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provisdom.math.special-functions


betaclj

(beta x y)

Returns the beta of x and y: integral[0, 1] (t ^ (x - 1) × (1 - t) ^ (y - 1) * dt.

Returns the beta of `x` and `y`:
integral[0, 1] (t ^ (`x` - 1) × (1 - t) ^ (`y` - 1) * dt.
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cdf-standard-normalclj

(cdf-standard-normal x)

Returns the standard Normal cdf.

Returns the standard Normal cdf.
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digammaclj

source

erfclj

(erf x)

Returns the error function: 2 / (sqrt PI) × integral[0, x] (e ^ - (t ^ 2) × dt).

Returns the error function:
2 / (sqrt PI) × integral[0, `x`] (e ^ - (t ^ 2) × dt).
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erf-derivativeclj

(erf-derivative x)

Returns the derivative of the error function.

Returns the derivative of the error function.
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erf-diffclj

(erf-diff x1 x2)

Returns the difference between erf(x1) and erf(x2).

Returns the difference between [[erf]](`x1`) and [[erf]](`x2`).
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erfcclj

(erfc x)

Returns the complementary error function.

Returns the complementary error function.
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euler-mascheroni-constantclj

source

exact-stirling-errorsclj

Expansion error for values 0.0 through 15.0 by 0.5.

Expansion error for values 0.0 through 15.0 by 0.5.
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gammaclj

(gamma a)

Returns the gamma function: integral[0, inf] (t ^ (a- 1) × e ^ -t × dt). Although gamma is defined for positive a, this function also allows for all non-roughly-round-non+ a.

Returns the gamma function: integral[0, inf] (t ^ (`a`- 1) × e ^ -t × dt).
Although gamma is defined for positive `a`, this function also allows for all
non-roughly-round-non+ `a`.
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gamma-derivativeclj

(gamma-derivative a)

Returns the derivative of the gamma of a. a > -3e8 because it becomes slow.

Returns the derivative of the gamma of `a`. `a` > -3e8 because it becomes
slow.
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incomplete-betaclj

(incomplete-beta c x y)

Returns the lower beta: integral[0, c] (t ^ (x - 1) × (1 - t) ^ (y - 1) × dt. If Apache's solution doesn't converge, then NaN is returned.

Returns the lower beta:
integral[0, `c`] (t ^ (`x` - 1) × (1 - t) ^ (`y` - 1) × dt. If Apache's
solution doesn't converge, then NaN is returned.
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inv-cdf-standard-normalclj

(inv-cdf-standard-normal cumulative-prob)

Returns the standard Normal inverse cdf.

Returns the standard Normal inverse cdf.
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inv-erfclj

(inv-erf x)

Returns the inverse error function.

Returns the inverse error function.
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inv-erfcclj

(inv-erfc x)

Returns the inverse complementary error function.

Returns the inverse complementary error function.
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inv-probitclj

See [[cdf-standard-normal]]
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lanczos-coefficientsclj

Lanczos Coefficients.

Lanczos Coefficients.
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log-betaclj

(log-beta x y)

Returns the log-beta of x and y.

Returns the log-beta of `x` and `y`.
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log-gammaclj

(log-gamma a)

Returns the log gamma of a.

Returns the log gamma of `a`.
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log-gamma-derivativeclj

(log-gamma-derivative a)

Returns the derivative of the log gamma of a. a > -3e8 because it becomes slow. Taken from Apache. Could use a better algorithm.

Returns the derivative of the log gamma of `a`. `a` > -3e8 because
it becomes slow. Taken from Apache. Could use a better algorithm.
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logisticclj

(logistic x)
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logitclj

(logit p)
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lower-gammaclj

(lower-gamma a x)

Returns the lower incomplete gamma function: integral[0, x] (t ^ (a - 1) × e ^ -t × dt).

Returns the lower incomplete gamma function: 
integral[0, `x`] (t ^ (`a` - 1) × e ^ -t × dt).
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multivariate-gammaclj

(multivariate-gamma a p)

Returns the multivariate gamma of a with dimension p.

Returns the multivariate gamma of `a` with dimension `p`.
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multivariate-log-gammaclj

(multivariate-log-gamma a p)

Returns the multivariate log gamma of a with dimension p.

Returns the multivariate log gamma of `a` with dimension `p`.
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probitclj

See [[inv-cdf-standard-normal]]
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rational-approximation-coefficientsclj

source

regularized-betaclj

(regularized-beta c x y)

Returns the regularized beta. Equal to incomplete beta function divided by beta function. If Apache's solution doesn't converge, then NaN is returned.

Returns the regularized beta. Equal to incomplete beta function divided by
beta function. If Apache's solution doesn't converge, then NaN is returned.
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regularized-gamma-pclj

(regularized-gamma-p a x)

Returns the regularized gamma function P(a, x) = 1 - Q(a, x). Equal to lower-gamma function (a, x) divided by gamma function (a).

Returns the regularized gamma function P(`a`, `x`) = 1 - Q(`a`, `x`). Equal
to [[lower-gamma]] function (a, x) divided by [[gamma]] function (`a`).
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regularized-gamma-qclj

(regularized-gamma-q a x)

Returns the regularized gamma function Q(a, x) = 1 - P(a, x). Equal to upper-gamma function (a, x) divided by gamma function (a).

Returns the regularized gamma function Q(`a`, `x`) = 1 - P(`a`, `x`). Equal
to [[upper-gamma]] function (`a`, `x`) divided by [[gamma]] function (`a`).
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trigammaclj

(trigamma a)

Returns the trigamma (2nd derivative of log-gamma) of a. Approximated for a < -1e7 because it becomes slow. Taken from Apache. Could use a better algorithm.

Returns the trigamma (2nd derivative of log-gamma) of `a`. Approximated for
`a` < -1e7 because it becomes slow. Taken from Apache. Could use a better
algorithm.
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upper-gammaclj

(upper-gamma a x)

Returns the upper incomplete gamma function: integral[x, inf] (t ^ (a - 1) × e ^ -t × dt).

Returns the upper incomplete gamma function: 
integral[`x`, inf] (t ^ (`a` - 1) × e ^ -t × dt).
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upper-gamma-derivative-xclj

(upper-gamma-derivative-x a x)

Returns the upper gamma derivative x.

Returns the upper gamma derivative `x`.
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