Symbolic differentiation of Bessel functions is incorrect

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?!?

Answer 1

[David Goodmanson]

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)   

Answer 2

[Selçuk Sehitoglu]

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

[David Goodmanson]

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.