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Special Functions Calculator

Real erf, erfc, Gamma, log-gamma, Bessel J_n, Y_n, I_n, and K_n, Lambert W_0 and W_{-1}, Euler beta, incomplete beta, 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. 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
Real erf, erfc, Gamma, log-gamma, integer-order J_n, Y_n, I_n, and K_n, Lambert W_0 and W_{-1}, complete Euler beta, incomplete beta, unnormalized sinc, and normalized sinc.
Scope
Real argument
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Embedded snapshot: unpublished · Build schema 1.10.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

Error functionerf(x) = (2/√π) ∫₀ˣ e^{−t²} dt
GammaΓ(x) = ∫₀^∞ t^{x−1} e^{−t} dt
Log-gammalgamma(x) = ln Γ(x), x > 0
Bessel JJ_n(x) = (x/2)^n Σ_k (−1)^k (x/2)^{2k} / (k! (n+k)!)
Bessel YY_n(x) Neumann, x > 0
Bessel II_n(x) = (x/2)^n Σ_k (x/2)^{2k} / (k! (n+k)!)
Bessel KK_n(x) Macdonald, x > 0
Lambert WW(x) e^{W(x)} = x
Lambert W₋₁W_{-1}(x) e^{W_{-1}(x)} = x, −1/e ≤ x < 0
BetaB(x, y) = Γ(x)Γ(y) / Γ(x+y)
Incomplete betaB_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt
sincsinc(x) = sin(x)/x, sinc(0)=1
Normalized sincsinc(x) = sin(πx)/(πx), sinc(0)=1
ierfc and lgamma are hub modes. Discovery pages lock erf, gamma, Bessel J, Bessel Y, Bessel I, Bessel K, Lambert W, Lambert W₋₁, Beta, incomplete beta, sinc, and normalized sinc. Γ(n)=(n−1)! for positive integers, but n! stays on /factorial. Not CAS.

How to use

1

Choose erf, erfc, Gamma, log-gamma, Bessel J, Bessel Y, Bessel I, Bessel K, Lambert W, Lambert W₋₁, Beta, incomplete beta, sinc, or normalized sinc

erf and erfc take any finite real x. Gamma is undefined at non-positive integers. log-gamma is real for x>0 on this seed. Bessel J and I need integer n=0…8 and |x|≤80. Bessel Y and K need integer n=0…8 and x>0. Lambert W_0 needs x ≥ −1/e. Lambert W_{-1} needs −1/e ≤ x < 0. Beta needs x>0, y>0. Incomplete beta needs a>0, b>0, 0≤x≤1. Unnormalized sinc(0)=1. Normalized sinc(1/2)=2/π.

2

Read the value

erf(1) ≈ 0.84270079295. Γ(1/2)=√π. Γ(6)=120. J_0(1) ≈ 0.765197686558. Y_0(1) ≈ 0.0882569642157. I_0(1) ≈ 1.26606587775. K_0(1) ≈ 0.421024438241. W_0(1) ≈ 0.56714329041. W_{-1}(−0.1) ≈ −3.57715206396. B(2,3)=1/12. B_{0.5}(2,3) ≈ 0.0572916666667. sinc(0)=1. normalized sinc(1/2)=2/π.

Example calculations

Common configurations with formula and result.

ϟ

erf(1)

x = 1

erf(1)
0.84270079295
ϟ

Half-integer Gamma

x = 0.5

Γ(1/2) = √π
√π
ϟ

Gamma of 6

x = 6

Γ(6) = 5!
120
ϟ

J_0(1)

n = 0, x = 1

J_0(1)
0.765197686558
ϟ

Y_0(1)

n = 0, x = 1

Y_0(1)
0.0882569642157
ϟ

I_0(1)

n = 0, x = 1

I_0(1)
1.26606587775
ϟ

K_0(1)

n = 0, x = 1

K_0(1)
0.421024438241
ϟ

W_0(1)

x = 1

W_0(1) = Ω
0.56714329041
ϟ

W_{-1}(−0.1)

x = −0.1

W_{-1}(−0.1)
−3.57715206396
ϟ

B(2, 3)

x = 2, y = 3

B(2, 3) = 1/12
0.0833333333333
ϟ

B_{0.5}(2, 3)

x = 0.5, a = 2, b = 3

B_{0.5}(2, 3)
0.0572916666667
ϟ

sinc(0)

x = 0

sinc(0) = 1
1
ϟ

Normalized sinc(1/2)

x = 0.5

sinc(1/2) = 2/π
0.636619772368

Special Functions calculator specification

Version 1.10.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Real special functions: erf(x)=(2/√π)∫₀ˣ e^{−t²} dt, erfc(x)=1−erf(x), Γ(x)=∫₀^∞ t^{x−1} e^{−t} dt except poles at 0,−1,−2,…, lgamma(x)=ln Γ(x) for x>0, integer-order J_n(x), integer-order Y_n(x) for x>0, integer-order I_n(x), integer-order K_n(x) for x>0, principal Lambert W_0(x) for x≥−1/e, secondary Lambert W_{-1}(x) for −1/e≤x<0, complete Euler B(x,y)=Γ(x)Γ(y)/Γ(x+y) for x>0, y>0, incomplete beta B_x(a,b)=∫₀ˣ t^{a−1}(1−t)^{b−1} dt for a>0, b>0, 0≤x≤1, unnormalized sinc(x)=sin(x)/x with sinc(0)=1, and normalized sinc(x)=sin(πx)/(πx) with sinc(0)=1. 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. Integer n! stays on factorial. Not CAS.
What it calculates
Real erf, erfc, Gamma, log-gamma, integer-order J_n, Y_n, I_n, and K_n, Lambert W_0 and W_{-1}, complete Euler beta, incomplete beta, unnormalized sinc, and normalized sinc.
Inputs
  • mode
  • x
  • y
  • n
  • a
  • b
Outputs
  • value
  • erf?
  • erfc?
  • gamma?
  • lgamma?
  • n?
  • branch?
  • y?
  • a?
  • b?
  • regularized?
Formula
erf(x); erfc(x)=1−erf(x); Γ(x); lgamma(x)=ln Γ(x); J_n(x); Y_n(x); I_n(x); K_n(x); W_0(x) e^{W_0}=x; W_{-1}(x) e^{W_{-1}}=x; B(x,y)=Γ(x)Γ(y)/Γ(x+y); B_x(a,b); sinc(x)=sin(x)/x; sinc(x)=sin(πx)/(πx)
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
Units
  • dimensionless
Boundary conditions
  • non-finite x → INVALID_NUMBER
  • Gamma at 0, −1, −2, … → INVALID_INPUT
  • lgamma with x ≤ 0 → INVALID_INPUT
  • Bessel n not in 0…8 or |x|>80 → INVALID_INPUT
  • Y_n or K_n with x ≤ 0 → INVALID_INPUT
  • Lambert W_0 x < −1/e → INVALID_INPUT
  • Lambert W_{-1} x < −1/e or x ≥ 0 → INVALID_INPUT
  • Beta x ≤ 0 or y ≤ 0 → INVALID_INPUT
  • Incomplete beta a ≤ 0, b ≤ 0, or x ∉ [0,1] → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
mode=erf x=1 → 0.84270079295
Validation cases

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

  • mode=erf x=1 → 0.84270079295
  • mode=gamma x=0.5 → √π
  • mode=gamma x=6 → 120
  • mode=bessel_j n=0 x=1 → 0.765197686558
  • mode=bessel_y n=0 x=1 → 0.0882569642157
  • mode=bessel_i n=0 x=1 → 1.26606587775
  • mode=bessel_k n=0 x=1 → 0.421024438241
  • mode=lambert_w x=1 → 0.56714329041
  • mode=lambert_wm1 x=−0.1 → −3.57715206396
  • mode=beta x=2 y=3 → 0.0833333333333
  • mode=incomplete_beta x=0.5 a=2 b=3 → 0.0572916666667
  • mode=sinc x=0 → 1
  • mode=normalized_sinc x=0.5 → 0.636619772368
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 K Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT 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 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. › → 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. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is Γ(6)=120 the factorial calculator?

No. Γ(n)=(n−1)! for positive integers, on a different model. Integer n! stays on /calc/math/factorial.

Is this incomplete beta or normalized sinc?

Both are on this engine: incomplete beta is mode=incomplete_beta (/incomplete-beta); normalized sinc is mode=normalized_sinc (/normalized-sinc). This seed is erf, erfc, Γ, ln Γ, integer-order J_n, Y_n, I_n, K_n, principal W_0, secondary W_{-1}, complete Euler B(x,y), incomplete B_x(a,b), unnormalized sinc, and normalized sinc. Not CAS.

Where does this run?

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