Wronskian of Bessel Functions of non-integral order v, -v

In summary, the textbook derivation of the equation J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = -\frac{2 \sin v \pi}{\pi x} is an identity for any order v and any variable x.
  • #1
mjordan2nd
177
1
My textbook states

[tex]
J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = -\frac{2 \sin v \pi}{\pi x}
[/tex]

My textbook derives this by showing that

[tex]
J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = \frac{C}{x}
[/tex]

where C is a constant. C is then ascertained by taking x to be very small and using only the first order of the power series expansion for Bessel functions. Does this mean that this computation for C is inexact? It seems that there should be some error terms in there from higher powers of x, or am I missing something?

By the way, I'm using Arfken/Weber and N.N. Lebedev as my guide here.

Thanks for any help.

Edit: Perhaps this would have been better in the differential equations section?
 
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  • #2
No it doesn't sound like an approximation.

If you have two power series that are equal for all x (in some interval), then their coefficients have to be equal at all orders. Strictly speaking these aren't power series since there is a term of order -1 but that doesn't change the fact that the coefficients on the right side have to match the coefficients on the left side. On the right side, C is the lowest order coefficient. So if you calculate the lowest order coefficient on the left side, it has to be equal to C.

The fact that the right side has only the one term C/x means that all the higher order terms cancel each other out on the left side.
 
  • #3
What math course is this? Just wondering
 
  • #4
[tex]J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = -\frac{2 \sin v \pi}{\pi x} [/tex]
is not an approximate. It is an identity for any variable x and any order v

Considering a constant order v, then [tex] C=-\frac{2 \sin v \pi}{\pi} [/tex] is constant. hence [tex] J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = \frac{C}{x} [/tex]
 
  • #5
I think I understand! Thank you for your explanations.

Pierce: This is my mathematical methods for physicists course.
 

1. What is the Wronskian of Bessel Functions of non-integral order v and -v?

The Wronskian of Bessel Functions of non-integral order v and -v is a mathematical quantity that measures the linear independence of two solutions to a differential equation. In this case, the Wronskian is specifically used to determine the linear independence of the Bessel Functions of non-integral order v and -v.

2. Why is the Wronskian important in studying Bessel Functions of non-integral order v and -v?

The Wronskian is important because it allows us to determine if the Bessel Functions of non-integral order v and -v are linearly independent, which is necessary for finding a complete set of solutions to a differential equation. Additionally, the Wronskian can provide information about the behavior of the Bessel Functions and their derivatives.

3. How do you calculate the Wronskian of Bessel Functions of non-integral order v and -v?

The Wronskian of Bessel Functions of non-integral order v and -v can be calculated using a formula that involves taking the determinant of a matrix containing the Bessel Functions and their derivatives. This formula can be found in most mathematics textbooks or online resources.

4. Can the Wronskian of Bessel Functions of non-integral order v and -v be zero?

Yes, it is possible for the Wronskian of Bessel Functions of non-integral order v and -v to be zero. If the Wronskian is zero, it indicates that the Bessel Functions are linearly dependent and therefore not a complete set of solutions to the differential equation.

5. What does the value of the Wronskian of Bessel Functions of non-integral order v and -v tell us?

The value of the Wronskian can tell us if the Bessel Functions of non-integral order v and -v are linearly independent or not. A non-zero Wronskian indicates linear independence, while a zero Wronskian indicates linear dependence. The Wronskian can also provide information about the behavior of the Bessel Functions and their derivatives.

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