Bessel Function Calculator
Calculate Bessel functions of first and second kind
Bessel functions are special mathematical functions that are solutions to Bessel's differential equation. They are named after Friedrich Wilhelm Bessel and are particularly important in solving problems involving cylindrical or spherical wave propagation. These functions commonly appear in various areas of physics and engineering.
First Kind (J)
Bessel functions of the first kind, denoted as Jn(x), are finite at x = 0 for non-negative n and are the most commonly used. They are defined by the series:
Jn(x) = Σ ((-1)^k / (k! * (n+k)!)) * (x/2)^(n+2k)
Second Kind (Y)
Bessel functions of the second kind, also known as Weber functions and denoted as Yn(x), are singular at x = 0. They are linearly independent solutions to Bessel's equation.
Bessel functions appear in many scientific and engineering applications:
- Wave propagation and vibration problems
- Heat conduction in cylindrical objects
- Electromagnetic radiation
- Signal processing and filter design
- Quantum mechanics and atomic physics
- Acoustics and sound wave propagation
Example 1: First Kind
- J0(0) = 1
- J0(π) ≈ -0.304
- J1(1) ≈ 0.440
Example 2: Second Kind
- Y0(1) ≈ 0.088
- Y1(1) ≈ -0.781
- Y2(2) ≈ -0.107
- Bessel functions are oscillatory and decay as x increases
- The zeros of Bessel functions are not evenly spaced
- They satisfy various recurrence relations and integral representations
- First kind functions are finite at x = 0 (except for negative orders)
- Second kind functions have a singularity at x = 0