Bessel Function Calculator
Calculate Bessel functions of the first, second, and third kind (Hankel functions)
Calculate Bessel Functions
Order must be a real number between -99 and 99
Value must be between -20 and 20 for accurate computation
Bessel Function Results
First Kind: J_ν(x)
Bessel function of the first kind
Second Kind: Y_ν(x)
Bessel function of the second kind
Hankel H_ν^(1)(x)
Hankel function of the first kind
Hankel H_ν^(2)(x)
Hankel function of the second kind
Formula Information:
J_ν(x): Calculated using power series expansion
Y_ν(x): Calculated using relationship with J_ν(x) and special formulas for integer orders
H_ν^(1)(x): J_ν(x) + i·Y_ν(x)
H_ν^(2)(x): J_ν(x) - i·Y_ν(x)
Example Calculations
J₀(0) = 1
Order: ν = 0
Input: x = 0
Result: J₀(0) = 1 (maximum value of Bessel function of first kind)
J₁(1) ≈ 0.440
Order: ν = 1
Input: x = 1
Result: Common value used in physics and engineering
J₁/₂(π) ≈ 0.450
Order: ν = 0.5 (half-integer order)
Input: x = π ≈ 3.14159
Result: Half-integer Bessel functions are related to elementary functions
Bessel Function Types
First Kind
J_ν(x) - Regular at origin
Finite for all finite x
Second Kind
Y_ν(x) - Singular at origin
Goes to -∞ as x → 0⁺
Third Kind
H_ν^(1,2)(x) - Hankel functions
Complex-valued combinations
Key Properties
Satisfy Bessel differential equation
Oscillatory behavior for large x
J₋ₙ(x) = (-1)ⁿJₙ(x) for integer n
Y₋ₙ(x) = (-1)ⁿYₙ(x) for integer n
Used in physics, engineering, and signal processing
Understanding Bessel Functions
What are Bessel Functions?
Bessel functions are solutions to Bessel's differential equation, which arises in many physical problems involving cylindrical or spherical symmetry. They appear in wave propagation, heat conduction, and vibrations of circular membranes.
The Bessel Differential Equation
x²(d²y/dx²) + x(dy/dx) + (x² - ν²)y = 0
This second-order differential equation has two linearly independent solutions: Bessel functions of the first and second kind.
Calculation Methods
First Kind (J_ν)
J_ν(x) = Σ[k=0 to ∞] (-1)ᵏ/(Γ(k+1)Γ(k+ν+1)) (x/2)^(2k+ν)
Power series expansion using gamma function
Second Kind (Y_ν)
Y_ν(x) = [J_ν(x)cos(νπ) - J₋_ν(x)]/sin(νπ)
For non-integer ν; special formulas for integer orders
Hankel Functions
H_ν^(1)(x) = J_ν(x) + iY_ν(x)
H_ν^(2)(x) = J_ν(x) - iY_ν(x)
Complex combinations useful for wave problems
Computational Limitations
• Order ν must be within [-99, 99] for computational efficiency
• Real part of x should be within [-20, 20] for numerical accuracy
• Large arguments require specialized asymptotic formulas
• Complex inputs are supported but may have reduced accuracy