1)^{k}x^{2k}}{2^{2k}(k!)^{2}}\bigg\}. (14)

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