Symbolic differentiation of Bessel functions is incorrect

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Sam Spedding
Sam Spedding 2021-3-11
评论： David Goodmanson 2022-5-16
The output of the below code gives an incorrect equation for the derivative of the modified Bessel function of the second kind.
syms x n
y = besselk(n,x);
diff(y,x)
It says the derivative of is
but as I understand, the formula for the derivative of the K bessel functions is given by
.
What's going on?!?

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David Goodmanson
David Goodmanson 2021-3-11
编辑：David Goodmanson 2021-3-11
Hi Sam,
There is really nothing going on. Both of those identities are correct, as you can check numerically. There are several recurrence relations available for Bessel functions. Another one is
K'(n,x) = -nK(n,x)/x - K(n-1,x)

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Selçuk Sehitoglu
Selçuk Sehitoglu 2022-5-15
Hello,
i am dealing with the same problem for besselj.
syms nu x
y=besselj(nu,x);
d_y=diff(y,x);
subs(d_y,x,0); % At this row i get "Division by zero." error, because derivative is defined as -nJ(n,x)/x - J(n-1,x) ans it is undefined at x=0.
However, the answer is available with the following expression:
-(J(n-1,x) - J(n+1,x))/2
Is there a way to use this expression?
Thanks in advance,
Ozi
  1 个评论
David Goodmanson
David Goodmanson 2022-5-16
Hi Ozi,
Yes there is. The expression works as is. It looks like n = 0 might be an exception because you will need J(-1,x) but for real n the besselj function works for negative n.

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