Bessel Function Calculator

Calculator

Function type

For spherical options, order is rounded to an integer n ≥ 0.

Order (ν or n)

Y and K are most stable when ν is not an integer.

Argument (x)

For real-valued Y, K, and spherical y, use x > 0.

Decimal precision

Higher precision may need more iterations.

Max iterations

Limits the series/identity evaluation loop.

Your last 200 computations are kept in this browser session.

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Time	Function	Order	x	Value	Precision	Notes
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Function	Order	x	Expected type	Typical use
Jν(x)	0	2.50	Oscillatory	Waves in circular domains
Yν(x)	0.50	3.00	Singular near 0	Second independent solution
Iν(x)	1	1.20	Grows for large x	Diffusion and heat kernels
Kν(x)	0.75	1.20	Decays for large x	Radial decay, screening
jₙ(x)	2	4.00	Oscillatory	Spherical wave expansions
yₙ(x)	2	4.00	Singular near 0	Second spherical solution

These rows are examples only; your computed values appear in the Saved Results table above.

Power-series definitions

Jν(x)= Σ (-1)^k (x/2)^{2k+ν} / (k! Γ(k+ν+1))
Iν(x)= Σ (x/2)^{2k+ν} / (k! Γ(k+ν+1))

Γ is computed using a Lanczos approximation for real inputs.

Identity-based forms

Yν(x) = (Jν(x)cos(νπ) − J−ν(x)) / sin(νπ)
Kν(x) = (π/2) (I−ν(x) − Iν(x)) / sin(νπ)

When ν is near an integer, sin(νπ) is small, so the calculator applies a tiny ν-shift for stability.

Spherical mapping: jₙ(x)=√(π/(2x))·J_{n+1/2}(x) and yₙ(x)=√(π/(2x))·Y_{n+1/2}(x).

For extreme inputs (very large |x| or high orders), specialized numerical libraries can provide higher stability.