All of these JavaScript functions are exposed in the global context for ease of use. Hyperlinks lead to plots in two dimensions of the real and imaginary parts of functions on the real and imaginary axes, as well as visualizations in three dimensions of the real and imaginary parts and their absolute value on the complex plane.
abs( x ) — absolute value of a real or complex number
arg( x ) — argument of a real or complex number
pow( x, y ) — power of a real or complex number to a real or complex exponent
root( x, y ) — root of a real or complex number with real or complex degree
surd( x, n ) — real-valued root of a real number
sqrt( x ) — square root of a real or complex number
exp( x ) — exponential of a real or complex number
logisticSigmoid( x ) — logistic sigmoid of a real or complex number
log( x ) — natural logarithm of a real or complex number
log( x, base ) — logarithm of a real or complex number to a real or complex base
ln( x ) — natural logarithm of a real or complex number
lambertW( x ) — principal branch of the Lambert W function of a real or complex number
lambertW( k, x ) — arbitrary branch of integral index k of the Lambert W function of a real or complex number
inverseLambertW( x ) — inverse of the Lambert W function of a real or complex number
wrightOmega( x ) — Wright omega function of a real or complex number
sin( x ) — sine of a real or complex number
cos( x ) — cosine of a real or complex number
tan( x ) — tangent of a real or complex number
cot( x ) — cotangent of a real or complex number
sec( x ) — secant of a real or complex number
csc( x ) — cosecant of a real or complex number
arcsin( x ) — inverse sine of a real or complex number
arccos( x ) — inverse cosine of a real or complex number
arctan( x ) — inverse tangent of a real or complex number
arccot( x ) — inverse cotangent of a real or complex number
arcsec( x ) — inverse secant of a real or complex number
arccsc( x ) — inverse cosecant of a real or complex number
sinh( x ) — hyperbolic sine of a real or complex number
cosh( x ) — hyperbolic cosine of a real or complex number
tanh( x ) — hyperbolic tangent of a real or complex number
coth( x ) — hyperbolic cotangent of a real or complex number
sech( x ) — hyperbolic secant of a real or complex number
csch( x ) — hyperbolic cosecant of a real or complex number
arcsinh( x ) — inverse hyperbolic sine of a real or complex number
arccosh( x ) — inverse hyperbolic cosine of a real or complex number
arctanh( x ) — inverse hyperbolic tangent of a real or complex number
arccoth( x ) — inverse hyperbolic cotangent of a real or complex number
arcsech( x ) — inverse secant of a real or complex number
arccsch( x ) — inverse hyperbolic cosecant of a real or complex number
sinc( x ) — cardinal sine of a real or complex number
haversine( x ) — haversine of a real or complex number
inverseHaversine( x ) — inverse haversine of a real or complex number
gudermannian( x ) — Gudermannian function of a real or complex number
inverseGudermannian( x ) — inverse Gudermannian function of a real or complex number
besselJ( n, x ) — Bessel function of the first kind of real or complex order n of a real or complex number
besselJZero( n, m ) — mth zero of the Bessel function of the first kind of real order n
besselJZero( n, m, true ) — mth zero of the first derivative of the Bessel function of the first kind of real positive order n
besselY( n, x ) — Bessel function of the second kind of real or complex order n of a real or complex number
besselYZero( n, m ) — mth zero of the Bessel function of the second kind of real order n
besselYZero( n, m, true ) — mth zero of the first derivative of the Bessel function of the second kind of real positive order n
besselI( n, x ) — modified Bessel function of the first kind of real or complex order n of a real or complex number
besselK( n, x ) — modified Bessel function of the second kind of real or complex order n of a real or complex number
hankel1( n, x ) — Hankel function of the first kind of real or complex order n of a real or complex number
hankel2( n, x ) — Hankel function of the second kind of real or complex order n of a real or complex number
airyAi( x ) — Airy function of the first kind of a real or complex number
airyAiPrime( x ) — derivative of the Airy function of the first kind of a real or complex number
airyBi( x ) — Airy function of the second kind of a real or complex number
airyBiPrime( x ) — derivative of the Airy function of the second kind of a real or complex number
sphericalBesselJ( n, x ) — spherical Bessel function of the first kind of real or complex order n of a real or complex number
sphericalBesselY( n, x ) — spherical Bessel function of the second kind of real or complex order n of a real or complex number
sphericalHankel1( n, x ) — spherical Hankel function of the first kind of real or complex order n of a real or complex number
sphericalHankel2( n, x ) — spherical Hankel function of the second kind of real or complex order n of a real or complex number
struveH( n, x ) — Struve function of real or complex order n of a real or complex number
struveL( n, x ) — modified Struve function of real or complex order n of a real or complex number
hermite( n, x ) — Hermite polynomial of real or complex index n of a real or complex number
laguerre( n, x ) — Laguerre polynomial of real or complex index n of a real or complex number
laguerre( n, a, x ) — associated Laguerre polynomial of real or complex index n and real or complex parameter a of a real or complex number
legendreP( l, x ) — Legendre polynomial of real or complex index l of a real or complex number
legendreP( l, m, x ) — associated Legendre polynomial of real or complex indices l and m of a real or complex number
legendreQ( l, x ) — Legendre function of the second kind of real or complex index l of a real or complex number
legendreQ( l, m, x ) — associated Legendre function of the second kind of real or complex indices l and m of a real or complex number
sphericalHarmonic( l, m, θ, φ ) — spherical harmonic of integer indices l and m and real numbers. Returns a complex number even if the result is purely real.
chebyshevT( n, x ) — Chebyshev polynomial of the first kind of real or complex index n of a real or complex number
chebyshevU( n, x ) — Chebyshev polynomial of the second kind of real or complex index n of a real or complex number
ellipticF( x, m ) — incomplete elliptic integral of the first kind of a real or complex number with real or complex elliptic parameter m
ellipticF( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
ellipticK( m ) — complete elliptic integral of the first kind of a real or complex elliptic parameter m
ellipticE( x, m ) — incomplete elliptic integral of the second kind of a real or complex number with real or complex elliptic parameter m
ellipticE( m ) — complete elliptic integral of the second kind of a real or complex elliptic parameter m
ellipticPi( n, x, m ) — incomplete elliptic integral of the third kind of a real or complex number with real or complex characteristic n and elliptic parameter m
ellipticPi( n, m ) — complete elliptic integral of the third kind of a real or complex elliptic characteristic n and parameter m
jacobiZeta( x, m ) — Jacobi zeta function of a real or complex number with real or complex elliptic parameter m, with the first argument of the same type as for elliptic integrals
carlsonRC( x, y ) — degenerate Carlson symmetric elliptic integral of the first kind of real or complex numbers
carlsonRD( x, y, z ) — degenerate Carlson symmetric elliptic integral of the third kind, or Carlson elliptic integral of the second kind, of real or complex numbers
carlsonRF( x, y, z ) — Carlson symmetric elliptic integral of the first kind of real or complex numbers
carlsonRG( x, y, z ) — Carlson completely symmetric elliptic integral of the second kind of real or complex numbers
carlsonRJ( x, y, z, w ) — Carlson symmetric elliptic integral of the third kind of real or complex numbers
jacobiTheta( n, x, q ) — Jacobi theta function n of a real or complex number with real or complex nome q
ellipticNome( m ) — elliptic nome q of a real or complex elliptic parameter m
am( x, m ) — Jacobi amplitude of a real or complex number with real or complex elliptic parameter m
sn( x, m ) — Jacobi elliptic sine of a real or complex number with real or complex elliptic parameter m
cn( x, m ) — Jacobi elliptic cosine of a real or complex number with real or complex elliptic parameter m
dn( x, m ) — Jacobi delta amplitude of a real or complex number with real or complex elliptic parameter m
weierstrassRoots( g2, g3 ) — Weierstrass roots e1, e2 and e3 for real or complex invariants. Returned as an array.
weierstrassHalfPeriods( g2, g3 ) — Weierstrass half periods w1 and w2 for real or complex invariants. Returned as an array. Consistent with evaluation of Weierstrass elliptic function in terms of Jacobi elliptic sine.
weierstrassInvariants( w1, w2 ) — Weierstrass invariants g2 and g3 for real or complex half periods. Returned as an array.
weierstrassP( x, g2, g3 ) — Weierstrass elliptic function ℘ of a real or complex number with real or complex invariants. Returned as a complex number for consistency.
weierstrassPPrime( x, g2, g3 ) — derivative of the Weierstrass elliptic function ℘ of a real or complex number with real or complex invariants. Returned as a complex number for consistency.
inverseWeierstrassP( x, g2, g3 ) — inverse Weierstrass elliptic function ℘ of a real or complex number with real or complex invariants. Returned as a complex number for consistency.
kleinJ( x ) — Klein j-invariant of a complex number
hypergeometric0F1( a, x ) — confluent hypergeometric function of a real or complex parameter a of a real or complex number
hypergeometric1F1( a, b, x ) — confluent hypergeometric function of the first kind of real or complex parameters a and b of a real or complex number
hypergeometricU( a, b, x ) — confluent hypergeometric function of the second kind of real or complex parameters a and b of a real or complex number
whittakerM( k, m, x ) — Whittaker function of the first kind of real or complex parameters k and m of a real or complex number
whittakerW( k, m, x ) — Whittaker function of the second kind of real or complex parameters k and m of a real or complex number
hypergeometric2F1( a, b, c, x ) — Gauss hypergeometric function of real or complex parameters a, b and c of a real or complex number
hypergeometric1F2( a, b, c, x ) — hypergeometric function of real or complex parameters a, b and c of a real or complex number
hypergeometricPFQ( A, B, x ) — generalized hypergeometric function of arrays of real or complex parameters A and B of a real or complex number
factorial( n ) — factorial of a real or complex number
factorial2( n ) — double factorial of a real or complex number
binomial( n, m ) — binomial coefficient of real or complex numbers
multinomial( n1, n2, … ) — multinomial coefficient of real or complex numbers
pochhammer( x, n ) — Pochhammer symbol of real or complex numbers
subfactorial( n ) — subfactorial of a real or complex number
logGamma( x ) — logarithm of the gamma function of a real or complex number
gamma( x ) — gamma function of a real or complex number
gamma( x, y ) — upper incomplete gamma function Γ(x,y) of real or complex numbers
gamma( x, 0, y ) — lower incomplete gamma function γ(x,y) of real or complex numbers
gamma( x, y, z ) — generalized incomplete gamma function γ(x,z) − γ(x,y) of real or complex numbers
gammaRegularized( x, y ) — regularized upper incomplete gamma function Q(x,y) of real or complex numbers
gammaRegularized( x, y, z ) — generalized regularized incomplete gamma function Q(x,z) − Q(x,y) of real or complex numbers
beta( x, y ) — beta function of real or complex numbers
beta( x, y, z ) — incomplete beta function Bx(y,z) of real or complex numbers
beta( x, y, z, w ) — generalized incomplete beta function By(z,w) − Bx(z,w) of real or complex numbers
betaRegularized( x, y, z ) — regularized incomplete beta function Ix(y,z) of real or complex numbers
betaRegularized( x, y, z, w ) — generalized regularized incomplete beta function Iy(z,w) − Ix(z,w) of real or complex numbers
digamma( x ) — digamma function of a real or complex number
polygamma( n, x ) — polygamma function of positive integer order of a real or complex number
erf( x ) — error function of a real or complex number
erfc( x ) — complementary error function of a real or complex number
erfi( x ) — imaginary error function of a real or complex number
fresnelS( x ) — Fresnel sine integral of a real or complex number
fresnelC( x ) — Fresnel cosine integral of a real or complex number
expIntegralEi( x ) — exponential integral Ei of a real or complex number
logIntegral( x ) — logarithmic integral of a real or complex number
sinIntegral( x ) — sine integral of a real or complex number
cosIntegral( x ) — cosine integral of a real or complex number
sinhIntegral( x ) — hyperbolic sine integral of a real or complex number
coshIntegral( x ) — hyperbolic cosine integral of a real or complex number
expIntegralE( n, x ) — generalized exponential integral En of a real or complex order n of a real or complex number
zeta( x ) — Riemann zeta function of a real or complex number
dirichletEta( x ) — Dirichlet eta function of a real or complex number
riemannXi( n ) — Riemman xi function of a real or complex number
bernoulli( n ) — Bernoulli number for index n. Returns a generalized value for fractional, negative or complex indices.
bernoulli( n, x ) — Bernoulli polynomial for real or complex index n of a real or complex number
harmonic( n ) — harmonic number for index n
hurwitzZeta( x, a ) — Hurwitz zeta function of a real or complex number with real or complex parameter a
polylog( n, x ) — polylogarithm function of real or complex order n of a real or complex number
chop( x ) — set real and complex parts smaller than 10−10 to zero
chop( x, tolerance ) — set real and complex parts smaller than tolerance to zero
round( x ) — closest integer to a real or complex number
round( x, y ) — closest integer multiple of y to a real or complex number
ceiling( x ) — closest integer greater than a real or complex number
floor( x ) — closest integer less than a real or complex number
sign( x ) — signum function of a real or complex number
integerPart( x ) — integer part of a real or complex number
fractionalPart( x ) — fractional part of a real or complex number
random() — random real number between zero and one
random( x ) — random real or complex number between zero and x
random( x, y ) — random real or complex number between x and y
kronecker( i, j ) — Kronecker delta δij for real or complex arguments
kronecker( i, j, k, … ) — Kronecker delta δijk… for an arbitrary number of real or complex arguments
piecewise( [ function, [begin,end] ], … ) — piecewise expression defined on an arbitrary number of subdomains returned as a function
doubleLambert( x, y ) — principle branch of a double Lambert function of two real or complex numbers
doubleLambert( n, x, y ) — arbitrary branch of integral index n of a double Lambert function of two real or complex numbers