Differentiating Bessel Functions

[physkid]

Hi all,

I was just wondering if anyone knew how to differentiate Bessel functions of the second kind? I've looked all over the net and in books and no literature seems to address this problem. I don't know if its just my poor search techniques but any assistance would be appreciated.

 

[Astronuc]

Try here -

http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html

if one can differentiate Bessel's function of first kind, then one can differentiate Bessel's function of first kind.

See bottom of -
http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

or try to find there references

Abramowitz, M. and Stegun, I. A. (Eds.). "Bessel Functions and ." §9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358-364, 1972.

Arfken, G. "Neumann Functions, Bessel Functions of the Second Kind, ." §11.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596-604, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 625-627, 1953.

Otherwise, I may have a reference elsewhere that might have exactly what you need.

 

[physkid]

I knew there'd be some web references to make me look silly. Thank you very much for your help.

 

[Astronuc]

Some identities for Bessel's functions in terms of J_n(z), but also valid for Y_n(z)

(2n/z) J_n(z) = J_n-1(z) + J_n+1(z),

2 d[J_n(z)]/dz = J_n-1(z) - J_n+1(z),

d J₀(z) / dz = - J₁(z)

J_n(z) Y_n-1(z) - J_n-1(z) Y_n(z) = 2/(pi z)

J_n(z) d Y_n(z) / dz - d J_n(z) /dz Y_n(z) = 2/(pi z)

 

[J77]

I'd add:

$$J_{-n}(x)=(-1)^nJ_n(x)$$

Pretty important if you want to have the differentiated form in terms of higher order functions.