Verified calculation capability

Special functions Calculation Capability

Real erf, erfc, Gamma, log-gamma, integer-order J_n, Y_n, I_n, and K_n, Lambert W_0 and W_{-1}, Euler beta B(x,y), incomplete beta B_x(a,b), unnormalized sinc, and normalized sinc. Discovery /error-function, /gamma-function, /bessel-function, /bessel-y, /bessel-i, /bessel-k, /lambert-w, /lambert-w-minus-1, /beta-function, /incomplete-beta, /sinc-function, /normalized-sinc. Not factorial, not CAS. · Math

Capability ID math.special · version 1.10.0

Open calculator REST API MCP Evidence Evidence JSON Review policy

Model

Error function

Error function: erf(x) = (2/√π) ∫₀ˣ e^{−t²} dt
Gamma: Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt
Log-gamma: lgamma(x) = ln Γ(x), x > 0
Bessel J: J_n(x) = (x/2)^n Σ_k (−1)^k (x/2)^{2k} / (k! (n+k)!)
Bessel Y: Y_n(x) Neumann, x > 0
Bessel I: I_n(x) = (x/2)^n Σ_k (x/2)^{2k} / (k! (n+k)!)
Bessel K: K_n(x) Macdonald, x > 0
Lambert W: W(x) e^{W(x)} = x
Lambert W₋₁: W_{-1}(x) e^{W_{-1}(x)} = x, −1/e ≤ x < 0
Beta: B(x, y) = Γ(x)Γ(y) / Γ(x+y)
Incomplete beta: B_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt
sinc: sinc(x) = sin(x)/x, sinc(0)=1
Normalized sinc: sinc(x) = sin(πx)/(πx), sinc(0)=1

Solve modes: erf · erfc · gamma · lgamma · bessel j · bessel y · bessel i · bessel k · lambert w · lambert wm1 · beta · incomplete beta · sinc · normalized sinc

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`x`	—	—	input / output
`y`	—	—	input / output
`a`	acceleration	—	input / output
`b`	—	—	input / output
`n`	amount of substance	—	input / output
`value`	—	—	output
`regularized`	—	—	output
`erf`	—	—	output
`erfc`	—	—	output
`gamma`	—	—	output
`lgamma`	—	—	output
`branch`	—	—	output
`algorithm`	—	—	output
`formula`	—	—	output

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

Real argument
Gamma poles at non-positive integers
lgamma is real for x > 0
Bessel J integer n=0…8, |x|≤80
Bessel Y integer n=0…8, x>0
Bessel I integer n=0…8, |x|≤80
Bessel K integer n=0…8, x>0
Lambert W_0 for x ≥ −1/e
Lambert W_{-1} for −1/e ≤ x < 0
Complete Euler B(x,y) for x>0, y>0
Incomplete B_x(a,b) for a>0, b>0, 0≤x≤1
Unnormalized sinc(x)=sin(x)/x, sinc(0)=1
Normalized sinc(x)=sin(πx)/(πx), sinc(0)=1
Not factorial
Not CAS

Example

One numerical check. The calculator page holds worked steps and FAQ.

mode=erf x=1 → 0.84270079295

Open this example in the calculator

Trust

Review-policy: engine tested — automated tests, sources, version lock, and evidence published. Supplemental to CVP; not Domain Expert Review. See CVP and Review policy.

Tier: Engine tested — catalog token engine_tested
Calculation version: 1.10.0
Evidence tests: 58/58 calculation tests · golden 13/13 · boundary 8/8 · property 37/37
Engine tested: Yes
Source checked: Yes
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.10.0 (workspace artifacts agree)

Production attestation: 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
Evidence attested at: 2026-09-18T21:41:55.460Z
Last production check: 2026-09-18T21:41:55.460Z
Public / cache: —
Origin: —

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 4/4.

Tool id: special-functions
Engine: special-functions
REST: POST /api/v1/calc/special-functions
MCP: execute_calculation with capability_id: "math.special"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.10.0)
Versioned output schema: Immutable pin (1.10.0)

References

Human-readable sources plus CVP reference evidence classes.

Sources

ISO 80000-2:2019 Quantities and units — Mathematics

Try with an agent

Inspect this capability via same-origin MCP, then call execute_calculation.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.special"}}}'

Inspect the trust tier and required assurance level before execution. Pin calculation_version (1.10.0) when reproducibility matters. Full protocol notes: MCP docs.