# On Bessel function's orthogonality

1. Dec 13, 2012

### samuelandjw

Use the orthogonality relation of Bessel function to argue whether the following two integrals are zero or not:
$\displaystyle\int_0^1J_1(x)xJ_2(x)dx$
$\displaystyle\int_0^1J_1(k_1x)J_1(k_2x)dx$, where $$k_1,k_2$$ are two distinct zeros of Bessel function of order 1.

The textbook we are using is Boas's Mathematical Methods in the Physical Sciences, so the formulas that are available to us are a set of recursion relations of Bessel function, and the orthogonality relations between Bessel function of the same order: http://upload.wikimedia.org/math/c/d/b/cdb1e8ba98f7855eba9777024cce03fd.png.

For the first integral, the two Bessel functions are of different order, and there is no zeros in the arguments of the two functions, so I have no idea how to link the first integral to the orthogonality relation of Bessel functions.

For the second integral, my argument is that since $\displaystyle\int_0^1J_1(k_1x)xJ_1(k_2x)dx$ is zero, the 2nd integral (note that there is no x between the two Bessel function) cannot be zero. I think this argument is quite weak. Would anyone give me a better argument?

Thanks.

2. Dec 13, 2012

### lurflurf

The first one integral(positive function)=positive number

Last edited: Dec 13, 2012
3. Dec 13, 2012

### samuelandjw

Thanks for your reply. We can surely say that $$J_1(x),J_2(x)$$ are positive functions between x=0 and x=1 because the first nontrivial zeros are larger than 1. One has to somehow use the information of the location of zeros to reach this conclusion. Suppose we don't have this information, is it still possible to argue that the integral is non-zero?

4. Dec 14, 2012

### vijay0

5. Dec 15, 2012