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Math calculator

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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 15/15 declared partitions (erf, erfc, gamma, lgamma, bessel-j, bessel-y, bessel-i, bessel-k, lambert-w, lambert-wm1, beta, incomplete-beta, sinc, normalized-sinc, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Modified Bessel function of the second kind K_n(x), integer n, x>0, with an engine-linked Live graph of y = K_n(x).
Scope
Integer order n=0…8
Verification
Engine tested · Specification checked · v1.10.2 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.10.2 · CVP protocol 1.0.0-proposed · Evidence 2026-10-08.math-response-consistency
Verification revision
2026-10-08.math-response-consistency · 1/1 property · digest 0b2836762d99
Legacy regression
58/58 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Special response contract also checks all 14 mode formulas and unrounded numeric summaries, including erfc/K tails and a subnormal erf input; UI is shared presentation/SSR, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
482/482 golden · 8/8 CVP boundary · 10/10 invalid · 1/1 property · 3/3 metamorphic · 24/24 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
13 legacy golden · 8 legacy boundary · legacy regression suite · 482/482 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.10.2 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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. The Live graph is y = K_n(x) for x>0.

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 1.10.2 · 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, with an engine-linked Live graph of y = K_n(x).
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
  • The Result Card includes an engine-linked Live graph of y = K_n(x) (Overview 0.18 ≤ x ≤ 6; singularity at 0). Same engine as REST/MCP. 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 6/6 · 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
1.10.2

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel Y Calculator, Beta Function Calculator, Error Function Calculator, Gamma Function Calculator, Incomplete Beta Calculator, Lambert W Calculator . Explore all Special Functions.

→ 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 Calculate the Bessel function J_n(x) for integer orders 0–8 and real |x|≤80. See the formula, graph, numerical method, references, and versioned evidence. › → 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. › → Error Function Calculator erf(x) = (2/√π) ∫₀ˣ e^{−t²} dt. Same engine as Special Functions. Odd, erf(0)=0, erf(∞)→1. Not CAS. Runs locally. › → Gamma Function Calculator Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt except poles at 0, −1, −2, …. Same engine as Special Functions. Not factorial. Not CAS. Runs locally. › → Incomplete Beta Calculator Incomplete beta B_x(a, b) = ∫₀ˣ t^{a−1}(1−t)^{b−1} dt for a>0, b>0, 0≤x≤1. Same engine as Special Functions. Not normalized sinc. Not CAS. Runs locally. › → Lambert W Calculator Principal real W_0(x) for x ≥ −1/e. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. ›

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.