Bessel Function Calculator

Math

Calculate the Bessel function of the first kind J₀(x) and J₁(x) using series expansion.

Formula

Jₙ(x) = Σ (−1)ᵐ / (m! Γ(m+n+1)) × (x/2)^(2m+n)
  • Bessel functions are solutions to Bessel's differential equation.
  • Used in physics for cylindrical waves, heat conduction, vibrations.
  • Convergence is fast — 20 terms gives excellent accuracy for |x| ≤ 20.

J₀(0) = 1, J₀(2.405) ≈ 0

Inputs
  • x value: 0
  • Order: 0

J₀(0) = 1 by definition. The first zero of J₀(x) occurs at x ≈ 2.405.

Frequently asked questions

What are Bessel functions used for?
Cylindrical wave propagation, heat conduction in cylinders, drumhead vibration modes, and many physics problems with cylindrical symmetry.
What is J₀(0)?
J₀(0) = 1 and Jₙ(0) = 0 for n ≠ 0.
What are the zeros of J₀?
J₀(x) = 0 at x ≈ 2.405, 5.520, 8.654, ...
What is the difference between J and Y Bessel functions?
Jₙ is finite at x=0 (first kind); Yₙ diverges at x=0 (second kind/Neumann functions).