Function Encyclopedia

Standard Trigonometric Functions

sin(x)

Sine is a function defined as sin(x), which gives the ratio of the opposite side to the hypotenuse in a right-angled triangle.

cos(x)

Cosine is a function defined as cos(x), which gives the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

tan(x)

Tangent is a function defined as tan(x), which gives the ratio of sine to cosine.

csc(x)

cscant is the reciprocal of sine, defined as csc(x) = 1/sin(x).

sec(x)

Secant is the reciprocal of cosine, defined as sec(x) = 1/cos(x).

cot(x)

Cotangent is the reciprocal of tangent, defined as cot(x) = 1/tan(x).

Arc Trigonometric Functions

asin(x)

Arcsine is the inverse of sine, providing the angle whose sine is x.

acos(x)

Arccosine is the inverse of cosine, providing the angle whose cosine is x.

atan(x)

Arctangent is the inverse of tangent, providing the angle whose tangent is x.

acsc(x)

Arccscant is the inverse of cscant, providing the angle whose cscant is x.

asec(x)

Arcsecant is the inverse of secant, providing the angle whose secant is x.

acot(x)

Arccotangent is the inverse of cotangent, providing the angle whose cotangent is x.

-c Analogous Trigonometric Functions

sinc(x)

The cardinal sine function, defined as sinc(x) = sin(x)/x.

cosc(x)

The analog of sinc of the cosine function, defined as cosc(x) = cos(x)/x.

tanc(x)

The analog of sinc of the tangent function, defined as tanc(x) = tan(x)/x.

cscc(x)

The analog of sinc of the cosecant function, defined as cscc(x) = csc(x)/x.

secc(x)

The analog of sinc of the secant function, defined as secc(x) = sec(x)/x.

cotc(x)

The analog of sinc of the cotangent function, defined as cotc(x) = cot(x)/x.

jinc(x)

The analog of sinc of the Bessel J function, defined as jinc(x) = J1(x)/x.

somb(x)

Somvrero function, alternate jinc function that is normalised also called besinc function, defined as somb(x,y) = 2*J1(p*pi)/p*pi [p^2=x^2+y^2].

exsecc(x)

The analog of sinc of the secant function, defined as exsecc(x) = sec(x)/x.

excscc(x)

The analog of sinc of the cosecant function, defined as excscc(x) = csc(x)/x.

versinc(x)

The analog of sinc of the versed sine function, defined as versinc(x) = versin(x)/x.

vercosinc(x)

The analog of sinc of the vercosine function, defined as vercosinc(x) = vercos(x)/x.

coversinc(x)

The analog of sinc of the coversed sine function, defined as coversinc(x) = coversin(x)/x.

covercosc(x)

The analog of sinc of the covercosine function, defined as covercosc(x) = covercos(x)/x.

haversinc(x)

The analog of sinc of the haversine function, defined as haversinc(x) = haversin(x)/x.

hacovercosc(x)

The analog of sinc of the hacovercoe function, defined as hacovercosc(x) = hacovercos(x)/x.

sinc(x)

The analog of sinc of the sine function, defined as sinc(x) = sin(x)/x.

sinc(x)

The analog of sinc of the sine function, defined as sinc(x) = sin(x)/x.

sinc(x)

The analog of sinc of the sine function, defined as sinc(x) = sin(x)/x.

Inverse Cardinal Trigonometric Functions

asinc(x)

Inverse of The analog of sinc of the sine function

acosc(x)

Inverse of The analog of sinc of the cosine function

acosq(x)

Quarter inverse of The analog of sinc of the cosine function. Defined as acosq(x) = acosc(i*x/4pi)

casin(x)

The analog of sinc of the arcsine function, defined as casin(x) = asin(x)/x.

cacos(x)

The analog of sinc of the arccosine function, defined as cacos(x) = acos(x)/x.

catan(x)

The analog of sinc of the arctangent function, defined as catan(x) = atan(x)/x.

cacsc(x)

The analog of sinc of the arccscant function, defined as cacsc(x) = acsc(x)/x.

casec(x)

The analog of sinc of the arcsecant function, defined as casec(x) = asec(x)/x.

cacot(x)

The analog of sinc of the arccotangent function, defined as cot(x) = acot(x)/x.

Specialized Trigonometric Functions

Exsec(x)

Exsec is defined as sec(x) - 1.

Excsc(x)

Excsc is defined as csc(x) - 1.

Versin(x)

Versine is defined as 1 - cos(x).

Vercosin(x)

Vercosine is defined as 1 + cos(x).

Coversin(x)

Coversine is defined as 1 - sin(x).

Covercosine(x)

Covercosine is defined as 1 + sin(x).

Haversin(x)

Haversine is defined as 0.5 * (1 - cos(x)).

hacoverco(x)

hacovercoe is defined as 0.5 * (1 + sin(x)).

Arcversin(x)

Arcversine is the inverse of versine.

Arcvercos(x)

Arcvercosine is the inverse of vercosine.

Arccoversin(x)

Arccoversine is the inverse of coversine.

Arccovercos(x)

Arccovercosine is the inverse of covercosine.

Archaversin(x)

Archaversine is the inverse of haversine.

Archacovercos(x)

Archacovercoe is the inverse of hacovercoe.

Standard Hyperbolic Functions

sinh(x)

Sinh is defined as sinh(x) = (e^x - e^(-x)) / 2.

cosh(x)

Cosh is defined as cosh(x) = (e^x + e^(-x)) / 2.

tanh(x)

Tanh is defined as tanh(x) = sinh(x) / cosh(x).

csch(x)

Csch is the reciprocal of sinh, defined as csch(x) = 1/sinh(x).

sech(x)

Sech is the reciprocal of cosh, defined as sech(x) = 1/cosh(x).

coth(x)

Coth is the reciprocal of tanh, defined as coth(x) = 1/tanh(x).

Arc Hyperbolic Functions (Area Functions)

asinh(x)

Arcsinh is the inverse of sinh, providing the area whose sinh is x.

acosh(x)

Arccosh is the inverse of cosh, providing the area whose cosh is x.

atanh(x)

Arctanh is the inverse of tanh, providing the area whose tanh is x.

acsch(x)

Arcscsch is the inverse of csch, providing the area whose csch is x.

asech(x)

Arcsech is the inverse of sech, providing the area whose sech is x.

acoth(x)

Arccoth is the inverse of coth, providing the area whose coth is x.

Cardinal Hyperbolic Functions

sinhc(x)

Cardinal sinh function defined as sinhc(x) = sinh(x)/x.

coshc(x)

Cardinal cosh function defined as coshc(x) = cosh(x)/x.

tanhc(x)

Cardinal tanh function defined as tanhc(x) = tanh(x)/x.

cschc(x)

Cardinal csch function defined as cschc(x) = csch(x)/x.

sechc(x)

Cardinal sech function defined as sechc(x) = sech(x)/x.

cothc(x)

Cardinal coth function defined as cothc(x) = coth(x)/x.

cis and sic Functions

cis(x)

Function for cos and i sin 'cis': cis(x) = cos(x) + i*sin(x).

sic(x)

Function for sin and i cos 'sic': sic(x) = sin(x) + i*cos(x).

cas(x)

Function for sin and cos 'cas': cas(x) = sin(x) + cos(x).

cish and sich Functions

cish(x)

Function for hyperbolic cos and i sin 'cish': cish(x) = cosh(x) + i*sinh(x).

sich(x)

Function for hyperbolic sin and i cos 'sich': sich(x) = sinh(x) + i*cosh(x).

cash(x)

Function for hyperbolic cos and sin 'cash': sich(x) = cosh(x) + sinh(x).

Cardinal Combined Functions

cisc(x)

Function for cardinal cos and i sin 'cisc': cisc(x) = cis(x)/x.

sicc(x)

Function for cardinal sin and i sin 'sicc': sicc(x) = sic(x)/x.

casc(x)

Function for cardinal cos and sin 'casc': casc(x) = cas(x)/x.

Cardinal Hyperbolic Combined Functions

cishc(x)

Function for cardinal hyperbolic cos and i sin 'cishc': cishc(x) = cish(x)/x.

sichc(x)

Function for cardinal hyperbolic sin and i cos 'sichc': sichc(x) = sich(x)/x.

cashc(x)

Function for cardinal hyperbolic cos and sin 'cashc': cashc(x) = cash(x)/x.

sinp(x)

Function for 'sinp': sinp(x) = 2 * sinh(asinh((3x - 4)/2)/3).

cosp(x)

Function for 'cosp': cosp(x) = (3 - 2 * cosh((2/3) * asinh((3x - 4)/2))) * 1i.

tanp(x)

Function for 'tanp': tanp(x) = sinh((1/3) * asinh((3x - 4)/2)) / (3 - 2 * cosh((2/3) * asinh((3x - 4)/2))).

cscp(x)

Function for 'cscp': cscp(x) = 1 / (2 * sinh((1/3) * asinh((3x - 4)/2))).

secp(x)

Function for 'secp': secp(x) = 1 / (3 - 2 * cosh((2/3) * asinh((3x - 4)/2))).

cotp(x)

Function for 'cotp': cotp(x) = sinh((1/3) * asinh((3x - 4)/2)) / cosh((1/3) * asinh((3x - 4)/2)).

asinp(x)

Function for 'asinp': asinp(x) = (2 * sinh(3 * asinh(x/2)) + 4)/3.

acosp(x)

Function for 'acosp': acosp(x) = (2 * sinh(3 * acosh((3x - 2)/2)) + 4)/3.

acscp(x)

Function for 'acscp': acscp(x) = (2 * sinh(3 * asinh(1/x)) + 4)/3.

asecp(x)

Function for 'asecp': asecp(x) = (2 * sinh(3 * acosh((3 - 1/x)/2)) + 4)/3.

Arithmetic Mean

Function: arithmeticmean(a, b) = (a + b) / 2.

Geometric Mean

Function: geometricmean(a, b) = sqrt(a * b).

Arithmetic-Geometric Mean

Function: arithmeticgeometricmean(a, b) = ... (elaborate formula here).

Arithmetic-Harmonic Mean

Function: arithmeticharmonicmean(a, b) = ... (elaborate formula here).

Geometric-Harmonic Mean

Function: geometricharmonicmean(a, b) = ... (elaborate formula here).

Harmonic Mean

Function: harmonicmean(a, b) = 1 / ((1/a) + (1/b)).

Quadratic Mean

Function: quadraticmean(a, b) = sqrt((a^2 + b^2) / 2).

Cubic Mean

Function: cubicmean(a, b) = (a^3 + b^3)^(1/3).

Heronian Mean

Function: heronianmean(a, b) = (a + b + sqrt(a*b)) / 3.

Contraharmonic Mean

Function: contraharmonicmean(a, b) = (a^2 + b^2) / (a + b).

Neuman-Sandor Mean

Function: neumansandormean(a, b) = (a - b) / (2 * asinh((a - b) / (a + b))).

Neuman-Sandor T Mean

Function: neumansandortmean(a, b) = (a - b) / (2 * atan((a - b) / (a + b))).

Root Mean

Function: rootmean(a, b) = sqrt((a^2 + b^2) / 2).

Logarithmic Mean

Function: logarithmicmean(a, b) = (a - b) / (log(a) - log(b)).

Identric Mean

Function: identricmean(a, b) = (a^a / b^b)^(1 / (a - b)) / euler.

tocomp(x)

Convert x to a complex number.

conj(x)

Conjugate of a complex number.

arg(x)

Argument of a complex number.

abs(x)

Magnitude of a complex number.

rabs(x)

Real absolute of a complex number.

iabs(x)

Imaginary absolute of a complex number.

cabs(x)

Takes absolute of each component of a complex number.

sabs(x)

sabs(x) is defined as sabs(x)=sqrt(x^2)

norm(x)

Magnitude of a complex number squared.

proj(x)

Projection of a complex number.

signum(x)

Signum (unit) of a complex number.

dot(y, x)

Dot product of two complex numbers.

cross(y, x)

Cross product of two complex numbers.

im(x)

Imaginary part of a complex number.

re(x)

Real part of a complex number.

round(x)

Rounds each component of an complex number

ceiling(x)

Takes ceiling each component of an complex number

floor(x)

Takes floor of each component of an complex number

Cullen Number (x)

Function for 'Cullen': cullen(x) = x * 2^x + 1.

Mersenne Number (x)

Complex Domain Coloring Functi

Function for 'Mersenne': mersenne(x) = 2^x - 1.

Double Mersenne Number (x)

Function for 'Double Mersenne': doubleMersenne(x) = 2^(2^x - 1) - 1.

Fermat Number (x)

Function for 'Fermat': fermat(x) = 2^(2^x) + 1.

Proth Number (a, x)

Function for 'Proth': proth(a, x) = a * 2^x + 1.

Thabit Number (x)

Function for 'Thabit': thabit(x) = 3 * 2^x - 1.

Woodall Number (x)

Function for 'Woodall': woodall(x) = x * 2^x - 1.

Hilbert Number (x)

Function for 'Hilbert': hilbert(x) = 4x + 1.

Fibonacci Number (x)

Function for 'Fibonacci': fibonacci(x) = ((1.61803399^x - (-0.61803399)^x) / sqrt(5)).

Tribonacci Number (x)

Function for 'Tribonacci': tribonacci(x) = tribonacci(x-1)+tribonacci(x-2)+tribonacci(x-3) ; t0== 0,1,1.

Tribonacci Number (x)

Function for alternate 'Tribonacci': tribonaccialt(x) = tribonaccialt(x-1)+tribonaccialt(x-2)+tribonaccialt(x-3) ; t0== 0,0,1.

Lucas Number (x)

Function for 'Lucas': lucas(x) = 1.61803399^x - (-0.61803399)^x.

Leonardo Number (x)

Function for 'Leonardo': leonardo(x)=2*fibonacci(x) - 1 ; l(x) = 2l(x-1)-l(x-3) = l(n-1)+l(n-2)+1 ; l0==0,1.

Naryana Number (x)

Function for 'Naryana ': naryana(x) = ...; n(x)=n(x-1)+n(x-3) ; n0==1,1,1.

Padovan Number (x)

Function for 'Padovan': padovan(x) = 1.61803399^x - (-0.61803399)^x ; p(x) = p(x-2)+p(x-3) = p(x-1)+p(x-5) ; p0==1,1,1.

Perrin Number (x)

Function for 'Perrin': perrin(x) = padovan(x+1)+padovan(x-10) ; p(x)=p(x-2)+p(x-3) ; p0==3,0,2.

Necklace Cover 3(x)

Function for 'Necklace Cover 3': necklacecover3 = naryana(x)+2*naryana(x-3); a(n) = a(n-1) + a(n-3) ; a0 == 3,1,1,1.

Necklace Cover 4(x)

Function for 'Necklace Cover 3': a(n) = a(n-1) + a(n-4) ; a0 == 4,1,1,1,1.

Necklace Cover 5(x)

Function for 'Necklace Cover 3': a(n) = a(n-1) + a(n-5) ; a0 == 5,1,1,1,1,1.

Necklace Cover 6(x)

Function for 'Necklace Cover 3': a(n) = a(n-1) + a(n-6) ; a0 == 6,1,1,1,1,1,1.

Necklace Cover 7(x)

Function for 'Necklace Cover 3': a(n) = a(n-1) + a(n-7) ; a0 == 7,1,1,1,1,1,1,1.

Necklace Cover 8(x)

Function for 'Necklace Cover 3': a(n) = a(n-1) + a(n-8) ; a0 == 8,1,1,1,1,1,1,1,1.

Necklace Cover 9(x)

Function for 'Necklace Cover 3': a(n) = a(n-1) + a(n-9) ; a0 == 9,1,1,1,1,1,1,1,1,1.

Colatz 3x+1 (x)

Function for 'Collatz': collatz(x) = (x/2)*cos(pi*x/2)^2+(3x+1)*sin(pi*x/2)^2.

Oriented Tree (x)

Function for 'Oriented Tree': orientedTree(x) = round(0.2257 * (5.6465^x / x^(5/2))).

Magic (x)

Function for 'Magic': magic(x) = 2 * (ncr(x, 1) + ncr(x, 2) + ncr(x, 3)).

Magic Constant (x)

Function for 'Magic Constant': magicConst(x) = (x * (x^2 + 1)) / 2.

Alucin (x)

Function for 'Alucin': alucin(x) = x^3 / [(1 - x^2)(1 - x^3)(1 - x^4)].

Metallic Ratio (x)

Function for 'Metallic Ratio': metallicRatio(x) = (x + sqrt(x^2 + 4)) / 2.

Joukowsky (x)

Function for 'Joukowsky': joukowsky(x) = x + (1 / x).

Kármán-Trefftz (a, x)

Function for 'Kármán-Trefftz': karmantrefftz(a, x) = [((x + a)^(2 - c/pi))] / [((x - a)^(2 - c/pi))].

Symmetrical Joukowsky (x, a)

Function for 'Symmetrical Joukowsky': symmetricalJoukowsky(x, a) = exp(i*c) * [x - a + (1 / (x - a)) + 2a^2 / (a + c)].

Cayley (x)

Function for 'Cayley': cayley(x) = (x - i) / (x + 1).

Bilinear (x)

Function for 'Bilinear': bilinear(x) = (x - 1) / (x + 1).

Poincaré Disc Metric (a, x)

Function for 'Poincaré Disc Metric': poincareDiscMetric(a, x) = 2 * atanh((a - x) / (1 - a * conj(x))).

Poincaré Metric (a, x)

Function for 'Poincaré Metric': poincareMetric(a, x) = 2 * atanh((a - x) / (a - conj(x))).

Super Ballot Function

Function: superballot(b) = (60 * Gamma(2 * b + 1)) / (Gamma(b + 1) * Gamma(b + 4))

Necklace Numbers (k, n)

The Necklace Numbers calculate the number of n-bead necklaces with up to k different colored beads.

Function definition: necklacel(k, n) = ...

Corrected Necklace Numbers (k, n)

The Corrected Necklace Numbers are a variation of the necklace function, the number of n-bead necklaces with exactly k different colored beads.

Function definition: necklaceb(k, n) = ...

Eulerian Number 2 (n, k)

Function for 'Eulerian Number 2': euleriannum2(n, k) = ....

Schroder-Hipparchus Function (x)

Function for 'Schroder-Hipparchus': schroderhipparchus(x) = ....

Generalized Schroder-Hipparchus Function (k, x)

Function for 'Generalized Schroder-Hipparchus': generalizedschroderhipparchus(k, x) = ....

Iverson Bracket (t)

Function for 'Iverson Bracket': iversonbracketeq0(t) = ....

Diagentringer Number (x)

Function for 'Diagentringer Number': diagentringernum(x) = ....

Number of De Bruijn (k, n)

Function for 'Number of De Bruijn': numberofdebrujin(k, n) = ....

Stirling 2 (a, b)

Function for 'Stirling 2': stirling2(a, b) = ....

Stirling (x, y)

Function for 'Stirling': stirling(x, y) = ....

Unsigned Stirling (x, y)

Function for 'Unsigned Stirling': unsignedstirling(x, y) = ....

Gregory Coefficient Derivative (t, n)

Function for 'Gregory Coefficient D': gregorycoefd(t, n) = ....

Gregory Coefficient (n)

Function for 'Gregory Coefficient': gregorycoef(x) = ....

Nielsen Ramanujan Derivative (b, a)

Function for 'Nielsen Ramanujan': nielsenramanujand(b, a) = ....

Rastrigin (x, y)

Function for 'Rastrigin': rastrigin(a, b) = 10 * 2 + (a^2 - 10 * cos(2πa)) + (b^2 - 10 * cos(2πb)).

Ackley (x, y)

Function for 'Ackley': ackley(a, b) = -20 * e^(-0.5 * sqrt(a^2 + b^2)) - e^(0.5 * (cos(2πa) + cos(2πb))) + e + 20.

Sphere (x, y)

Function for 'Sphere': sphere(a, b) = a^2 + b^2.

Rosenbrock (x, y)

Function for 'Rosenbrock': rosenbrock(a, b) = 100 * (b - a^2)^2 + (1 - a)^2.

Beale (x, y)

Function for 'Beale': beale(a, b) = (1.5 - a + ab)^2 + (2.25 - a + ab^2)^2 + (2.625 - a + ab^3)^2.

Goldstein Price (x, y)

Function for 'Goldstein Price': goldsteinprice(a, b) = (1 + (a + b + 1)^2 * (19 - 14a + 3a^2 - 14b + 6ab + 3b^2)) * (30 + (2a - 3b)^2 * (18 - 32a + 12a^2 + 48b - 36ab + 27b^2)).

Booth (x, y)

Function for 'Booth': booth(a, b) = (a + 2b - 7)^2 + (2a + b - 5)^2.

Bukin (x, y)

Function for 'Bukin': bukin(a, b) = 100 * sqrt(|b - 0.01a^2|) + 0.01 |a + 10|.

Matyas (x, y)

Function for 'Matyas': matyas(a, b) = 0.26(a^2 + b^2) - 0.48ab.

Levi (x, y)

Function for 'Levi': levi(a, b) = sin(3πa)^3 + (1 - a)^2(1 + sin(3πb)^3) + (1 - b)^2(1 + sin(2πb)^2).

Himmelblau (x, y)

Function for 'Himmelblau': himmelblau(a, b) = (a^2 - 11)^2 + (a*b^2 - 7)^2.

Three Hump (x, y)

Function for 'Three Hump': threehump(a, b) = 2a^2 - 1.05a^4 + a^2b^2 + b^2.

Easom(x, y)

Function for 'Easom': easom(a, b) = cos(a) * cos(b) * e^(-((a - π)^2 + (b - π)^2)).

Cross Intray (x, y)

Function for 'Cross Intray': crossintray(a, b) = (-0.0001 - sin(a) * sin(b) * e^(|100 - sqrt(a^2 + b^2)|/π))^0.1.

Egg Holder (x, y)

Function for 'Egg Holder': eggholder(a, b) = -((b + 47) * sin(sqrt(|a/2 + 47|)) + sin(sqrt(|a - (b + 47)|))).

Holder Table (x, y)

Function for 'Holder Table': holdertable(a, b) = -|sin(a) * cos(b) * e^(1 - sqrt(a^2 + b^2)/π)|.

McCormick (x, y)

Function for 'McCormick': mccormick(a, b) = sin(a + b) + (a - b)^2 - 1.5a + 2.5b + 1.

Schaffer N.2 (a, b)

Function for 'Schaffer N.2': schaffern2(a, b) = \frac{\sin^2(a^2 - b^2) - 0.5}{(1 + 0.001(a^2 + b^2))^2}.

Schaffer N.4 (a, b)

Function for 'Schaffer N.4': schaffern4(a, b) = \frac{\cos^2(\sin(|a^2 - b^2|)) - 0.5}{(1 + 0.001(a^2 + b^2))^2}.

Styblinski-Tang (a, b)

Function for 'Styblinski-Tang': styblinskitang(a, b) = \frac{(a^4 - 16a) + (b^4 - 16b) + 5}{2}.

Mihrasbird (a, b)

Function for 'Mihrasbird': mihrasbird(a, b) = \sin(b) e^{(1 - \cos(a))^2} + \sin(a) e^{(1 - \cos(b))^2} + (a - b)^2.

Townsend (a, b)

Function for 'Townsend': townsend(a, b) = -1 - \cos((a - 0.1)b)^2 - a \sin(3a + b).

Gomez-Levi (a, b)

Function for 'Gomez-Levi': gomezlevi(a, b) = 4a^2 - 2.1a^4 + \frac{a^5}{6} + ab + 4b^2 - 4b^4.

Simionescu (a, b)

Function for 'Simionescu': simionescu(a, b) = 0.1ab.

Griewank (a, b)

Function for 'Griewank': griewank(a, b) = 1 + \frac{a^2 + b^2}{4000} - \cos(a) \cos\left(\frac{b}{\sqrt{2}}\right).

Schwefel 2.21 (a, b)

Function for 'Schwefel 2.21': schwefel221(a, b) = 0.01(|a| + |b|).

Schwefel 2.22 (a, b)

Function for 'Schwefel 2.22': schwefel222(a, b) = \max(|a|, |b|).

Bird (a, b)

Function for 'Bird': bird(a, b) = \sin(a) e^{(1 - \cos(b))^2} + \cos(b) e^{(1 - \sin(a))^2} + (a - b)^2.

Alpine (a, b)

Function for 'Alpine': alpine(a, b) = |a \sin(a) + 0.1a| + |b \sin(b) + 0.1b|.

SDP (a, b)

Function for 'SDP': sdp(a, b) = |a|^2 + |b|^3.

Sum of Squares on Sphere (a, b)

Function for 'Sum of Squares on Sphere': sumsquaresonsphere(a, b) = a^2 + b^2 - \cos(18a\pi) - \cos(18b\pi).

Michalewicz (a, b)

Function for 'Michalewicz': michalewicz(a, b) = -\sum_{i=1}^{\lfloor a \rfloor} i \sin(b) \sin\left(\frac{i b}{\lfloor a \rfloor}\right)^{2 \lfloor a \rfloor}.

Booth (a, b)

Function for 'Booth': booths(a, b) = (a + 2b - 7)^2 + (2a + b - 5)^2.

Sum of Squares (a, b)

Function for 'Sum of Squares': sumsquares(a, b) = a^2 + b^2.

Bohachevsky (a, b)

Function for 'Bohachevsky': bohachevsky(a, b) = a^2 + 2b^2 - 0.3\cos(3\pi a) - 0.4\cos(4\pi b) + 0.7.

Six Hump Camel (a, b)

Function for 'Six Hump Camel': sixhumpcamel(a, b) = (4 + a^2 - \frac{2.1a^2}{3} + a^4) + (b^2 - 4b^2)(-4 + 4b^2).

Shubert (a, b)

Function for 'Shubert': shubert(a, b) = \sum_{i=1}^{5}\sin((i + 1)a + i) \sin((i + 1)b + i).

Levy (a, b)

Function for 'Levy': levy(a, b) = \sin^2(3\pi a) + (a - 1)^2(1 + \sin^2(3\pi b))\.

Rosenbrock (a, b)

Function for 'Rosenbrock': rosenbrock(a, b) = (1 - a)^2 + 100(b - a^2)^2.

Rastrigin (a, b)

Function for 'Rastrigin': rastrigin(a, b) = 10n + (a^2 + b^2 - 10\cos(2\pi a) - 10\cos(2\pi b)).

Dedekind Eta Function (n)

Function for 'Dedekind Eta': dedekindeta(z) = e^{2\pi i z} \prod_{n=1}^{\infty} (1 - e^{2\pi i n z}).

Einstein Series (n, x)

Function for 'Einstein Series': einsteinseries(a, b) = \sum_{i=-5}^{5} \sum_{j=-5}^{5} \frac{1}{(i + jb)^a}, \quad (i, j) \neq (0, 0).

Fourier Einstein Function (n, x)

Function for 'Fourier Einstein': fouriereinstein(a, b) = 2 \cdot \sum_{i=-5}^{5} \sum_{j=-5}^{5} \frac{1}{(i + jb)^a} \cdot ZZZ(a).

Jacobi Theta 1 (n, x)

Function for 'Jacobi Theta 1': jacobitheta1(z, q) = 2 \cdot \sum_{n=0}^{\infty} (-1)^n q^{(n + 0.5)^2} \sin((2n + 1)z).

Jacobi Theta 2 (n, x)

Function for 'Jacobi Theta 2': jacobitheta2(z, q) = 2 \cdot \sum_{n=0}^{\infty} q^{(n + 0.5)^2} \cos((2n + 1)z).

Jacobi Theta 3 (n, x)

Function for 'Jacobi Theta 3': jacobitheta3(z, q) = 2 \cdot \sum_{n=1}^{\infty} q^{n^2} \cos(2nz).

Jacobi Theta 4 (n, x)

Function for 'Jacobi Theta 4': jacobitheta4(z, q) = 2 \cdot \sum_{n=1}^{\infty} (-1)^n q^{n^2} \cos(2nz).

Elliptic Modulus (n, x)

Function for 'Elliptic Modulus': ellipticModulus(a, b) = \left(\frac{\theta_2(a, b)}{\theta_1(a, b)}\right)^2.

Comp Elliptic Modulus (n, x)

Function for 'Comp Elliptic Modulus': compEllipticModulus(a, b) = \left(\frac{\theta_4(a, b)}{\theta_1(a, b)}\right)^2.

Elliptic Lambda (n, x)

Function for 'Elliptic Lambda': ellipticLambda(a, b) = \left(\frac{\theta_2(a, b)}{\theta_3(a, b)}\right)^4.

Klein J Invariant(x)

Function for 'J Invariant': jinvariant(b) = ((dedekindeta(x)^24+(256*dedekindeta(x*2)^24))^3)/(1728*dedekindeta(b)^48*dedekindeta(x*2)^24).

Picard-Fuchs J (x)

Function for 'Picard-Fuchs J': picardFuchsJ(b) = \frac{g_2(b)^3}{g_2(b)^3 - 27g_3(b)^2}.

Elliptic Discriminant ( x)

Function for 'Elliptic Discriminant': ellipticDiscriminant(b) = g_2(b)^3 - 27g_3(b)^2.

Elliptic Lambda Star (n, x)

Function for 'Elliptic Lambda Star': ellipticLambdaStar(a, b) = \left(\frac{\theta_2(a, b)}{\theta_3(a, b)}\right)^2.

Lacunary (a, b)

Function for 'Lacunary': lacunary(a, b) = Σ (b^(a^i)) for i = 0 to ∞.

Weber F (b)

Function for 'Weber F': weberf(b) = (dedekindeta(b)^2) / (dedekindeta(b/2) * dedekindeta(2b)).

Weber F1 (b)

Function for 'Weber F1': weberf1(b) = dedekindeta(b/2) / dedekindeta(b).

Weber F2 (b)

Function for 'Weber F2': weberf2(b) = (√2 * dedekindeta(2b)) / dedekindeta(b).

Weber R (a, b)

Function for 'Weber R': weberr(a, b) = (2^(a-1)/4 * qpocinf(b^a, b^(2a), bign)) / (qpocinf(b, b^2, bign)^a).

Weber R5 (a, b)

Function for 'Weber R5': weberr5(a, b) = (2^(5-1)/4 * qpocinf(b^a, b^10, bign)) / (qpocinf(b, b^2, bign)^5).

Polygonal Functions

Polygonal Function

Function: polygonal(a, b) = ((a - 2) * b^2 - (a - 4) * b) / 2

Antiside Polygonal Function

Function: antisidepolygonal(a, b) = (sqrt(8 * (a - 2) + b^2 - 4 * b) + (a - 4)) / (2 * a - 4)

Antipolygonal Function

Function: antipolygonal(a, b) = antisidepolygonal(a, b)

Centered Polygonal Function

Function: centeredpolygonal(a, b) = (a / 2 * b * (b - 1)) + 1

Pyramidal Function

Function: pyramidal(a, b) = (3 * b^2 + (b^2 * (a - 2)) - b - (a - 5)) / 6

Star Function

Function: star(b) = 6 * b * (b - 1) + 1

Star Prime Function

Function: starprime(b)

Superstar Prime Function

Function: superstarprime(b)

Reverse Superstar Function

Function: reversesuperstar(b)

Trigonal (b)

Function for 'Trigonal': trigonal(b) is a geometric function involving complex numbers.

Pentagonal (b)

Function for 'Pentagonal': pentagonal(b) is a geometric function involving complex numbers.

Hexagonal (b)

Function for 'Hexagonal': hexagonal(b) is a geometric function involving complex numbers.

Septagonal (b)

Function for 'Septagonal': septagonal(b) is a geometric function involving complex numbers.

Octagonal (b)

Function for 'Octagonal': octagonal(b) is a geometric function involving complex numbers.

Nonagonal (b)

Function for 'Nonagonal': nonagonal(b) is a geometric function involving complex numbers.

Decaagonal (b)

Function for 'Decaagonal': decaagonal(b) is a geometric function involving complex numbers.

Dodecagonal (b)

Function for 'Dodecagonal': dodecagonalgonal(b) is a geometric function involving complex numbers.

Icosagonal (b)

Function for 'Icosagonal': icosagonal(b) is a geometric function involving complex numbers.

Myriagonal (b)

Function for 'Myriagonal': myriagonal(b) is a geometric function involving complex numbers.

Gnomon (b)

Function for 'Gnomon': gnomon(b) = 2b + 1.

Pronic

Function for 'Pronic': pronic(b) = b(b + 1).

Polyhedral Functions

Hauy Octahedral

Function for 'Hauy Octahedral': hauyoctahedral(b) = (2b - 1)((2b²) - (2b + 3)) / 3.

Hauy Rhombic Dodecahedronal

Function for 'Hauy Rhombic Dodecahedronal': hauyrhombicdodecahedronal(b) = (2b - 1)((8b²) - (14b + 7)).

Hauy Square Pyramid

Function for 'Hauy Square Pyramid': hauysquarepyramid(b) = b(4b² - 1) / 3.

Octahedral

Function for 'Octahedral': octahedral(b) = b(2b² + 1) / 3.

Rhombic Dodecahedronal (b)

Function for 'Rhombic Dodecahedronal': rhombicdodecahedronal(b) is a geometric function involving complex numbers.

Truncoctahedral (b)

Function for 'Truncoctahedral': truncoctahedral(b) is a geometric function involving complex numbers.

Trunc Tetrahedral (b)

Function for 'Trunc Tetrahedral': trunctetrahedral(b) is a geometric function involving complex numbers.

Polytope+ Functions

Tetrahedral (b)

Function for 'Tetrahedral': tetrahedral(b) is a geometric function involving binomial coefficients.

Pentachoric (b)

Function for 'Pentachoric': pentachoric(b) is a geometric function involving binomial coefficients.

Simplex (b, a)

Function for 'Simplex': simplex(b, a) simplex(b,a)=ncr(b+a-1,a)

Perfect Power Functions

Dot

Function for 'Dot': x⁰ = 1.

Line

Function for 'Line': x¹ = x.

Square

Function for 'Square': sqr(x) = b².

Cube

Function for 'Cube': cum(x) = b³.

Tesseract

Function for 'Tesseract': tesseract(x) = b⁴.

Penteract

Function for 'Penteract': Penteract(x) = b⁵.

Biquadratic

Function for 'Biquadratic': biquadratic(x) = b⁴.

Surfolide

Function for 'Surfolide': surfolide(x) = b⁵.

Second Surfolide

Function for 'Second Surfolide': secondsurfolide(x) = b⁷.

Third Surfolide

Function for 'Third Surfolide': thirdsurfolide(x) = b¹¹.

Fourth Surfolide

Function for 'Fourth Surfolide': fourthsurfolide(x) = b¹³.

Fifth Surfolide

Function for 'Fifth Surfolide': fifthsurfolide(x) = b¹⁷.

Sixth Surfolide

Function for 'Sixth Surfolide': sixthsurfolide(x) = b¹⁹.

Seventh Surfolide

Function for 'Seventh Surfolide': seventhsurfolide(x) = b²³.

Nth Surfolide

Function for 'Nth Surfolide': nthsurfolide(x, a) = x^{nthPrime(a+2)}.

Zenzicube (x)

Function for 'Zenzicube': zenzicube(x) = x^6.

Cubicube (x)

Function for 'Cubicube': cubicube(x) = x^9.

Zenzizenzizenzic (x)

Function for 'Zenzizenzizenzic': zenzizenzizenzic(x) = x^8.

Zenzizenzicube (x)

Function for 'Zenzizenzicube': zenzizenzicube(x) = x^12.

Zenzizenzizenzizenzic (x)

Function for 'Zenzizenzizenzizenzic': zenzizenzizenzizenzic(x) = x^16.

Zenzicubicube (x)

Function for 'Zenzicubicube': zenzicubicube(x) = x^18.

Zenzizenzizenzicube (x)

Function for 'Zenzizenzizenzicube': zenzizenzizenzicube(x) = x^24.

Nth Zenzic (x, a)

Function for 'Nth Zenzic': nthzenzic(x, a) = x^(2^a).

Bessel K (a, x)

Function for 'Bessel K': besselk(a, x) = (Bessel I(-a, x) - Bessel I(a, x)) / sin(pi * a).

Bessel I (a, x)

Function for 'Bessel I': besseli(a, x) = Σ (x/2)^a / (Γ(n+1) * Γ(a+n+1)).

Bessel J (a, x)

Function for 'Bessel J': besseli(a, x) = idk.

Bessel Y (a, x)

Function for 'Bessel Y': bessely(a, x) = csc(pi * a) * (a * pi * J(a, x) - J(0, x)).

Hankel Function (1)

Function for 'Hankel Function 1': hankel1(a, x) = Bessel Y(a, x) + i * Bessel J(a, x).

Hankel Function (2)

Function for 'Hankel Function 2': hankel2(a, x) = Bessel J(a, x) - i * Bessel Y(a, x).

Spherical Bessel I (a, x)

Function for 'Spherical Bessel I': sphbesseli(a, x) = sqrt(π/(x+x)) * besseli(a + 0.5, x).

Spherical Bessel J (a, x)

Function for 'Spherical Bessel J': sphbesselj(a, x) = sqrt(π/(x+x)) * besselj(a + 0.5, x).

Spherical Bessel K (a,x)

Function for 'Spherical Bessel K': sphbesselk(a,x) = sqrt(π / (x + x)) * besselk(a + 0.5, x).

Spherical Bessel Y (a,x)

Function for 'Spherical Bessel Y': sphbessely(a,x) = sqrt(π / (x + x)) * bessely(a + 0.5, x).

Spherical Hankel 1 (a,x)

Function for 'Spherical Hankel 1': sphhankel1(a,x) = sqrt(π / (x + x)) * hankel1(a + 0.5, x).

Spherical Hankel 2 (a,x)

Function for 'Spherical Hankel 2': sphhankel2(a,x) = sqrt(π / (x + x)) * hankel2(a + 0.5, x).

Ricatti Bessel S (n, x)

Function for 'Ricatti Bessel S': ricattibessels(n, x) = x * sphbesselj(n, x).

Ricatti Bessel C (n, x)

Function for 'Ricatti Bessel C': ricattibesselc(n, x) = -1 * x * sphbessely(n, x).

Ricatti Bessel Xi (n,x)

Function for 'Ricatti Bessel Xi': ricattibesselxi(n,x) = x * sphhankel1(n,x).

Ricatti Bessel Zeta (n,x)

Function for 'Ricatti Bessel Zeta': ricattibesselzeta(n,x) = x * sphhankel2(n,x).

Neuman (a,x)

Function for 'Neuman': neuman(a,x) = (besselj(a, x) * cos(pi * a) - besselj(-a, x)) / sin(pi * a).

Struve H (a,x)

Function for 'Struve H': struve(a,x) = (x / 2)^(a + 1) * Σ ((-1)^n * (x / 2)^(2n) / (Γ(n + 3/2) * Γ(a + n + 3/2))).

Struve L (a,x)

Function for 'Struve L': struvel(a,x) = (x / 2)^(a + 1) * Σ ((x / 2)^(2n) / (Γ(n + 3/2) * Γ(a + n + 3/2))).

Struve K (a,x)

Function for 'Struve K': struvek(a,x) = struve(a,x) - struvel(a,x).

Struve M (a,x)

Function for 'Struve M': struvem(a,x) = struvevel(a,x) - besseli(a,x).

Scorer Gi' (t, x)

Function for 'Scorer Gi Derivative': scorergid(t, x) = ....

Scorer Hi' (t, x)

Function for 'Scorer Hi Derivative': scorerhid(t, x) = ....

Scorer Gi (x)

Function for 'Scorer G': scorergi(x) = ....

Scorer Hi (x)

Function for 'Scorer Hi': scorerhi(x) = ....

Ai (x)

Function for 'Ai': ai(x) = ....

Bi (x)

Function for 'Bi': bi(x) = ....

Ai' (x)

Function for 'Ai'': ai(x) = ....

Bi' (x)

Function for 'Bi'': bi(x) = ....

Airy C (x)

Function for 'Airy C': airyc(x) = bi(x) / ai(x).

Airy Zeta (x)

Function for 'Airy Zeta': airyzeta(x) = ....

Airy Bi Zeta (x)

Function for 'Airy Bi Zeta': airybizeta(x) = ....

Sinc Zeta (x)

Function for 'Sinc Zeta': sinczeta(x) = ....

Logarithm of Cosine (x,n)

Function for 'Logarithm of Cosine': lncos(x,n) = (log(cos(x)))^n.

Logarithm of Sine (x,n)

Function for 'Logarithm of Sine': lnsin(x,n) = (log(sin(x)))^n.

Logarithm of Tangent (x,n)

Function for 'Logarithm of Tangent': lntan(x,n) = (log(tan(x)))^n.

Logarithm of Secant (x,n)

Function for 'Logarithm of Secant': lnsec(x,n) = (log(sec(x)))^n.

Logarithm of Cotangent (x,n)

Function for 'Logarithm of Cotangent': lncot(x,n) = (log(cot(x)))^n.

Logarithm of cscant (x,n)

Function for 'Logarithm of cscant': lncsc(x,n) = (log(csc(x)))^n.

Log Integral Sine (x)

Function for 'Log Integral Sine': logsin(x) = ∫ ln(sin(t)) dt from 0 to π/2.

Log Integral Cosine (x)

Function for 'Log Integral Cosine': logcos(x) = ∫ ln(cos(t)) dt from 0 to π/2.

Log Integral Tangent (x)

Function for 'Log Integral Tangent': logtan(x) = ∫ ln(tan(t)) dt from 0 to π/2.

Log Integral Secant (x)

Function for 'Log Integral Secant': logsec(x) = ∫ ln(sec(t)) dt from 0 to π/2.

Log Integral cscant (x)

Function for 'Log Integral cscant': logcsc(x) = ∫ ln(csc(t)) dt from 0 to π/2.

Log Integral Cotangent (x)

Function for 'Log Integral Cotangent': logcot(x) = ∫ ln(cot(t)) dt from 0 to π/2.

Trigonometric Integral Auxiliary (x)

Function for 'Trigonometric Integral Auxiliary': triintgauxg(x) = -∫ cos(t) dt from x to bign + (π/2 - ∫ sin(t) dt from 0 to x) * sin(x).

Entire Exponential Integral (x)

Function for 'Entire Exponential Integral': ein(x) = ∫ e^(i*t) dt from 0 to x.

Si Integral (x)

Function for 'Si Integral': ssi(x) = -∫ sinh(t) dt from 0 to x.

Shi Integral (x)

Function for 'Shi Integral': shi(x) = -∫ sin(t) dt from 0 to x.

Reciprocal (x)

Function for 'Reciprocal': rec(x) = 1/x.

Chi Integral (x)

Function for 'Chi Integral': chi(x) = 0.5772156649 + log(x) * ∫ cosh(t) dt from 0 to x - ∫ (1/t) dt from 0 to x.

Exponential Integral Auxilary function (t,n)

Function for 'Exponential Integral Auxilary': expcp(t,nx) = exp(-nx[1]/t) * t^(n[0]-2).

Exponential Integral (n,x)

Function for 'Exponential Integral': en(n,x) = ∫ exp(t) dt from 1 to bign.

Ei Integral (x)

Function for 'Ei Integral': ei(x) = ∫ exp(t) dt from -bign to x.

Fresnel C (x)

Function for 'Fresnel C': fresnelc(x) = ∫ cos(t^2) dt from 0 to x.

Fresnel S (x)

Function for 'Fresnel S': fresnels(x) = ∫ sin(t^2) dt from 0 to x.

Fresnel T (x)

Function for 'Fresnel T': fresnelt(x) = ....

Fresnel CT (x)

Function for 'Fresnel CT': fresnelct(x) = ....

Fresnel SC (x)

Function for 'Fresnel SC': fresnelsc(x) = ....

Fresnel CS (x)

Function for 'Fresnel CS': fresnelcs(x) = ....

Gudermannian (x)

Function for 'Gudermannian': gudermannian(x) = integral(sech, 0, x).

Inverse Gudermannian (x)

Function for 'Inverse Gudermannian': invguderman(x) = integral(sec, 0, x).

Cosine Integral (x)

Function for 'Cosine Integral': ci(x) = -integral(cosc, x, bign).

Si (x)

Function for 'Si': si(x) = -integral(sinc, 0, x).

Trigonometric Integral Auxiliary Function (x)

Function for 'Triangular Integral Auxiliary': triintgauxf(x) = ....

Nielsen Cosine Integral (a, x)

Function for 'Nielsen Integral': nielsenci(a, x) = ....

Nielsen Sine Integral (a, x)

Function for 'Nielsen Sine Integral': nielsensi(a, x) = ....

Incomplete Elliptic Integral of the First Kind (F)

Function for 'F': ellint1(p, a) = integral(compellint1d, 0, p, a).

Complete Elliptic Integral of the First Kind (K)

Function for 'K': compellint1(a) = integral(compellint1d, 0, 1, a).

Incomplete Elliptic Integral of the Second Kind (E)

Function for 'E': ellint2(p, a) = integral(compellint2d, 0, p, a).

Complete Elliptic Integral of the Second Kind (E)

Function for 'E': compellint2(a) = integral(compellint2d, 0, 1, a).

Incomplete Elliptic Integral of the Third Kind (Π)

Function for 'Π': ellint3(n, p, a) = integral(compellint3d, 0, p, [n, a]).

Complete Elliptic Integral of the Third Kind (Π)

Function for 'Π': compellint3(n, a) = integral(compellint3d, 0, 1, [n, a]).

Partitions

Partition(x)

Durfee Square Generating Function Denominator Series(x) (why)

Durfee Square Generating Function(x)

Durfee Square of a Partition(x)

Least Durfee Partition(x)

Durfee Partition(x)

Durfee Polynomial(x)

Thomea

thomea(x)

getdenomeinstein(x)

getdenom(x)

getdenomangle(x,t)

getdenompolyangle(x,T)

getnom(x)

approxfraction(x)

approxfractionvalue(x)

fractionerror(x)

dirichlet(x)

Totient

mobius(x)

totient(x)

radical(x)

generalizedradical(x, t)

popovici(k, x)

sumofdivisors(x)

sumofdivisorsk(k, x)

jordantotient(k, x)

carmichael(x)

mangoldt(x)

gcd(a, x)

lcm(a, x)

sumofproperdivisors(x)

countdivisors(x)

liouville(x)

distinctprimefactors(x)

totalprimefactors(x)

Functions Related to Astronomy

Initial Mass

Initial Mass 2

Kroupa Initial Mass

Kroupa Tout Gilmore Initial Mass

Plank Radiation P

Plank Radiation

Larson Initial Mass A

Larson Initial Mass B

Salpeter Initial Mass

Initial Mass 2

Sigmoid

Function for 'Sigmoid': sigmoid(a) = 1 / (1 + e^-x).

Generalized Logistic

Function for 'Generalized Logistic': generalizedlogistic(b,a) = (1+e^-b)^-a.

Logistic Phi

Function for 'Logistic Phi': logisticphi(b,a) = (1-b*a)^(1/b).

Logistic Regression

Function for 'Logistic Regression': logisticregression(b,m,s) = 1 / (1 + e^-(x-m)/s).

Softplus

Function for 'Softplus': softplus(b) = log(1 + e^b).

Sobolev Tanh

Function for 'Sobolev Tanh': sobolevatanh(b) = tanh(b) + b / cosh(b)^2.

Swish

Function for 'Swish': swish(b) = b / (1 + e^-b).

Fermi-Dirac

Function for 'Fermi-Dirac': fermidirac(b) = 1 / (1 + e^b).

Bose-Einstein

Function for 'Bose-Einstein': boseeinstein(b, a) = b^a / (e^(b - globalc) - 1).

Einstein 1

Function for 'Einstein 1': einstein1(b) = b^2 e^b / (e^b - 1)^2.

Einstein 2

Function for 'Einstein 2': einstein2(b) = b / (e^b - 1).

Einstein 3

Function for 'Einstein 3': einstein3(b) = log(1 - e^-b).

Einstein 4

Function for 'Einstein 4': einstein4(b) = (b / (e^b - 1)) - log(1 - e^-b).

Probit

Function for 'Probit': h.

Logit Logistic

Function for 'Logit Logistic': logitlogistic(a, b) = a / (1 + e^-b).

Complementary Log-Log

Function for 'Complementary Log-Log': cloglog(b) = -log(-log(1 - e^-b)).

Gompertz

Function for 'Gompertz': gompertz(a, b) = e^(-e^(-a * (b - 1))).

Log-Logistic

Function for 'Log-Logistic': loglogistic(a, b) = 1 / (1 + (b / a)^-1).

Logistic Exponential

Function for 'Logistic Exponential': logisticexponential(a, b) = (a * e^b) / (1 + e^b).

Log Odds

Function for 'Log Odds': logodds(b) = log(b / (1 - b)).

Zex(x)

Inverse of lambertw: zex(x) = x*e^x.

Based Zex(x)

Inverse of based lambertw: zex(b,x) = x*b^x.

Lambert w(x)

Inverse of zex (x*e^x): lambertw(x) = integral{a 0->pi}(log(1+b*sinc(a)*exp(a/tan(a)))) da.

Based Lambert w(x)

Inverse of based zex (x*b^x): lambertw(x) = lambertw(a*log(b))/log(b).

Lambert t(x)

lambertt(x) = -lambertw(-x).

Lambert u(x)

lambertu(x) = lambertt(x)-lambertt(x)^2/2.

Lambert v(x)

lambertv(x) = log(1-lambertt(x))/2.

Peritet(x)

Period of base x tetrationperitet(x) = (pi*-2i)/(log(-lambertw(-log(b)))).

Filog(x)

Fixed Logarithm log_x(y)=yfilog(x) = -lambertw(-log(b))/log(b).

Wexzal(x)

wexzal(x) = ....

Superroot(x)

Inverse of x^xssrt(x) = log(x)/lambertw(log(x)).

Supercuberoot(x)

Inverse of x^x^xscbrt(x) = e^w(w( e^w(w(...e^w(w(b*log(b)))...*log(b))) *log(b))).

Dilbert Lambda(b)

dilbertlambda(b) = sqrt(2 * b^2) / sqrt(2).

Olga(b)

olga(b) = b / (b^2 + 1).

GLog(b)

glog(b) = lambertw(-1 / b).

Arcshoka(b)

arcshoka(b) = log(exp(b) - 1) / log(e - 1).

Arctania(b)

arctania(b) = b + log(b) - 1.

Anka(b)

anka(b) = b * exp(b - 1).

Logit(b)

logit(b) = -log(1 / b - 1).

Wright-W(b)

wrightw(b) = lambertw(exp(b)).

Tania(b)

tania(b) = lambertw(exp(b + 1)).

ArcTrappmann(b)

arctrappmann(b) = b - lambertw(exp(b)).

Doya(b)

doya(b) = lambertw(b * exp(b + 1)).

Bell Lambda(b)

belllambda(b) = b/lambertw(b).

Helper/Auxilary Functions

xpsin(x)

Function for 'xpsin': xpsin(x) = x + sin(x).

xmsin(x)

Function for 'xmsin': xmsin(x) = x - sin(x).

arcxmsin(x)

Function for 'arcxmsin': arcxmsin(x) = newtoninv("xmsin(x)", ...).

arcxpsin(x)

Function for 'arcxpsin': arcxpsin(x) = newtoninv("xpsin(x)", ...).

arcxmdsin(x, d)

Function for 'arcxmdsin': arcxmdsin(x, d) = newtoninv("x - d*sin(x)", ...).

tcospcost(x, t)

Function for 'tcospcost': tcospcost(x, t) = t*cos(x) + cos(t*x).

arctcospcost(x, d)

Function for 'arctcospcost': arctcospcost(x, d) = newtoninv("tcospcost(x, d)", ...).

tcosmcost(x, t)

Function for 'tcosmcost': tcosmcost(x, t) = t*cos(x) - cos(t*x).

arctcosmcost(x, d)

Function for 'arctcosmcost': arctcosmcost(x, d) = newtoninv("tcosmcost(x, d)", ...).

tcospdcost(x, d, t)

Function for 'tcospdcost': tcospdcost(x, d, t) = t*cos(x) + d*cos(t*x).

arctcospdcost(x, d, t)

Function for 'arctcospdcost': arctcospdcost(x, d, t) = newtoninv("tcospdcost(x, d, t)", ...).

wittgensteinsc(x, h, v, l)

Function for 'wittgensteinsc': wittgensteinsc(x, h, v, l) = ....

arcwittgensteinsc(x, h, v, l)

Function for 'arcwittgensteinsc': arcwittgensteinsc(x, h, v, l) = newtoninv("wittgensteinsc(x, h, v, l)", ...).

dtanpi4pcos(x, d)

Function for 'dtanpi4pcos': dtanpi4pcos(x, d) = d*tan(x + π/4) + cos(x).

arcdtanpi4pcos(x, d)

Function for 'arcdtanpi4pcos': arcdtanpi4pcos(x, d) = newtoninv("dtanpi4pcos(x, d)", ...).

sin2tan(x)

Function for 'sin2tan': sin2tan(x) = sin^3(x) / cos(x).

arcsin2tan(x)

Function for 'arcsin2tan': arcsin2tan(x) = newtoninv("sin^3(x)/cos(x)", ...).

Functions That Draw the Curve

Circle (x, p=1, parity=0)

Function for 'Circle': circle(x, p, parity) = sqrt(1-x*x).

Ellipse (x, px=1, py=1, parity=0)

Function for 'Ellipse': ellipse(x, px, py, parity) = ....

Tractrix (x, a=1)

Function for 'Tractrix': circle(x, p, parity) = inverse of a*cosh(a/x)-sqrt(a*a-x*x).

Superellipse (x, p=1, parity=0, n=2)

Function for 'Superellipse': superellipse(x, p, parity, n) = ....

Cycloid (x, p=1)

Function for 'Cycloid': cycloid(x, p) = ....

Tautochrone (x, p=1)

Function for 'Tautochrone': tautochrone(x, p) = ....

Trochoid (x, d=1, p=1)

Function for 'Trochoid': trochoid(x, d, p) = ....

Hypocycloid (x, k=1)

Function for 'Hypocycloid': hypocycloid(x, k) = ....

Hypotrochoid (x, d=1, k=1)

Function for 'Hypotrochoid': hypotrochoid(x, d, k) = ....

Epitrochoid (x, d=1, k=1)

Function for 'Epitrochoid': epitrochoid(x, d, k) = ....

Epicycloid (x, k=1)

Function for 'Epicycloid': epicycloid(x, k) = ....

Lissajous (x, wx=1, wy=1, dx=0, dy=0, ax=1, bx=1)

Function for 'Lissajous': lissajous(x, wx, wy, dx, dy, ax, bx) = ....

Lissajous Simple (x, wx=1, dx=0, ax=1, bx=1)

Function for 'Lissajous Simple': lissajoussimple(x, wx, dx, ax, bx) = ....

Wittgenstein's Function (x, h=1, v=1, l=1)

Function for 'Wittgenstein's': wittgensteins(x, h, v, l) = ....

Cardioid (x, parity=1, parity2=0)

Function for 'Cardioid': cardioid(x, parity, parity2) = ....

Deltoid (x, parity=1)

Function for 'Deltoid': deltoid(x, parity) = ....

Astroid (x, parity=1)

Function for 'Astroid': astroid(x, parity) = ....

Generalized Cardioid (x, parity=1, limach=1, parity2=0)

Function for 'Generalized Cardioid': generalizedcardioid(x, parity, limach, parity2) = ....

Nephroid (x, parity=2)

Function for 'Nephroid': nephroid(x, parity) = ....

Nephroid Y (x, parity=2)

Function for 'Nephroid Y': nephroidy(x, parity) = ....

Conchoid of de Sluze (x, a=1, parity=0)

Conchoid of Durer (y) (x, a=3, b=1, parity=0)

Conchoid of Durer (r) (x, a=3, b=1, parity=0)

Strophoid (x, a=1, parity=0)

Lemniscate of Bernoulli (x, a=1, parity=0, parity2=0)

Cissoid of Diocles (x, a=1, parity=0)

Cissoid of Diocles (y) (x, a=1, parity=0)

Devil's Curve (x, a=1, b=1, parity=0, parity2=0)

Devil's Curve (y) (x, a=1, b=1, parity=0, parity2=0)

Electric Motor (x, parity=0, parity2=0)

Electric Motor (y) (x, a=1, b=1, parity=0, parity2=0)

Lemniscate Gerono (x, parity=0, parity2=0)

Lemniscate Gerono (y) (x, parity=0)

cis Analouges

Cissoidal Limacon (x, b=2, a=1)

Cissoidal Cardioid (x, b=2, a=1)

Cissoidal Nephroid (x, a=1, n=3)

Cissoidal Deltoid (x, a=1)

Properties of the Curve

Conchoid of de Sluze - Curvature (x, a=1)

Conchoid of de Sluze - Tangential Angle (x, a=1)

Conchoid of de Sluze - Loops Area (x, a=1)

Conchoid of Nicomedes - Curvature (x, a=1, b=3)

Conchoid of Nicomedes - Tangential Angle (x, a=1, b=3)

Cissoid of Diocles - Arc Length (x, a=1)

Cissoid of Diocles - Curvature (x, a=1)

Cissoid of Diocles - Tangential Angle (x, a=1)

Hypergeometric Functions

Defined by Mellis-Barnes Integral

Generalized Hypergeometric pFq

Hypergeometric 0F0 Exponential Function = e^z

Hypergeometric 0F1 Confluent Hypergeometric Limit Function

Hypergeometric 0F2

Hypergeometric 0F3

Hypergeometric 1F0 Binomial Series

Hypergeometric 1F1

Hypergeometric 1F2

Hypergeometric 2F0

Hypergeometric 2F1 Gauss Hypergeometric Function

Hypergeometric 2F2

Hypergeometric 3F2

Hypergeometric 3F0

...

Humbert Series

Humbert Phi_2(b1,b2,c;x,y) = F_1(-,b1,b2,c;x,y)

Humbert Phi_3(b,c;x,y) = F_1(-,b,-,c;x,y)

Humbert Psi_1(a,b,c1,c2;x,y) = F_2(a,b,-,c1,c2;x,y)

Humbert Psi_2(a,c1,c2;x,y) = F_2(a,-,-,c1,c2;x,y)

Humbert Xi_1(a1,a2,b,c;x,y) = F_3(a1,a2,b,-,c;x,y)

Humbert Xi_2(a,b,c;x,y) = F_3(a,-,b,-,c;x,y)

Apell Series

Apell F_1(a,b1,b2,c;x,y) = sum{m,n=0->inf} ((a)_{m+n}(b1)_{m}(b2)_{n}/(c)_{m+n}m!n!) x^my^n

Apell F_2(a,b1,b2,c1,c2;x,y) = sum{m,n=0->inf} ((a)_{m+n}(b1)_{m}(b2)_{n}/(c)_{m+n}m!n!) x^my^n

Apell F_3(a1,a2,b1,b2,c;x,y) = sum{m,n=0->inf} ((a1)_{m}(a2)_{n}(b1)_{m}(b2)_{n}/(c1)_{m}(c2)_{n}m!n!) x^my^n

Apell F_4(a,b,c1,c2;x,y) = sum{m,n=0->inf} ((a)_{m+n}(b)_{m+n}/(c1)_{m}(c2)_{n}m!n!) x^my^n

Apell F_1 Integral Representation(a,b1,b2,c;x,y) = gamma(c)/(gamma(a)gamma(c-a)) integral{t=0->1} t^a-1 * (1-t)^c-a-1 * (1-ct)^-b1 * (1-yt)^-b2 dt

Lauricel Series

Lauricel D(a,b1,b2,b3,c;x,y,z) = sum{m,n,k=0->inf} ((a)_{m+n+k}(b1)_{m}(b2)_{n}(b3)_{k}/(c)_{m+n+k}m!n!k!) x^my^n*z^k

Lauricel A(a,b1,b2,b3,c1,c2,c3;x,y,z) = sum{m,n,k=0->inf} ((a)_{m+n}(b1)_{m}(b2)_{n}(b3)_{k}/(c)_{m+n+k}m!n!k!) x^my^n*z^k

Lauricel B(a1,a2,a3,b1,b2,b3,c;x,y,z) = sum{m,n,k=0->inf} ((a1)_{m}(a2)_{n}(a3)_{k}(b1)_{m}(b2)_{n}(b3)_{k}/(c)_{m+n+k}m!n!k!) x^my^n*z^k

Lauricel C(a,b,c1,c2,c3;x,y,z) = sum{m,n,k=0->inf} ((a)_{m+n+k}(b)_{m+n+k}/(c1)_{m}(c2)_{n}(c3)_{k}m!n!k!) x^my^n*z^k

Multiparameter Factorial Functions

Multi q-pochhammer symbol - multiqpochhammer([a,b,c],q,n) = (a,b,c;q)_n = (a;q)_n(b;q)_n(c;q)_n

Multi infinite q-pochhammer symbol - multiqpochhammer([a,b,c],q) = (a,b,c;q)_inf = (a;q)_inf(b;q)_inf(c;q)_inf

Multi Modified Jacobi Theta - polymodifiedjacobitheta([a,b,c],q) = theta(a,b,c;q) = theta(a;q)theta(b;q)theta(c;q)

Multi Elliptic Shifted Factorial - polyellipticshiftedfactorial([a,b,c],q,p,n) = (a,b,c;q,p)_n = (a;q,p)_n(b;q,p)_n(c;q,p)_n

Selberg Delta - selbergdelta(A) = prod(1<=j<N)prod(j<k<=N) (A[j]-A[k])

Generalized Tribonacci (Binet's Formula n=3) (x , a,b,c,r,s,t)

Function for 'Generalized Tribonacci': r1*A^x+r2*B^x+r3*C^x ; f(x)=f(x-1)*r+f(x-2)*s+f(x-3)*t ; f0 == a,b,c

Escapetime Fractals

Mandelbrot

Function for 'Mandelbrot': z_n+1 = z_n^2+c.

Perpendicular Mandelbrot

Function for 'perpendicularmandelbrot': z_n+1 = (conj(rabs(z_n)))^2+c.

Tricorn

Function for 'tricorn': z_n+1 = conj(z_n^2)+c.

Celtic

Function for 'celtic': z_n+1 = rabs(z_n^2)+c.

Perpendicular Celtic

Function for 'perpendicularceltic': z_n+1 = conj(rabs(z_n^2))+c.

Burning Ship

Function for 'burningship': z_n+1 = iabs(z_n^2)+c.

Perpendicular Burning Ship

Function for 'perpendicularburningship': z_n+1 = iabs(z_n)^2+c.

Buffalo

Function for 'buffalo': z_n+1 = cabs(z_n^2)+c.

Perpendicular Buffalo

Function for 'perpendicularbuffalo': z_n+1 = iabs(rabs(z_n)^2)+c.

Simonbrot

Function for 'simonbrot': z_n+1 = (z*|z|)^2+c.

Perpendicular Simonbrot

Function for 'perpendicularsimonbrot': z_n+1 = (z_n^2*|z_n^2|)+c.

Lambda Fractal

Function for 'lambdafractal': z_n+1 = c*z*(1-z).

Duck Fractal

Function for 'duckfractal': z_n+1 = log(iabs(z)+c).

Custom Function

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