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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 