Special functions

Collection of special functions for real arguments and real returned value. Most of the functions are implemented natively in Clojure, some are based on Apache Commons Math.

Native implementation is based on Julia packages (SpecialFunctions.jl, Bessel, HypergeometricFunctions) or scientific papers and books (NIST, Meshfree Approximation Methods with Matlab by G. E. Fasshauer]).

(require '[fastmath.special :as special]
         '[fastmath.core :as m])

Gamma

Gamma and related functions

Defined functions
  • gamma, log-gamma
  • inv-gamma-1pm1, log-gamma-1p
  • upper-incomplete-gamma, lower-incomplete-gamma
  • regularized-gamma-p, regularized-gamma-q
  • digamma, trigamma, polygamma

Gamma function

gamma \(\Gamma(x)\) function is an extension of the factorial.

\[\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}\,dt\]

For positive integer n

\[\Gamma(n) = (n-1)!\]

Gamma for negative integers is not defined.

Examples 
(special/gamma 1.5) ;; => 0.886226925452758
(special/gamma -1.5) ;; => 2.3632718012073544
(special/gamma -2.0) ;; => ##NaN
(special/gamma 15) ;; => 8.71782912E10
(m/factorial 14) ;; => 8.71782912E10

Additionally reciprocal gamma function inv-gamma-1pm1 is defined as:

\[\frac{1}{\Gamma(x+1)}-1\text{ for } -0.5\leq x\leq 1.5\]

Examples 
(special/inv-gamma-1pm1 -0.5) ;; => -0.4358104164522437
(special/inv-gamma-1pm1 0.5) ;; => 0.12837916709551256

Log gamma

Logartihm of gamma log-gamma \(\log\Gamma(x)\) with derivatives: digamma \(\psi\), trigamma \(\psi_1\) and polygamma \(\psi^{(m)}\).

Examples 
(special/log-gamma 1.01) ;; => -0.005690307946069651
(special/log-gamma 0.5) ;; => 0.5723649429247001
(m/exp (special/log-gamma 5)) ;; => 24.000000000000004

log-gamma-1p is more accurate function defined as \(\log\Gamma(x+1)\) for \(-0.5\leq x 1.5\)

Examples 
(special/log-gamma-1p -0.1) ;; => 0.06637623973474298
(special/log-gamma-1p 0.01) ;; => -0.005690307946069646
(special/log-gamma-1p 1.01) ;; => 0.004260022907098441

Derivatives of log-gamma function. First derivative digamma.

\[\operatorname{digamma}(x)=\psi(x)=\psi^{(0)}(x)=\frac{d}{dx}\log\Gamma(x)=\frac{\Gamma'(x)}{\Gamma(x)}\]

Second derivative trigamma

\[\operatorname{trigamma}(x)=\psi_1(x)=\psi^{(1)}(x)=\frac{d}{dx}\psi(x)=\frac{d^2}{dx^2}\log\Gamma(x)\]

polygamma as mth derivative of digamma

\[\operatorname{polygamma}(m,x)=\psi^{(m)}=\frac{d^m}{dx^m}\psi(x)=\frac{d^{m+1}}{dx^{m+1}}\log\Gamma(x)\]

Examples 
(special/digamma 0.5) ;; => -1.9635100260214235
(special/trigamma 0.5) ;; => 4.93480220054468
(special/polygamma 0 0.5) ;; => -1.9635100260214235
(special/polygamma 1 0.5) ;; => 4.93480220054468
(special/polygamma 2 0.5) ;; => -16.828796644234316
(special/polygamma 3 0.5) ;; => 97.40909103400247

Incomplete and regularized

upper-incomplete-gamma \(\Gamma(s,x)\) and lower-incomplete-gamma \(\gamma(s,x)\) are versions of gamma function with parametrized integral limits.

Upper incomplete gamma is defined as:

\[\Gamma(s,x) = \int_x^\infty t^{s-1}e^{-t}\,dt\]
Examples 
(special/upper-incomplete-gamma 2 0.5) ;; => 0.9097959895689501
(special/upper-incomplete-gamma -2 0.5) ;; => 0.886417457100714

Lower incomplete gamma is defined as:

\[\gamma(s,x) = \int_0^x t^{s-1}e^{-t}\,dt\]
Examples 
(special/lower-incomplete-gamma 0.5 3) ;; => 1.7470973415820528
(special/lower-incomplete-gamma -0.5 3) ;; => -3.5516838378128024

regularized-gamma-p \(P(s,x)\) and regularized-gamma-q \(Q(s,x)\) are normalized incomplete gamma functions by gamma of s. s can be negative non-integer.

\[P(s,x)=\frac{\gamma(s,x)}{\Gamma(x)}\]

Examples 
(special/regularized-gamma-p 2.5 0.5) ;; => 0.03743422675270362
(special/regularized-gamma-p -2.5 0.5) ;; => 2.134513839251947

\[Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(x)}=1-P(s,x)\]

Examples 
(special/regularized-gamma-q 2.5 0.5) ;; => 0.9625657732472964
(special/regularized-gamma-q -2.5 0.5) ;; => -1.134513839251947

Beta

Beta and related functions

Defined functions
  • beta, log-beta
  • incomplete-beta, regularized-beta

Beta function

beta \(B(p,q)\) function, defined also for negative non-integer p and q.

\[B(p,q) = \int_0^1 t^{p-1}(1-t)^{q-1}\,dt = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}\]

Examples 
(special/beta 2 3) ;; => 0.08333333333333334
(special/beta -1.2 0.1) ;; => 4.750441365819471
(special/beta -1.2 -0.1) ;; => -15.574914582341846

log-beta is log of \(B(p,q)\) for positive p and q

Examples 
(special/log-beta 2 3) ;; => -2.4849066497880004
(special/log-beta 2 3) ;; => -2.4849066497880004
(m/log (special/beta 2 3)) ;; => -2.4849066497880004

Incomplete and regularized

incomplete-beta \(B(x,a,b)\) and regularized-beta \(I_x(a,b)\). Both are defined also for negative non-integer a and b.

\[B(x,a,b)=\int_0^x t^{a-1}(1-t)^{b-1}\,dt\]

Examples 
(special/incomplete-beta 0.5 0.1 0.2) ;; => 9.789912164505285
(special/incomplete-beta 0.5 -0.1 -0.2) ;; => -9.707992848052843

\[I_x(a,b)=\frac{B(x,a,b)}{B(a,b)}\]

Examples 
(special/regularized-beta 0.99 -0.5 -0.7) ;; => 12.086770426418143
(special/regularized-beta 0.01 0.5 0.5) ;; => 0.06376856085851987

Bessel

  • Bessel functions of the first (\(J_\alpha\)) and the second (\(Y_\alpha\)) kind
  • Modified Bessel functions of the first (\(I_\alpha\)) and the second (\(K_\alpha\)) kind
  • Spherical Bessel functions of the first (\(j_\alpha\)) and the second (\(y_\alpha\)) kind
  • Modified spherical Bessel functions of the first (\(i_\alpha^{(1)}\), \(i_\alpha^{(2)}\)) and the second (\(k_\alpha\)) kind
  • Sombrero function jinc
Defined functions
  • bessel-J0, bessel-J1, bessel-J, jinc
  • bessel-Y0, bessel-Y1, bessel-Y
  • bessel-I0, bessel-I1, bessel-I
  • bessel-K0, bessel-K1, bessel-K, bessel-K-half-odd
  • spherical-bessel-j0, spherical-bessel-j1, spherical-bessel-j2,spherical-bessel-j
  • spherical-bessel-y0, spherical-bessel-y1, spherical-bessel-y2,spherical-bessel-y
  • spherical-bessel-1-i0, spherical-bessel-1-i1, spherical-bessel-1-i2,spherical-bessel-1-i
  • spherical-bessel-2-i0, spherical-bessel-2-i1, spherical-bessel-2-i2,spherical-bessel-2-1
  • spherical-bessel-k0, spherical-bessel-k1, spherical-bessel-k2,spherical-bessel-k

Bessel J, j

Bessel functions of the first kind, bessel-J \(J_\alpha(x)\). bessel-J0 and bessel-J1 are functions of orders 0 and 1. An order should be integer for negative arguments.
Examples 
(special/bessel-J0 2.3) ;; => 0.055539784445602064
(special/bessel-J1 2.3) ;; => 0.5398725326043137
(special/bessel-J 2.1 3) ;; => 0.4761626361699597
(special/bessel-J -3 -3.2) ;; => 0.3430663764006682
(special/bessel-J 3 -3.2) ;; => -0.3430663764006682
(special/bessel-J -3.1 -3.2) ;; => ##NaN
(special/bessel-J 3.1 -3.2) ;; => ##NaN

Spherical Bessel functions of the first kind spherical-bessel-j \(j_\alpha(x)\), spherical-bessel-j0, spherical-bessel-j1 and spherical-bessel-j2 are functions of orders 0,1 and 2. Functions are defined for positive argument (only functions with orders 0, 1 and 2 accept non positive argument).

\[j_\alpha(x)=\sqrt{\frac{\pi}{2x}}J_{\alpha+\frac{1}{2}}(x)\]
Examples 
(special/spherical-bessel-j0 2.3) ;; => 0.3242196574681393
(special/spherical-bessel-j1 2.3) ;; => 0.43065029510781005
(special/spherical-bessel-j2 2.3) ;; => 0.23749811875943916
(special/spherical-bessel-j 3.1 3.2) ;; => 0.1579561007291703
(special/spherical-bessel-j -3.1 3.2) ;; => 0.12796785869167607

additional jinc (sombrero) function is defined as:

\[\operatorname{jinc}(x)=\frac{2J_1(\pi x)}{\pi x}\]

Examples 
(special/jinc 0.0) ;; => 1.0
(special/jinc -2.3) ;; => 0.01707103495964295
(special/jinc 2.3) ;; => 0.01707103495964295

Bessel Y, y

Bessel functions of the second kind, bessel-Y, \(Y_\alpha(x)\). bessel-Y0 and bessel-Y1 are functions of orders 0 and 1. They are defined for positive argument only and any order.
Examples 
(special/bessel-Y0 2.3) ;; => 0.5180753962076221
(special/bessel-Y1 2.3) ;; => 0.05227731584422471
(special/bessel-Y 2 2.3) ;; => -0.47261686069090497
(special/bessel-Y -2.1 2.3) ;; => -0.36752629274516924
(special/bessel-Y 3 -1) ;; => ##NaN

Spherical Bessel functions of the second kind spherical-bessel-y \(y_\alpha(x)\), spherical-bessel-y0, spherical-bessel-y1 and spherical-bessel-y2 are functions of orders 0,1 and 2. Functions are defined for positive argument.

\[y_\alpha(x)=\sqrt{\frac{\pi}{2x}}Y_{\alpha+\frac{1}{2}}(x)\]
Examples 
(special/spherical-bessel-j0 2.3) ;; => 0.3242196574681393
(special/spherical-bessel-j1 2.3) ;; => 0.43065029510781005
(special/spherical-bessel-j2 2.3) ;; => 0.23749811875943916
(special/spherical-bessel-j 3.1 3.2) ;; => 0.1579561007291703
(special/spherical-bessel-j -3.1 3.2) ;; => 0.12796785869167607

Bessel I, i

Modified Bessel functions of the first kind, bessel-I, \(I_\alpha(x)\), bessel-I0 and bessel-I1 are functions of orders 0 and 1. An order should be integer for negative arguments.
Examples 
(special/bessel-I0 2.3) ;; => 2.8296056006275854
(special/bessel-I1 2.3) ;; => 2.097800027517421
(special/bessel-I 2 2.3) ;; => 1.0054316636559146
(special/bessel-I -2.1 2.3) ;; => 0.9505851207098388
(special/bessel-I -3 -1) ;; => -0.0221684249243319
(special/bessel-I -3.1 -1) ;; => ##NaN

Two modfified spherical Bessel functions of the first kind spherical-bessel-1-i \(i_\alpha^{(1)}(x)\) and spherical-bessel-2-i \(i_\alpha^{(2)}(x)\). spherical-bessel-1-i0, spherical-bessel-1-i1, spherical-bessel-1-i2, spherical-bessel-2-i0, spherical-bessel-2-i1 and spherical-bessel-2-i2 are functions of orders 0,1 and 2. Functions are defined for positive argument.

\[i_\alpha^{(1)}(x)=\sqrt{\frac{\pi}{2x}}I_{\alpha+\frac{1}{2}}(x)\] \[i_\alpha^{(2)}(x)=\sqrt{\frac{\pi}{2x}}I_{-\alpha-\frac{1}{2}}(x)\]
Examples

\(i_\alpha^{(1)}\)

(special/spherical-bessel-1-i0 2.3) ;; => 2.1465051328460687
(special/spherical-bessel-1-i1 2.3) ;; => 1.256832833227258
(special/spherical-bessel-1-i2 2.3) ;; => 0.5071579590713844
(special/spherical-bessel-1-i 3.1 3.2) ;; => 0.48382455936204793
(special/spherical-bessel-1-i -3.1 3.2) ;; => 1.271418391309527

\(i_\alpha^{(2)}\)

(special/spherical-bessel-2-i0 2.3) ;; => 2.1900959344646793
(special/spherical-bessel-2-i1 2.3) ;; => 1.1942895091657733
(special/spherical-bessel-2-i2 2.3) ;; => 0.6323270094658442
(special/spherical-bessel-2-i 3.1 3.2) ;; => 0.42032145323101916
(special/spherical-bessel-2-i -3.1 3.2) ;; => 1.2425414548936577

Bessel K, k

Modified Bessel functions of the second kind, bessel-K, \(K_\alpha(x)\), bessel-K0 and bessel-K1 are functions of orders 0 and 1. They are defined for positive argument only and any order.
Examples 
(special/bessel-K0 2.3) ;; => 0.07913993300209364
(special/bessel-K1 2.3) ;; => 0.09498244384536267
(special/bessel-K 2 2.3) ;; => 0.1617333624328438
(special/bessel-K -2.1 2.3) ;; => 0.17365527243516982
(special/bessel-K 3 -1) ;; => ##NaN

Additionally bessel-K-half-odd function is optimized version for order of the half of odd integer, ie 1/2, 3/2, 5/2 and so on. First argument is an odd numerator.

Examples 
[(special/bessel-K-half-odd 1 2.3) (special/bessel-K 0.5 2.3)] ;; => [0.0828549981836159 0.0828549981836159]
[(special/bessel-K-half-odd 3 2.3) (special/bessel-K 1.5 2.3)] ;; => [0.11887891043736196 0.11887891043736196]
[(special/bessel-K-half-odd 5 2.3) (special/bessel-K 2.5 2.3)] ;; => [0.23791444658017497 0.23791444658017497]

Modified spherical Bessel functions of the second kind spherical-bessel-k \(k_\alpha(x)\), spherical-bessel-k0, spherical-bessel-k1 and spherical-bessel-k2 are functions of orders 0,1 and 2. Functions are defined for positive argument.

\[k_\alpha(x)=\sqrt{\frac{\pi}{2x}}K_{\alpha+\frac{1}{2}}(x)\]
Examples 
(special/spherical-bessel-k0 2.3) ;; => 0.06847227106455815
(special/spherical-bessel-k1 2.3) ;; => 0.09824282370132256
(special/spherical-bessel-k2 2.3) ;; => 0.19661508458802238
(special/spherical-bessel-k 3.1 3.2) ;; => 0.10488382566292166
(special/spherical-bessel-k -3.1 3.2) ;; => 0.047694101111715716

Erf

Defined functions
  • erf, erfc
  • inv-erf, inv-erfc

Error functions

\[\operatorname{erf}(x)=\frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,dt\] \[\operatorname{erfc}(x)=1-\operatorname{erf}(x)\]

When two arguments are passed, difference between erf of two values is calculated \(\operatorname{erf}(x_2)-\operatorname{erf}(x_1)\)

Examples 
(special/erf 0.4) ;; => 0.4283923550466685
(special/erfc 0.4) ;; => 0.5716076449533315
[(special/erf 0.5 0.4) (- (special/erf 0.4) (special/erf 0.5))] ;; => [-0.09210752276637812 -0.09210752276637812]

Inverse of error functions.

  • inv-erf - inverse of erf defined on \((-1,1)\)
  • inv-erfc- inverse of erfc defined on \((0,2)\)
Examples 
(special/inv-erf 0.42839235504666856) ;; => 0.4
(special/inv-erfc (- 1 0.42839235504666856)) ;; => 0.4000000000000001

Airy

Airy functions and derivatives

Defined functions
  • airy-Ai, airy-Bi
  • airy-Ai' airy-Bi'
Examples 
(special/airy-Ai 2.3) ;; => 0.02183199318062265
(special/airy-Bi 2.3) ;; => 4.885061581835644
(special/airy-Ai' 2.3) ;; => -0.03517312272081809
(special/airy-Bi' 2.3) ;; => 6.709740812723825

Integrals

Trigonometric, exponential and logarithmic integrals

Defined functions
  • Si, si, Ci, Cin
  • E0, E1, Ei, Ein, En
  • li, Li (offset)

Trigonometric

Sine and cosine integrals

\[\operatorname{Si}(x)=\int_0^x\frac{\sin t}{t}\, dt\] \[\operatorname{si}(x)=-\int_x^\infty\frac{\sin t}{t}\, dt = \operatorname{Si}(x)-\frac{\pi}{2}\] \[\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}{t}\, dt\] \[\operatorname{Cin}(x)=\int_0^x\frac{1-\cos t}{t}\, dt = \gamma + \ln x- \operatorname{Ci}(x)\]
Examples 
(special/Si 0.5) ;; => 0.49310741804306674
(special/si 0.5) ;; => -1.0776889087518298
(special/Ci 0.5) ;; => -0.17778407880661298
(special/Cin 0.5) ;; => 0.06185256314820056

Exponential

Exponential integrals

\[E_0(x)=\frac{e^{-x}}{x}\] \[E_1(x)=\int_x^\infty\frac{e^{-t}}{t}\,dt\] \[E_i(x)=-\int_{-x}^\infty\frac{e^{-t}}{t}\,dt\] \[E_{in}(x)=\int_0^x\frac{1-e^{-t}}{t}\,dt\] \[E_n(x)=\int_1^\infty\frac{e^{-xt}}{t^n}\,dt\]
Examples 
(special/E0 0.6) ;; => 0.9146860601567108
(special/E1 0.6) ;; => 0.4543795031894021
(special/Ei 0.6) ;; => 0.7699269875786519
(special/Ein 0.6) ;; => 0.5207695443249443
(special/En 2 0.6) ;; => 0.2761839341803851
(special/En -2 0.6) ;; => 9.04522881710525

Logarithmic

Logarithmic integrals

\[\operatorname{li}(x)=\int_0^x\frac{dt}{\ln t}\] \[\operatorname{Li}(x)=\int_2^x\frac{dt}{\ln t}=\operatorname{li}(x)-\operatorname{li}(2)\]

Examples 
(special/li 0.5) ;; => -0.378671043061088
(special/Li 0.5) ;; => -1.423834823178581

Zeta

Zeta function and related

Defined functions
  • zeta - Riemann and Hurwitz zeta
  • xi - Riemann (Landau) xi
  • eta - Dirichlet eta
  • dirichlet-beta - Dirichlet beta

Riemann zeta

\[\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}\]

Examples 
(special/zeta 0.0) ;; => -0.5
(special/zeta 2.2) ;; => 1.4905432565068941
(special/zeta -2.2) ;; => 0.0048792123593036025

Hurwitz zeta

\[\zeta(s,z)=\sum_{n=1}^\infty\frac{1}{(n+z)^s}\]
Examples 
(special/zeta 0.0 3) ;; => -2.5
(special/zeta 2.2 3) ;; => 0.27290561568286237
(special/zeta -2.2 3) ;; => -5.589914207628901
(special/zeta 2.2 -3) ;; => 2.7973744041931727
(special/zeta -2.2 -3) ;; => 16.811251088887037
(special/zeta 0.0 -3) ;; => 2.5

xi

Landau’s Xi function, symmetrical along \(x=0.5\)

\[\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s)\] \[\xi(s)=\xi(1-s)\]

Examples 
(special/xi 0.0) ;; => 0.5
(special/xi 3.5) ;; => 0.6111280074951515
(special/xi (- 1.0 3.5)) ;; => 0.6111280074951515

eta

Dirichlet eta function

\[\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}=(1-2^{1-s})\zeta(s)\]

Examples 
(special/eta 0.0) ;; => 0.5
(special/eta 3.5) ;; => 0.927553577773949
(special/eta -3.5) ;; => -0.09604760404512332

beta

Dirichlet (Catalan) beta function

\[\beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}\]

Examples 
(special/dirichlet-beta 0.0) ;; => 0.5
(special/dirichlet-beta 3.5) ;; => 0.9814025112714404
(special/dirichlet-beta -3.5) ;; => 1.0708860161983005

Hypergeometric

Selection of hypergeometric functions \({}_pF_q\)

\[{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};x\right)=\sum_{k=0}^{\infty}\frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{{\left(b_{1}\right)_{k}}\cdots{\left(b_{q}\right)_{k}}}\frac{x^{k}}{k!}.\]

where \((a_p)_k\) and \((b_q)_k\) are kth rising factorials

Defined functions
  • hypergeometric-pFq, hypergeometric-pFq-ratio
  • hypergeometric-0F0, hypergeometric-0F1, hypergeometric-0F2
  • hypergeometric-1F0, hypergeometric-1F1
  • hypergeometric-2F0, hypergeometric-2F1
  • kummers-M, tricomis-U
  • whittaker-M, whittaker-W

Functions are implemented using various recursive formulas, Maclaurin series and Weniger acceleration.

pFq, generalized

Two implementations of general \({}_pF_q\) hypergeometric functions using Maclaurin series. One implementation operates on doubles (hypergeometric-pFq), second on Clojure ratio type which is accurate but slow (hypergeometric-pFq-ratio).

Examples 
(special/hypergeometric-2F0 0.1 0.1 0.01) ;; => 1.0001006141146784
(special/hypergeometric-pFq [0.1 0.1] [] 0.01) ;; => 1.0001006141146789
(special/hypergeometric-pFq-ratio [0.1 0.1] [] 0.01) ;; => 1.000100614114679

Both functions accept optional max-iters parameter to control number of iterations.

Every implementation but ratio is unstable. Take a look at the example of \({}_1F_1(-50;3;19.5)\), only ratio version gives a valid result.

Examples 
(special/hypergeometric-1F1 -50 3 19.5) ;; => 864.8061702264724
(special/hypergeometric-pFq [-50] [3] 19.5) ;; => -1072.5532046613325
(special/hypergeometric-pFq-ratio [-50] [3] 19.5) ;; => -1.195066852171838

Following plot shows stable (but slow) implementation hypergeometric-pFq-ratio vs unstable (but fast) hypergeometric-1F1 and Maclaurin seriers hypergeometric-pFq.

0F0, exp

\({}_0F_0\) is simply exponential function.

\[{}_0F_0(;;x)=\sum_{k=0}^\infty\frac{x^k}{k!}=e^x\]

Examples 
(special/hypergeometric-0F0 2.3) ;; => 9.97418245481472
(m/exp 2.3) ;; => 9.97418245481472

0F1

\({}_0F_1\) is called confluent hypergeometric limit function.

\[{}_0F_1(;a;x)=\sum_{k=0}^\infty\frac{x^k}{(a)_k k!}= \begin{cases} 1.0 & x=0 \\ \frac{J_{a-1}\left(2\sqrt{|x|}\right)\Gamma(a)}{|x|^\frac{a-1}{2}} & x<0 \\ \frac{I_{a-1}\left(2\sqrt{|x|}\right)\Gamma(a)}{|x|^\frac{a-1}{2}} & x>0 \end{cases} \]

Examples 
(special/hypergeometric-pFq-ratio [] [-0.5] -2) ;; => -0.08000465565839093
(special/hypergeometric-0F1 -0.5 -2) ;; => -0.08000465565839159

0F2

\[{}_0F_2(;a,b;x)=\sum_{k=0}^\infty\frac{x^k}{(a)_k(b)_k k!}\]

Examples 
(special/hypergeometric-pFq-ratio [] [-0.5 0.5] -2) ;; => 0.1239850666953995
(special/hypergeometric-0F2 -0.5 0.5 -2) ;; => 0.12398506669539959

1F0

\[{}_1F_0(a;;x)=\sum_{k=0}^\infty\frac{(a)_k x^k}{k!}=(1-x)^{-a}\]

Examples 
(special/hypergeometric-pFq-ratio [0.5] [] -0.5) ;; => 0.8164965809277258
(special/hypergeometric-1F0 0.5 -0.5) ;; => 0.816496580927726

1F1, M

Confluent hypergeometric function of the first kind, Kummer’s M.

\[{}_1F_1(a;b;x)=M(a,b,x)=\sum_{k=0}^\infty\frac{(a)_k x^k}{(b)_k k!}\]

Examples 
(special/hypergeometric-pFq-ratio [0.5] [1] -0.5) ;; => 0.7910171621397194
(special/hypergeometric-1F1 0.5 1 -0.5) ;; => 0.7910171621397188
(special/kummers-M 0.5 1 -0.5) ;; => 0.7910171621397188

2F0, U

\({}_2F_0\) is related to the confluent hypergeometric function of the second kind, Tricomi’s U

\[{}_2F_0(a,b;;x)=\sum_{k=0}^\infty\frac{(a)_k (b)_k x^k}{k!}\]

Examples 
(special/hypergeometric-pFq-ratio [0.1 -0.1] [] 0.01) ;; => 0.9998994982635961
(special/hypergeometric-2F0 0.1 -0.1 0.01) ;; => 0.9998994982635958
(special/hypergeometric-2F0 0.1 -0.1 1.2) ;; => 0.987133788261332

\[U(a,b,x) \sim x^{-a}{}_2F_0(a,b;;-\frac{1}{x})\]

Examples 
(special/tricomis-U 0.5 0.2 0.1) ;; => 1.090844545544952
(special/tricomis-U -0.5 0.2 0.1) ;; => 0.5507462877526579

2F1, Gauss

Gauss’ hypergeometric function \({}_2F_1\).

\[{}_2F_1(a,b;c;x)=\sum_{k=0}^\infty\frac{(a)_k (b)_k x^k}{(c)_k k!}\]

Examples 
(special/hypergeometric-pFq-ratio [1.1 -0.1] [0.2] 0.5) ;; => 0.5254580717634016
(special/hypergeometric-2F1 1.1 -0.1 0.2 0.5) ;; => 0.5254580717633999

Whittaker M and W

Modified hypergeometric functions by Whittaker

\[M_{\kappa,\mu}\left(x\right)=e^{-\frac{1}{2}x}x^{\frac{1}{2}+\mu}M\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,x\right)\] \[W_{\kappa,\mu}\left(x\right)=e^{-\frac{1}{2}x}x^{\frac{1}{2}+\mu}U\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,x\right)\]
Examples 
(special/whittaker-M 0.3 0.4 1.2) ;; => 1.0250053521045672
(special/whittaker-W 0.3 0.4 1.2) ;; => 0.6219095691834272

Other

Defined functions
  • lambert-W (\(W_0\)), lambert-W-1 (\(W_{-1}\))
  • harmonic-number
  • minkowski - \(?(x)\)

Lambert W

Lambert W is a function for which \(W(xe^x)=x\). There are two branches \(W_0\) (lambert-W) and \(W_{-1}\) (lambert-W-1).

\[ \begin{align} W_0(xe^x)=x & \text{ for } x\ge -1 \\ W_{-1}(xe^x)=x & \text{ for } x\le -1 \end{align} \]

domain of functions

\[ \begin{align} W_0(t) & \text{ for } t\in(-1/e,\infty) \\ W_{-1}(t) & \text{ for } t\in(-1/e,0) \end{align} \]

Examples 
(special/lambert-W m/E) ;; => 1.0
(special/lambert-W (* 2.3 (m/exp 2.3))) ;; => 2.3
(special/lambert-W-1 (* -2 (m/exp -2))) ;; => -2.0

Harmonic H

Harmonic numbers

\[H_n=\int_0^1\frac{1-x^n}{1-x}\,dx=\operatorname{digamma}(x+1)+\gamma\]

For non-negative integers

\[H_n=\sum_{k=1}^n\frac{1}{k}\]

Examples 
(special/harmonic-number 2) ;; => 1.5
(special/harmonic-number 2.5) ;; => 1.680372305546776
(special/harmonic-number 3) ;; => 1.8333333333333335
(special/harmonic-number -0.5) ;; => -1.3862943611198908

Generalized harmonic numbers for \(m\neq0\) or \(m\neq1\)

\[H_{n,m}=\zeta(m)-\zeta(m,n+1)\] \[H_{n,0}=n\text{, }H_{n,1}=H_n\]

For non-negative integer n

\[H_{n,m}=\sum_{k=1}^n\frac{1}{k^m}\]
Examples 
(special/harmonic-number 2.2 -0.5) ;; => 2.737173754376224
(special/harmonic-number 2.2 0.5) ;; => 1.8306098144389147

Minowski

Minkowski’s question mark \(?(x)\) function.

Examples 
(special/minkowski 0.5) ;; => 0.5
[(special/minkowski 0.2) (- 1.0 (special/minkowski 0.8))] ;; => [0.0625 0.0625]
[(special/minkowski (/ 0.5 1.5)) (/ (special/minkowski 0.5) 2)] ;; => [0.25 0.25]

Reference

fastmath.special

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

Ci

  • (Ci x)

Cosine integral

Cin

  • (Cin x)

Cosine integral, alternative definition

E0

  • (E0 x)

Exponential integral E0

E1

  • (E1 x)

Exponential integral E1 for positive real numbers

Ei

  • (Ei x)

Exponential integral

Ein

  • (Ein x)

Exponential integral, alternative definition

En

  • (En n x)

Generalized exponential integral En

Li

  • (Li x)

Offset logarythmic integral

Si

  • (Si x)

Sine integral

airy-Ai

  • (airy-Ai x)

Airy Ai function

airy-Ai’

  • (airy-Ai' x)

First derivative of the Airy Ai function

airy-Bi

  • (airy-Bi x)

Airy Bi function

airy-Bi’

  • (airy-Bi' x)

First derivative of the Airy Bi function

bessel-I

  • (bessel-I order x)

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

bessel-I0

  • (bessel-I0 x)

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

bessel-I1

  • (bessel-I1 x)

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

bessel-J

  • (bessel-J order x)

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

bessel-J0

  • (bessel-J0 x)

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

bessel-J1

  • (bessel-J1 x)

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

bessel-K

  • (bessel-K order x)

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

bessel-K-half-odd

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

Bessel K_a function for a = order/2

Function accepts only odd integers for order

bessel-K0

  • (bessel-K0 x)

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

bessel-K1

  • (bessel-K1 x)

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

bessel-Y

  • (bessel-Y order x)

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

bessel-Y0

  • (bessel-Y0 x)

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

bessel-Y1

  • (bessel-Y1 x)

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

beta

  • (beta p q)

Beta function

digamma

  • (digamma x)

First derivative of log of Gamma function.

dirichlet-beta

  • (dirichlet-beta x)

Dirichlet Beta function

erf

  • (erf x)
  • (erf x1 x2)

Error function.

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

erfc

  • (erfc x)

Complementary error function.

eta

  • (eta x)

Dirichlet Eta function

gamma

  • (gamma x)

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

harmonic-number

  • (harmonic-number n)
  • (harmonic-number n m)

Harmonic number H_n or generalized harmonic number H_n,m

hypergeometric-0F0

  • (hypergeometric-0F0 x)

Hypergeometric ₀F₀ function, exp(x)

hypergeometric-0F1

  • (hypergeometric-0F1 a x)

Confluent hypergeometric ₀F₁ limit function.

hypergeometric-0F2

  • (hypergeometric-0F2 a b x)

Generalized hypergeometric ₀F₂ function.

hypergeometric-1F0

  • (hypergeometric-1F0 a x)

Hypergeometric ₁F₀ function.

hypergeometric-1F1

  • (hypergeometric-1F1 a b x)

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

hypergeometric-2F0

  • (hypergeometric-2F0 a b x)

Generalized hypergeometric ₂F₀ function.

hypergeometric-2F1

  • (hypergeometric-2F1 a b c x)

Gauss’s hypergeometric ₂F₁ function.

hypergeometric-pFq

  • (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

  • (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

  • (incomplete-beta x a b)

Incomplete Beta B(x,a,b)

inv-erf

  • (inv-erf x)

Inverse of erf function.

inv-erfc

  • (inv-erfc x)

Inverse of erfc function.

inv-gamma-1pm1

  • (inv-gamma-1pm1 x)

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

jinc

  • (jinc x)

Besselj1 devided by x

kummers-M

  • (kummers-M a b x)

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

lambert-W

  • (lambert-W x)

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

lambert-W-1

  • (lambert-W-1 x)

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

li

  • (li x)

Logarythmic integral

log-beta

  • (log-beta p q)

Logarithm of Beta function.

log-gamma

  • (log-gamma x)

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

log-gamma-1p

  • (log-gamma-1p x)

\(\ln\Gamma(1+x)\) for \(-0.5≤x≤1.5\).

lower-incomplete-gamma

  • (lower-incomplete-gamma s x)

Lower incomplete gamma function

minkowski

  • (minkowski x)

Minkowski’s question mark function ?(x)

polygamma

  • (polygamma m x)

Polygamma function of order m and real argument.

regularized-beta

  • (regularized-beta x a b)

Regularized Beta I_x(a,b)

regularized-gamma-p

  • (regularized-gamma-p a x)

Regularized gamma P(a,x)

regularized-gamma-q

  • (regularized-gamma-q a x)

Regularized gamma Q(a,x)

si

  • (si x)

Sine integral, Si shifted by -pi/2

spherical-bessel-1-i

  • (spherical-bessel-1-i order x)

First modified spherical Bessel function of the first kind.

spherical-bessel-1-i0

  • (spherical-bessel-1-i0 x)

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

spherical-bessel-1-i1

  • (spherical-bessel-1-i1 x)

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

spherical-bessel-1-i2

  • (spherical-bessel-1-i2 x)

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

spherical-bessel-2-i

  • (spherical-bessel-2-i order x)

Second modified spherical Bessel function of the first kind.

spherical-bessel-2-i0

  • (spherical-bessel-2-i0 x)

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

spherical-bessel-2-i1

  • (spherical-bessel-2-i1 x)

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

spherical-bessel-2-i2

  • (spherical-bessel-2-i2 x)

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

spherical-bessel-j

  • (spherical-bessel-j order x)

Spherical Bessel function of the first kind.

spherical-bessel-j0

  • (spherical-bessel-j0 x)

Spherical Bessel function of the first kind and order 0.

spherical-bessel-j1

  • (spherical-bessel-j1 x)

Spherical Bessel function of the first kind and order 1.

spherical-bessel-j2

  • (spherical-bessel-j2 x)

Spherical Bessel function of the first kind and order 2.

spherical-bessel-k

  • (spherical-bessel-k order x)

Modified spherical Bessel function of the second kind.

spherical-bessel-k0

  • (spherical-bessel-k0 x)

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

spherical-bessel-k1

  • (spherical-bessel-k1 x)

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

spherical-bessel-k2

  • (spherical-bessel-k2 x)

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

spherical-bessel-y

  • (spherical-bessel-y order x)

Spherical Bessel function of the second kind.

spherical-bessel-y0

  • (spherical-bessel-y0 x)

Spherical Bessel function of the second kind and order 0.

spherical-bessel-y1

  • (spherical-bessel-y1 x)

Spherical Bessel function of the second kind and order 1.

spherical-bessel-y2

  • (spherical-bessel-y2 x)

Spherical Bessel function of the second kind and order 2.

tricomis-U

  • (tricomis-U a b x)

Confluent hypergeometric function U of the second kind.

trigamma

  • (trigamma x)

Second derivative of log of Gamma function.

upper-incomplete-gamma

  • (upper-incomplete-gamma s x)

Upper incomplete gamma function

whittaker-M

  • (whittaker-M kappa mu x)

Whittaker’s M

whittaker-W

  • (whittaker-W kappa mu x)

Whittaker’s W

xi

  • (xi s)

Riemann (Landau’s) Xi function

zeta

  • (zeta s)
  • (zeta s z)

Riemann and Hurwitz zeta functions for real arguments