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Bessel K Calculator

K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 58/58 tests · Production surface contract 4/4 · v1.10.0
Input interpretation
Enter values to calculate.
Result
—
Model
Modified Bessel function of the second kind K_n(x), integer n, x>0.
Scope
Integer order n=0…8
Verification
Engine tested · 58/58 tests · Production surface contract 4/4 · Specification checked · v1.10.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
13 golden · 8 boundary · 37 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.10.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

SeriesK_n(x) = ½ (x/2)^{−n} Σ_{k<n} (−1)^k + (−1)^{n+1} I_n ln(x/2) + (−1)^n ½ (x/2)^n Σ ψ
Domainx > 0
iSame engine as /special-functions. Not a second engine. Not /kn. W_{-1}, incomplete beta, and normalized sinc later on this engine.

How to use

1

Enter n and x

Integer order n=0…8. Finite real x>0 with x≤80. Defaults n=0, x=1.

2

Read K_n(x)

K_0(1) ≈ 0.421024438241. K_1(1) ≈ 0.601907230197. x=0 is singular.

Example calculations

Common configurations with formula and result.

ϟ

K_0(1)

n = 0, x = 1

K_0(1)
0.421024438241
ϟ

K_1(1)

n = 1, x = 1

K_1(1)
0.601907230197

Bessel K calculator specification

Version 1.10.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The modified Bessel function of the second kind K_n(x) for integer order n=0…8 and x>0. Logarithmic singularity at x=0. Exponential decay for large x. This page is mode=bessel_k on math.special. Not W_{-1}. Not CAS.
What it calculates
Modified Bessel function of the second kind K_n(x), integer n, x>0.
Inputs
  • n
  • x
Outputs
  • value
  • n
  • x
Formula
K_n(x) = ½ (x/2)^{−n} Σ_{k<n} (−1)^k + (−1)^{n+1} I_n ln(x/2) + (−1)^n ½ (x/2)^n Σ ψ
Assumptions
  • Integer order n=0…8
  • Real x with 0 < x ≤ 80
  • Not W_{-1}
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • non-finite x → INVALID_NUMBER
  • n not in 0…8 → INVALID_INPUT
  • x ≤ 0 → INVALID_INPUT
  • x > 80 → INVALID_INPUT
  • unknown mode (W_{-1} / CAS) → INVALID_MODE
Example
n=0 x=1 → 0.421024438241
Validation cases

2 published on this page · 58/58 tests · Production surface contract 4/4 · View evidence

  • n=0 x=1 → 0.421024438241
  • n=1 x=1 → 0.601907230197
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.10.0

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT Calculator, Discrete Fourier Transform Calculator . Explore all Numerical Calculus.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator J_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → DFT Calculator Forward DFT X[k]=Σ x[n] e^{−2πi kn/N}, N≤64, unnormalized. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not inverse, not Laplace. Runs locally. › → Discrete Fourier Transform Calculator Direct DFT / IDFT and radix-2 FFT / IFFT of a length-N sequence, N≤64. FFT requires N=2^k. Numpy backward: unnormalized forward, 1/N inverse. Discovery /dft, /inverse-dft, /fft, /ifft. Not Laplace. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second special-functions engine?

No. It is mode=bessel_k on math.special, the same engine as /calc/math/special-functions.

Does this compute W_{-1}?

No. This seed is integer-order K_n for x>0. W_{-1} comes later on this engine. Not CAS.

Where does this run?

Locally in the browser by default. REST and MCP call the same special-functions engine.