fastmath.special — generateme/fastmath 3.0.0-alpha2

Special functions for real arguments and value.

• Bessel J, Y, jinc
• Modified Bessel I, K
• Spherical Bessel j, y
• Modified spherical Bessel i1, i2, k
• Gamma, log, digamma, trigamma, polygamma, regularized, lower/upper incomplete
• Beta, log, regularized, incomplete
• Erf, inverse
• Airy A, B with derivatives
• Zeta (Riemann, Hurwitz), Eta (Dirichlet), Xi (Landau), Beta (Dirichlet)
• Integrals: Si, Ci, li/Li, Ei, En, Ein
• Hypergeometric 0F0, 0F1, 1F0, 1F1, 2F1, 2F0, 0F2, pFq, Kummers M, Tricomis U, Whittaker M and W
• Lambert W (0 and -1)
• Minkowski
• Harmonic H

airy-Ai clj

(airy-Ai x)

Airy Ai function

airy-Ai' clj

(airy-Ai' x)

First derivative of the Airy Ai function

airy-Bi clj

(airy-Bi x)

Airy Bi function

airy-Bi' clj

(airy-Bi' x)

First derivative of the Airy Bi function

bessel-I clj

(bessel-I order x)

Modified Bessel function of the first kind of order v, I_v(x)

bessel-I0 clj

(bessel-I0 x)

Modified Bessel function of the first kind of order 0, I_0(x)

bessel-I1 clj

(bessel-I1 x)

Modified Bessel function of the first kind of order 1, I_0(x)

bessel-J clj

(bessel-J order x)

Bessel function of the first kind of order v, J_v(x)

bessel-J0 clj

(bessel-J0 x)

Bessel function of the first kind of order 0, J_0(x)

bessel-J1 clj

(bessel-J1 x)

Bessel function of the first kind of order 1, J_1(x)

bessel-K clj

(bessel-K order x)

Modified Bessel function of the second kind and real order v, K_v(x)

bessel-K-half-odd clj

(bessel-K-half-odd odd-numerator x)

Bessel K_a function for a = order/2

Function accepts only odd integers for order

bessel-K0 clj

(bessel-K0 x)

Modified Bessel function of the second kind of order 0, K_0(x)

bessel-K1 clj

(bessel-K1 x)

Modified Bessel function of the second kind of order 1, K_1(x)

bessel-Y clj

(bessel-Y order x)

Bessel function of the second kind of order v, Y_v(x)

bessel-Y0 clj

(bessel-Y0 x)

Bessel function of the second kind of order 0, Y_0(x)

bessel-Y1 clj

(bessel-Y1 x)

Bessel function of the second kind of order 1, Y_1(x)

beta clj

(beta p q)

Beta function

Ci clj

(Ci x)

Cosine integral

Cin clj

(Cin x)

Cosine integral, alternative definition

digamma clj

(digamma x)

First derivative of log of Gamma function.

dirichlet-beta clj

(dirichlet-beta x)

Dirichlet Beta function

E0 clj

(E0 x)

Exponential integral E0

E1 clj

(E1 x)

Exponential integral E1 for positive real numbers

Ei clj

(Ei x)

Exponential integral

Ein clj

(Ein x)

Exponential integral, alternative definition

En clj

(En n x)

Generalized exponential integral En

erf clj

(erf x)

(erf x1 x2)

Error function.

For two arguments returns a difference between (erf x2) and (erf x1).

erfc clj

(erfc x)

Complementary error function.

eta clj

(eta x)

Dirichlet Eta function

gamma clj

(gamma x)

Gamma function $\Gamma(x)$. Extension of the factorial.

harmonic-number clj

(harmonic-number n)

(harmonic-number n m)

Harmonic number H_n or generalized harmonic number H_n,m

hypergeometric-0F0 clj

(hypergeometric-0F0 x)

Hypergeometric ₀F₀ function, exp(x)

hypergeometric-0F1 clj

(hypergeometric-0F1 a x)

Confluent hypergeometric ₀F₁ limit function.

hypergeometric-0F2 clj

(hypergeometric-0F2 a b x)

Generalized hypergeometric ₀F₂ function.

hypergeometric-1F0 clj

(hypergeometric-1F0 a x)

Hypergeometric ₁F₀ function.

hypergeometric-1F1 clj

(hypergeometric-1F1 a b x)

Confluent hypergeometric ₁F₁ function of the first kind, Kummer's M.

hypergeometric-2F0 clj

(hypergeometric-2F0 a b x)

Generalized hypergeometric ₂F₀ function.

hypergeometric-2F1 clj

(hypergeometric-2F1 a b c x)

Gauss's hypergeometric ₂F₁ function.

hypergeometric-pFq clj

(hypergeometric-pFq ps qs z)

(hypergeometric-pFq ps qs z max-iters)

hypergeometric-pFq using MacLaurin series.

max-iters is set to 1000000 by default.

hypergeometric-pFq-ratio clj

(hypergeometric-pFq-ratio ps qs z)

(hypergeometric-pFq-ratio ps qs z max-iters)

Hypergeometric-pFq using MacLaurin series on ratios. Can be very slow.

max-iters is set to 10000 by default.

incomplete-beta clj

(incomplete-beta x a b)

Incomplete Beta B(x,a,b)

inv-erf clj

(inv-erf x)

Inverse of erf function.

inv-erfc clj

(inv-erfc x)

Inverse of erfc function.

inv-gamma-1pm1 clj

(inv-gamma-1pm1 x)

$\frac{1}{\Gamma(1+x)}-1$ for $-0.5≤x≤1.5$.

jinc clj

(jinc x)

Besselj1 devided by x

kummers-M clj

(kummers-M a b x)

Kummer's (confluent hypergeometric, 1F1) function for real arguments.

lambert-W clj

(lambert-W x)

Lambert W_0 function. W(xe^x)=x for x>=-1.0.

lambert-W-1 clj

(lambert-W-1 x)

Lambert W_1 function. W_1(xe^x)=x for x<=-1.0.

Li clj

(Li x)

Offset logarythmic integral

li clj

(li x)

Logarythmic integral

log-beta clj

(log-beta p q)

Logarithm of Beta function.

log-gamma clj

(log-gamma x)

Log of Gamma function $\log\Gamma(x)$.

log-gamma-1p clj

(log-gamma-1p x)

$\ln\Gamma(1+x)$ for $-0.5≤x≤1.5$.

lower-incomplete-gamma clj

(lower-incomplete-gamma s x)

Lower incomplete gamma function

minkowski clj

(minkowski x)

Minkowski's question mark function ?(x)

polygamma clj

(polygamma m x)

Polygamma function of order m and real argument.

regularized-beta clj

(regularized-beta x a b)

Regularized Beta I_x(a,b)

regularized-gamma-p clj

(regularized-gamma-p a x)

Regularized gamma P(a,x)

regularized-gamma-q clj

(regularized-gamma-q a x)

Regularized gamma Q(a,x)

Si clj

(Si x)

Sine integral

si clj

(si x)

Sine integral, Si shifted by -pi/2

spherical-bessel-1-i clj

(spherical-bessel-1-i order x)

First modified spherical Bessel function of the first kind.

spherical-bessel-1-i0 clj

(spherical-bessel-1-i0 x)

First modified spherical Bessel function of the first kind and order 0.

spherical-bessel-1-i1 clj

(spherical-bessel-1-i1 x)

First modified spherical Bessel function of the first kind and order 1.

spherical-bessel-1-i2 clj

(spherical-bessel-1-i2 x)

First modified spherical Bessel function of the first kind and order 2.

spherical-bessel-2-i clj

(spherical-bessel-2-i order x)

Second modified spherical Bessel function of the first kind.

spherical-bessel-2-i0 clj

(spherical-bessel-2-i0 x)

Second modified spherical Bessel function of the first kind and order 0.

spherical-bessel-2-i1 clj

(spherical-bessel-2-i1 x)

Second modified spherical Bessel function of the first kind and order 1.

spherical-bessel-2-i2 clj

(spherical-bessel-2-i2 x)

Second modified spherical Bessel function of the first kind and order 2.

spherical-bessel-j clj

(spherical-bessel-j order x)

Spherical Bessel function of the first kind.

spherical-bessel-j0 clj

(spherical-bessel-j0 x)

Spherical Bessel function of the first kind and order 0.

spherical-bessel-j1 clj

(spherical-bessel-j1 x)

Spherical Bessel function of the first kind and order 1.

spherical-bessel-j2 clj

(spherical-bessel-j2 x)

Spherical Bessel function of the first kind and order 2.

spherical-bessel-k clj

(spherical-bessel-k order x)

Modified spherical Bessel function of the second kind.

spherical-bessel-k0 clj

(spherical-bessel-k0 x)

Modified spherical Bessel function of the second kind and order 0.

spherical-bessel-k1 clj

(spherical-bessel-k1 x)

Modified spherical Bessel function of the second kind and order 1.

spherical-bessel-k2 clj

(spherical-bessel-k2 x)

Modified spherical Bessel function of the second kind and order 2.

spherical-bessel-y clj

(spherical-bessel-y order x)

Spherical Bessel function of the second kind.

spherical-bessel-y0 clj

(spherical-bessel-y0 x)

Spherical Bessel function of the second kind and order 0.

spherical-bessel-y1 clj

(spherical-bessel-y1 x)

Spherical Bessel function of the second kind and order 1.

spherical-bessel-y2 clj

(spherical-bessel-y2 x)

Spherical Bessel function of the second kind and order 2.

tricomis-U clj

(tricomis-U a b x)

Confluent hypergeometric function U of the second kind.

trigamma clj

(trigamma x)

Second derivative of log of Gamma function.

upper-incomplete-gamma clj

(upper-incomplete-gamma s x)

Upper incomplete gamma function

whittaker-M clj

(whittaker-M kappa mu x)

Whittaker's M

whittaker-W clj

(whittaker-W kappa mu x)

Whittaker's W

xi clj

(xi s)

Riemann (Landau's) Xi function

zeta clj

(zeta s)

(zeta s z)

Riemann and Hurwitz zeta functions for real arguments