Dear RG members,
I have a doubt in my derivation of Bessel functions and modified Bessel functions of asymptotic expansions in the limit zero.
When I derive the asymptotic expansion of x*J'(m,x)/J(m,x) in the limit x tends to 0 for all m>0 I have the value m and in the modified Bessel functions
x*K'(m,x) /K(m,x) in the limit x tends to zero for all m>0 I have the value -m
I just want to know whether this values are correct or not.
In each of the above Bessel function numerator involving a first derivative.
Thanking you
This is the bible of mathematics! Everyone should have this reference--and it's free!
http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf
Use the corresponding hypergeometric function defined in terms of power series, and you immediately get the desired answer.
This is the bible of mathematics! Everyone should have this reference--and it's free!
http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf
Just use Mathematica:
In:
Series[(x D[BesselJ[m, x], x])/BesselJ[m, x], {x, 0, 2}]//Normal
Out:
Gamma[1 + m]/Gamma[m] - (
x^2 Gamma[1 + m] (m Gamma[m] + m^2 Gamma[m] + Gamma[2 + m]))/(
4 m (1 + m) Gamma[m] Gamma[2 + m])
In:
Series[(x D[BesselK[m, x], x])/BesselK[m, x], {x, 0, 2}] // Normal
Out:
-(((-1 + m) (1 + m) (x^(2 + 2 m) Gamma[-1 - m] +
4 (x^(2 m) Gamma[1 - m] + 4^m Gamma[1 + m])))/((-1 + m) x^(
2 m) (4 + 4 m + x^2) Gamma[-m] +
4^m (1 + m) (-4 + 4 m - x^2) Gamma[m]))
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics