Bessel functions are the solutions of Bessel’s differential equation x²y″ + xy′ + (x² − ν²)y = 0. They describe vibrating drumheads, heat flow in cylinders, electromagnetic waves in circular waveguides and FM radio spectra.
How to use the Bessel Function Calculator
- Order ν
- Argument x
Results update as you type, and everything is calculated in your browser.
How it works
J is computed from its power series using the gamma function; Y uses the standard combination of J₊ν and J₋ν for non-integer orders and the limiting series (with harmonic numbers) for integer orders. The series is accurate for |x| up to about 20.
Tips
- Rewrite the equation in standard form before reading off the coefficients.
- Check your answer by substituting it back into the original equation.
- Keep exact forms (fractions and square roots) until the final step to avoid rounding errors.
- For inequalities, flip the sign whenever you multiply or divide by a negative number.
Common mistakes to avoid
- Dropping a negative sign when copying a coefficient.
- Forgetting the ± when taking a square root.
- Dividing by an expression that could be zero.
Inputs explained
| Field | What to enter |
|---|---|
| Order ν | A number |
| Argument x | A number |
What each result means
- J_ν(x) – first kind = besselj([Order ν], [Argument x])
- J_ν₊₁(x) = besselj([Order ν] + 1, [Argument x])
Formula
J_ν(x) = Σₖ (−1)ᵏ / (k! Γ(k + ν + 1)) · (x/2)^(2k+ν)Y_ν(x) = (J_ν(x) cos νπ − J₋ν(x)) / sin νπWorked example
Inputs — Order ν: 0; Argument x: 2.5.
Results — J_ν(x) – first kind = -0.04838378; Y_ν(x) – second kind (x > 0) = 0.49807036; J_ν₊₁(x) = 0.4970941.
Key terms
- Order ν
- The parameter in Bessel's equation.
- Gamma function
- The extension of the factorial used in the series.
Frequently asked questions
What are Bessel functions used for?
Problems with cylindrical symmetry: vibrating membranes, heat conduction in rods, waveguides and diffraction.
Why is Y undefined for x ≤ 0?
Y_ν(x) has a logarithmic singularity at 0 and is real only for positive x.
What is J₀(0)?
1. All other J_ν(0) with ν > 0 are 0.
What are the Hankel functions?
Complex combinations H⁽¹,²⁾ = J ± iY, used for travelling waves.
What formula does the Bessel Function Calculator use?
J_ν(x) = Σₖ (−1)ᵏ / (k! Γ(k + ν + 1)) · (x/2)^(2k+ν).