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# Prove that the Bessel functions are solutions of the Bessel equation

In the previous exercise (Section 11.16, Exercise #10) we defined the Bessel functions of the first kind of orders zero and one by,

Prove that these Bessel functions are solutions of the differential equation

when and , respectively.

Proof. In the previous exercise (linked above) we proved the following

For the case we have the differential equation

Plugging in we then have

So, is indeed a solution in the case .

Now, from the previous exercise we have the relations

Starting with the case we differentiate,

Using the relations above from the previous problem, we substitute

Hence, is indeed a solution of the differentiation equation

# Prove some properties of the Bessel functions of the first kind of orders zero and one

We define the Bessel functions of the first kind of orders zero and one by

1. Prove that both and converge for all .
2. Prove that .
3. If we define two new functions

prove that .

1. Proof. For the order zero Bessel function of the first kind we have and so using the ratio test we have

Hence, converges for all .

For we have and so

Hence, converges for all

2. Proof. We compute the derivative of directly,

3. Proof. First, we have

On the other hand we have,

Therefore,

Hence, we indeed have