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Math calculator

Bessel Y Calculator

Evaluate Bessel Y functions for positive x and orders 0–8. Explore a live graph, worked examples, input limits and shareable results computed 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
Bessel function of the second kind Y_n(x), integer n, x>0, with an engine-linked Live graph of y = Y_n(x).
Scope
Integer order n=0…8
Verification
Engine tested · Source checked · v1.10.6 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.10.6 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.metric-scope-v3
Verification revision
2026-10-28.metric-scope-v3 · 1/1 property · digest c70f9a6399c7
Legacy regression
57/58 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 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
1000/1000 golden · 8/8 CVP boundary · 24/24 invalid · 1/1 property · 3/3 metamorphic · 24/24 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
13 legacy golden · 8 legacy boundary · legacy regression suite · 1000/1000 oracle-backed golden · 24/24 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.10.6 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.

NeumannY_n(x) = (2/π) J_n(x) ln(x/2) − (x/2)^{−n}/π Σ_{k<n} − (x/2)^n/π Σ ψ
Domainx > 0
iSame engine as /special-functions. Not a second engine. Not /yn. 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 Y_n(x)

Y_0(1) ≈ 0.0882569642157. Y_1(1) ≈ −0.7812128213. x=0 is singular. The Live graph is y = Y_n(x) for x>0.

Example calculations

Common configurations with formula and result.

ϟ

Y_0(1)

n = 0, x = 1

Y_0(1)
0.0882569642157
ϟ

Y_1(1)

n = 1, x = 1

Y_1(1)
−0.7812128213

Bessel Y calculator specification

Version 1.10.6 · Engine tested

Calculation status

Review policy · Evidence

Definition
The Bessel function of the second kind (Neumann) Y_n(x) for integer order n=0…8 and x>0. Logarithmic singularity at x=0. This page is mode=bessel_y on math.special. Not W_{-1}. Not CAS.
What it calculates
Bessel function of the second kind Y_n(x), integer n, x>0, with an engine-linked Live graph of y = Y_n(x).
Inputs
  • n
  • x
Outputs
  • value
  • n
  • x
Formula
Y_n(x): small-argument Neumann series or integral quadrature
Assumptions
  • Integer order n=0…8
  • Real x with 0 < x ≤ 80
  • Singular at x=0
  • The Result Card includes an engine-linked Live graph of y = Y_n(x) (Overview 0.18 ≤ x ≤ 12; singularity at 0). Same engine as REST/MCP. Not W_{-1} (I_n is on /bessel-i; K_n is on /bessel-k). Not CAS.
Units
  • dimensionless
Boundary conditions
  • non-finite x → INVALID_NUMBER
  • n not in 0…8 → INVALID_INPUT
  • x ≤ 0 or x > 80 → INVALID_INPUT
  • unknown mode (K / CAS) → INVALID_MODE
Example
n=0 x=1 → 0.0882569642157
Validation cases

2 published on this page · 57/58 tests · Production surface contract 6/6 · View evidence

  • n=0 x=1 → 0.0882569642157
  • n=1 x=1 → −0.7812128213
Sources
Calculation version
1.10.6

Background

Interpretation and common distinctions.

What does this bessel y calculator compute?

The Bessel function of the second kind (Neumann) Y_n(x) for integer order n=0…8 and x>0. Logarithmic singularity at x=0.

The published formula or method is Y_n(x): small-argument Neumann series or integral quadrature. The following references support the stated mathematics or method:

Which inputs and limits should you check?

The mathematical function can have a wider domain than this calculator supports. Choose the intended function and branch, then check its input domain and published precision limits.

The worked example uses n = 0, x = 1 with Y_0(1). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.special v1.10.6 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical precision and input policy

  • Real, dimensionless binary64 inputs; publishes12 significant digits, no arbitrary-precision/exact claim.
  • Default mode erf and Bessel order0 apply only when absent; explicit empty/null rejects. All supplied numeric aliases must agree; known irrelevant numeric parameters are parsed then ignored.
  • Gamma poles reject. Nonzero Gamma that underflows rejects NUMERICAL_RANGE_ERROR; positive overflow suggests lgamma. lgamma is computed directly for x>0, using a local Taylor expansion near its zeros1 and2.
  • Bessel integer n0–8, |x|≤80, Y/K require x>0; subnormal error and zeros limit relative accuracy.
  • Lambert W0 and W-1 explicit real branches; binary64 -1/e designated endpoint, no tolerance band absorbs adjacent inputs.
  • Beta shapes positive and at most1e6, except exact B(x,1)=1/x or B(1,y)=1/y. Incomplete beta shapes at most1e6. CF capped10000 steps; failure NO_CONVERGENCE.
  • erf/erfc and beta tails can round to zero only on binary64 underflow, not an absolute display cutoff. Regularized incomplete beta evaluated separately in logs.
  • Normalized sinc uses sinpi reduction and returns zero at representable nonzero integer x. Other tiny nonzero results retain significance.
  • Numeric references have relative1e-10, with J/Y near zeros absolute2e-14,4 minimum subnormals otherwise. No global error proof for all real inputs.
  • Browser interaction computes locally; parameterized share pages, REST and MCP send inputs to server. Logs/retention/region are deployment concerns; no zero-log promise.

Discovery pages lock their advertised function; the hub selects all14 operations. A shared link replays with the current engine and identifies its calculation version; it does not preserve a historical runtime. CSV and JSON exports retain the original decimal inputs, complete function result, formula and version.

Definitions and methods: NIST DLMF Gamma, incomplete beta, and Lambert W. CVP verification is separate from expert review and production freshness.

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

→ Bessel I Calculator Evaluate modified Bessel I functions for integer orders 0–8. Explore a live graph, worked examples, input limits and shareable results computed locally. › → Bessel J Calculator Evaluate integer-order Bessel J functions for orders 0–8. Explore a live graph, worked examples, input limits and shareable results computed locally. › → Bessel K Calculator Evaluate modified Bessel K functions for positive x, orders 0–8. Explore a live graph, worked examples, input limits and shareable results computed locally. › → Beta Function Calculator Calculate the complete Euler beta function B(x,y). Explore a live graph, worked examples, input limits and shareable results computed locally. › → Error Function Calculator Calculate erf(x) and the complementary error function erfc(x). Explore a live graph, worked examples, input limits and shareable results computed locally. › → Gamma Function Calculator Calculate Gamma values for real inputs away from its poles. Explore a live graph, worked examples, input limits and shareable results computed locally. › → Incomplete Beta Calculator Calculate incomplete beta Bx(a,b) and regularized Ix(a,b). Explore a live graph, worked examples, input limits and shareable results computed locally. › → Lambert W Calculator Calculate the principal real Lambert W branch W0(x). Explore a live graph, worked examples, input limits and shareable results computed locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second special-functions engine?

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

Does this compute I or K?

I_n is on /calc/math/bessel-i. K_n is on /calc/math/bessel-k. This page is integer-order Y_n for x>0. Not CAS.

Where does this run?

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