# Differentiating Bessel Functions

1. Apr 4, 2006

### 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.

2. Apr 4, 2006

### Astronuc

Staff Emeritus
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.

3. Apr 4, 2006

### physkid

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

4. Apr 4, 2006

### Astronuc

Staff Emeritus
Some identities for Bessel's functions in terms of Jn(z), but also valid for Yn(z)

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

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

d J0(z) / dz = - J1(z)

Jn(z) Yn-1(z) - Jn-1(z) Yn(z) = 2/(pi z)

Jn(z) d Yn(z) / dz - d Jn(z) /dz Yn(z) = 2/(pi z)

5. May 2, 2006

### J77

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