Bessel functions: solutions to x²y''+xy'+(x²−n²)y=0. J_n(x) = Σ (-1)^k/(k!Γ(k+n+1))·(x/2)^{2k+n}. Arise in any problem with cylindrical symmetry: vibrating drums, heat flow in pipes, fiber optics.
How to Use
Enter order n and argument x:
- J_n(x): First kind
- Y_n(x): Second kind
- Plot: Function values
Physics
Vibrating circular drum: u(r,θ,t)=J_m(α_{mn}r/a)·{cos,sin}(mθ)·cos(ωt). Zeros of J_m determine resonant frequencies. FM synthesis uses J_n for modulation index spectra. Diffraction: Airy pattern = [2J_1(x)/x]².
Key Properties
Recurrence: J_{n-1}+J_{n+1}=2n/x·J_n. Orthogonality: ∫₀¹ rJ_n(α_{nk}r)J_n(α_{nl}r)dr = δ_{kl}J_{n+1}²(α_{nk})/2. Zeros interlace between orders.
Step-by-Step Instructions
- 1Enter order n.
- 2Enter x value.
- 3Compute J_n(x).
- 4Compute Y_n(x).
- 5View function values.