Utility File Functions

The following is an alphabetical list of the functions defined in the utility files in Math directory distributed with Derive (see the Utility File LibraryG5BS2R). These functions are automatically load on demand.

ADJOINTI._DEZ(A) adjoint of square matrix A

AI_SERIES3JDX.V4(z,m) m+1 terms of series approximation for Airy function Ai(z)

ALMOST_LINOA7SLO(r,b,p,q,x,y,x0,y0) implicit solution of r(x,y)y'+p(x)b(y)=q(x) if almost linear

ALMOST_LIN_GENOA7SLO(r,b,p,q,x,y,c) general solution of r(x,y)y'+p(x)b(y)=q(x) if almost linear

APPEND_COLUMNSI._DEZ(A,B) append columns of A and B

APPROX_EIGENVECTORI._DEZ(A,μ) approximate eigenvector of A for approximate eigenvalue μ

ARC_LENGTH2W1Y_YU(y,x,x1,x2) arc length of y(x) from x=x1 to x2

ARC_LENGTH2W1Y_YU(y,x,x1,x2,μ) integral of μ(x) along arc y(x) from x=x1 to x2

ARCSPlotting_Complex_valued_Expressions(w,z,r0,rm,m,θ0,θn,n) w-plane map of polar z-plane grid r=r0...rm, θ=θ0...θn

AREA2W1Y_YU(x,x1,x2,y,y1,y2) area of region x=x1 to x2 and y=y1(x) to y2(x)

AREA2W1Y_YU(x,x1,x2,y,y1,y2,μ) integral of μ(x,y) over region

AREA_CENTROID2W1Y_YU(x,x1,x2,y,y1,y2) area centroid of region

AREA_CENTROID2W1Y_YU(x,x1,x2,y,y1,y2,μ) centroid of density μ(x,y) over region

AREA_INERTIA2W1Y_YU(x,x1,x2,y,y1,y2) area inertia tensor of region

AREA_INERTIA2W1Y_YU(x,x1,x2,y,y1,y2,μ) inertia tensor of density μ(x,y)

AREA_OF_REVOLUTION2W1Y_YU(y,x,x1,x2) area of y(x) revolved about x-axis

AreaBetweenCurvesPlotting_Areas_and_Integrals(u,v,x,a,b) plots area between u(x) and v(x) from x = a to b (a < b)

AreaOverCurvesPlotting_Areas_and_Integrals(u,x,a,b) plots area over u(x) and under the x-axis from x = a to b (a < b)

AreaUnderCurvesPlotting_Areas_and_Integrals(u,x,a,b) plots area under u(x) and above the x-axis from x = a to b (a < b)

AREAY_OF_REVOLUTION2W1Y_YU(y,x,x1,x2) area of y(x) revolved about y-axis

ASSOCIATED_LEGENDRE_PHBCBN(n,m,x) nth associated Legendre polynomial Pn(m)(x)

AUTONOMOUSOB7SLO(r,v) dv/dy, given y"=r(y,v), reducing to sequence of two 1st order

AUTONOMOUS_CONSERVATIVEOB7SLO(q,x,y,x0,y0,v0) solves y"=q(y), y=y0 and y'=v0 at x=x0

BELL1338PND(n) nth Bell or exponential number

BERNOULLI1338PND(n) nth Bernoulli number

BERNOULLI_ODEOA7SLO(p,q,k,x,y,x0,y0) implicit solution of Bernoulli equation y'+p(x)y=q(x)y^k

BERNOULLI_ODE_GENOA7SLO(p,q,k,x,y,c) general solution of Bernoulli equation y'+p(x)y=q(x)y^k

BERNOULLI_POLY1338PND(n,x) nth Bernoulli polynomial evaluated at x

BESSEL_I3JDX.V4(n,z) modified Bessel function of 1st kind In(z)

BESSEL_I_ASYMP3JDX.V4(n,z) 2-term asymptotic approximation for In(z)

BESSEL_I_SERIES3JDX.V4(n,z,m) m+1 terms of series approximation for In(z)

BESSEL_J3JDX.V4(n,z) Bessel function of 1st kind Jn(z)

BESSEL_J_ASYMP3JDX.V4(n,z) 1-term asymptotic approximation for Jn(z)

BESSEL_J_LIST3JDX.V4(n,z) vector of Bessel functions of 1st kind J0(z) through Jn(z)

BESSEL_J_SERIES3JDX.V4(n,z,m) m+1 terms of series approximation for Jn(z)

BESSEL_K3JDX.V4(n,z) modified Bessel function of 2nd kind Kn(z)

BESSEL_K_ASYMP3JDX.V4(n,z) 2-term asymptotic approximation for Kn(z)

BESSEL_Y3JDX.V4(n,z) Bessel function of 2nd kind Yn(z), n fractional

BESSEL_Y_ASYMP3JDX.V4(n,z) 1-term asymptotic approximation for Yn(z)

BESSEL_Y_SERIES3JDX.V4(n,z,m) m+1 terms of series approximation for Yn(z), n integer

BI_SERIES3JDX.V4(z,m) m+1 terms of series approximation for Airy function Bi(z)

BINOMIAL_DENSITY0FMTUG(k,n,p) binomial probability density function

BINOMIAL_DISTRIBUTION0FMTUG(k,n,p) cumulative probability binomial distribution function

CATALAN1338PND(n) nth Catalan number

CENTER_OF_CURVATUREAUKXVG(y,x) center of curvature of y(x)

CENTERED1338PND(n,p) nth p-sided centered number

CENTERED_CUBE1338PND(n,d) nth d-dimensional centered cube number

CENTERED_HEX1338PND(n,d) nth d-dimensional centered hex number

CENTERED_PYRAMID1338PND(n,p) nth p-sided d-dimensional centered pyramid number

CHEBYCHEV_THBCBN(n,x) nth Chebychev polynomial of 1st kind Tn(x)

CHEBYCHEV_T_LISTHBCBN(n,x) vector of first n Chebychev polynomials of 1st kind

CHEBYCHEV_UHBCBN(n,x) nth Chebychev polynomial of 2nd kind Un(x)

CHEBYCHEV_U_LISTHBCBN(n,x) vector of first n Chebychev polynomials of 2st kind

CHI_SQUARE0FMTUG(u,v) Chi-square distribution P(u|v), u = χ²

CI7RN4D(z) cosine integral Ci(z), -π < phase z < π

CLAIRAUTOA7SLO(p,q,x,y,v,c) vector of general and implicit solutions of generalized Clairaut differential equation

CLAIRAUT_DIFOKAKSN(p,q,d,x,y,c) implicit solution of Clairaut difference equation

COFACTORI._DEZ(A,i,j) numerator of element i,j of inverse of square matrix A

CONE2WQ_YS1(ϕ,θ,z) 3D coordinate vector of cone at an angle of ϕ radians from z-axis

CONTINUED_FRACTION19ICN1P(x,n) vector of n+1 partial quotients of continued fraction of x

CONVERGENT19ICN1P(x,k) kth convergent of x based on continued fraction of x

CONVERGENTS19ICN1P(x,k) vector of first k+1 convergents of x based on continued fraction of x

COPROJECTION2WQ_YS1(A) convert matrix A from lines of constant t to constant s

COVARIANT_METRIC_TENSORI._DEZ(A) covariant metric tensor of Jacobian matrix A

CRT19ICN1P(a,m) solution of system of linear congruence equations x = ai mod mi

CURVATUREAUKXVG(y,x) curvature of y(x)

CYCLOTOMICY5C_1X(n,x) simplifies to the nth cyclotomic polynomial in x

CYLINDER2WQ_YS1(r,θ,z) 3D coordinate vector of cylinder of radius r from z-axis

CYLINDRICAL_VOLUME2W1Y_YU(r, r1,r2,θ,θ1,θ2,z,z1,z2) volume of region in cylindrical coordinates

CYLINDRICAL_VOLUME2W1Y_YU(r,r1,r2,θ,θ1,θ2,z,z1,z2,μ) integral of μ(r,θ,z) over region in cylindrical coordinates

DAWSONXYKLH9(x) Dawson’s integral F(x)

DEF_INT_PARTS144O15P(u,v,x,a,b) integral of u(x)·v(x) from x=a to b using integration by parts

DEF_INT_SUBST144O15P(y,x,u,a,b) integral of y(x) from x=a to b using substitution

DIF_DATAY5C_1X(A) 1st derivative of 2-column numeric data matrix A

DIF_NUMERICY5C_1X(y,x,x0,h,n) nth derivative of y wrt x at x0 using step size h

DIF2_DATAY5C_1X(A) 2nd derivative of 2-column numeric data matrix A

DIGAMMA_PSI0FMTUG(z) approximation for digamma function ψ(z)

DILOG1CKZXC(x) dilogarithm function of x

DIRECTION_FIELD1NBVQQO(r,x,x0,xm,m,y,y0,yn,n) vector that plots as direction field for y'=r(x,y)

DISTINCT_PARTS1338PND(n) number of decompositions of n into integer summands

DIVISOR_SIGMA19ICN1P(k,n) sum of kth powers of positive divisors of n

DIVISOR_TAU19ICN1P(n) number of divisors of n

DIVISORS19ICN1P(n) ordered vector of all positive divisors of n

DSOLVE1OA7SLO(p,q,x,y,x0,y0) specific solution of p(x,y)+q(x,y)y'=0 with initial conditions y=y0 at x=x0

DSOLVE1_GENOA7SLO(p,q,x,y,c) general solution of p(x,y)+q(x,y)y'=0 in terms of c

DSOLVE2OB7SLO(p,q,r,x,c1,c2) general solution of y"+p(x)y'+q(x)y=r(x) in terms of c1 and c2

DSOLVE2_BVOB7SLO(p,q,r,x,x0,y0,x2,y2) specific solution of y"+p(x)y'+q(x)y=r(x) with boundary conditions

DSOLVE2_IVOB7SLO(p,q,r,x,x0,y0,v0) specific solution of y"+p(x)y'+q(x)y=r(x) with initial conditions

EI7RN4D(x,m) m terms of series approximation for exponential integral Ei(x)

EI17RN4D(z,m) m terms of series approximation for E1(z), -π < phase z < π

EIF296HRX(Φ,m,n) n iteration approximation of elliptic integral of 1st kind F(Φ|m)

ELLIPTIC_E296HRX(Φ,m) approximation of elliptic integral of 2nd kind E(Φ|m)

ELLIPTIC_F296HRX(Φ,m) approximation of elliptic integral of 1st kind F(Φ|m)

ELLIPTIC_PI296HRX(Φ,m,n) approximation of elliptic integral of 3rd kind II(n;Φ|m)

EN7RN4D(n,z) nth exponential integral En(z), real part of z > 0

EN_ASYMP7RN4D(n,z,m) m+1 terms of asymptotic approximation for En(z)

EULER1338PND(n) nth Euler number

EULER_ODE1NBVQQO(r,x,y,x0,y0,h,n) vector of n+1 solution points of y'=r(x,y) using Euler's method

EULER_BETA0FMTUG(z,w) Euler's beta function B(z,w)

EULER_PHI19ICN1P(n) Euler's totient function ϕ(n)

EULER_POLY1338PND(n,x) nth Euler polynomial evaluated at x

EXACTOA7SLO(p,q,x,y,x0,y0) implicit specific solution of p(x,y)+q(x,y)y'=0, if it is exact

EXACT_EIGENVECTORI._DEZ(A,μ) eigenvector of matrix A corresponding to exact eigenvalue μ

EXACT_GENOA7SLO(p,q,x,y,c) implicit general solution of p(x,y)+q(x,y)y'=0, if it is exact

EXACT2OB7SLO(p,q,x,y,v,c) reduces order of p(x,y,v)y"+q(x,y,v)=0 with v=y', if it is exact

EXTENDED_GCD19ICN1P(a,b) vector [g, [x, y]] of integers such that g = gcd(a, b) = x·a+y·b

EXTRACT_2_COLUMNS1NBVQQO(A,j,k) matrix composed of jth and kth columns of matrix A

F_DISTRIBUTION0FMTUG(F,v1,v2) cumulative probability F-distribution P(F|v1,v2)

FAREY19ICN1P(n) vector of Farey fractions of order n

FIBONACCI19ICN1P(n) nth Fibonacci number

FIXED_POINT.S02QQ(g,x,x0,n) n iterations of vector x=g(x), starting at x=x0

FORCE0I._DEZ(A,i,j,p) force element i,j of A to 0 using pivot row p

FOURIER2W1Y_YU(y,t,t1,t2,n) nth harmonic Fourier series of y(t) from t=t1 to t2

FRESNEL_COSXYKLH9(z) approximates to Fresnel cosine integral C(z)

FRESNEL_COS_ASYMPXYKLH9(z) 5-term asymptotic approximation for cosine integral C(z)

FRESNEL_COS_JXYKLH9(z,m) approximation of C(z) based on sum of m+1 spherical Bessel functions

FRESNEL_COS_SERIESXYKLH9(z,m) m+1 terms of series approximation for cosine integral C(z)

FRESNEL_SINXYKLH9(z) approximates to Fresnel sine integral S(z)

FRESNEL_SIN_ASYMPXYKLH9(z) 5-term asymptotic approximation for sine integral S(z)

FRESNEL_SIN_JXYKLH9(z,m) approximation of S(z) based on sum of m+1 spherical Bessel functions

FRESNEL_SIN_SERIESXYKLH9(z,m) m+1 terms of series approximation for sine integral S(z)

FUN_LIN_CCFOA7SLO(r,p,q,k,x,y,x0,y0) implicit solution of y'=r(p·x+q·y+k), if p,q,k constant

FUN_LIN_CCF_GENOA7SLO(r,p,q,k,x,y,c) general solutions of y'=r(p·x+q·y+k), if p,q,k constant

GAUSS3XSHYO3(a,b,c,z) Gauss hypergeometric function F(a,b;c;z)

GAUSS_SERIES3XSHYO3(a,b,c,z,m) m+1 terms of series for Gauss hypergeometric function

GEGENBAUER_CHBCBN(n,a,x) nth Gegenbauer ultraspherical polynomial Cn(a)(x)

GEGENBAUER_C_LISTHBCBN(n,a,x) vector of first n Gegenbauer ultraspherical polynomials

GEN_HOMOA7SLO(r,x,y,x0,y0) implicit specific solution of y'=r(x,y), if r is generalized homogeneous

GEN_HOM_GENOA7SLO(r,x,y,c) implicit general solution of y'=r(x,y), if r is generalized homogeneous

GEN_LUCAS19ICN1P(n,p,q,L0,L1) nth term of the generalized Lucas sequence L(n)

GENERALIZED_LAGUERREHBCBN(n,a,x) nth generalized Laguerre polynomial Ln(a)(x)

GENERALIZED_LAGUERRE_LISTHBCBN(n,a,x) vector of first n generalized Laguerre a polynomials

GEOMETRIC1OKAKSN(k,p,q,x,x0,y0) solution of 1st order linear-geometric recurrence equation

GEOMETRY_MATRIXI._DEZ(θ,G) geometry matrix for coordinate vector θ having metric tensor G

GOODNESS_OF_FIT144O15P(u,x,A) standard deviation of u(x) from data in matrix A

GRID_LINESPlotting_Grid_Lines_and_Points(a,b,s,o) plots as three sets of grid lines in a 3D-plot window

GRID_POINTSPlotting_Grid_Lines_and_Points(a,b,s,o) plots as three sets of grid points in a 3D-plot window

HERMITE_HHBCBN(n,x) nth Hermite polynomial Hn(x)

HERMITE_H_LISTHBCBN(n,x) vector of first n Hermite polynomials

HERMITE_HEHBCBN(n,x) nth associated Hermite polynomial HEn(x)

HERMITE_HE_LISTHBCBN(n,x) vector of first n associated Hermite polynomials

HOMOGENEOUSOA7SLO(r,x,y,x0,y0) specific solution of y'=r(x,y), if r is homogeneous

HOMOGENEOUS_GENOA7SLO(r,x,y,c) general solution of y'=r(x,y), if r is homogeneous

HORIZONTALSPlotting_Complex_valued_Expressions(w,z,z00,zmn,m,n) 2D plots as w-plane map of z-plane grid from z00 to zmn

HURWITZ_ZETA1CKZXC(s,a,m) m+1 terms of series approximation for Hurwitz generalized zeta function

HYPERGEOMETRIC_DENSITY0FMTUG(k,n,m,j) hypergeometric probability density function

HYPERGEOMETRIC_DISTRIBUTION0FMTUG(k,n,m,j) cumulative hypergeometric probability distribution function

HYPERGEOMETRIC_SERIES3XSHYO3(plist,qlist,z,m) m+1 terms of series for generalized hypergeometric function

IMP_CENTER_OF_ CURVATUREAUKXVG(u,x,y) center of curvature of implicit function u(x,y)=0

IMP_CURVATUREAUKXVG(u,x,y) curvature of implicit function u(x,y)=0

IMP_DIFAUKXVG(u,x,y,n) nth derivative of implicit function u(x,y)=0

IMP_OSCULATING_CIRCLEAUKXVG(u,x,y,x0,y0,θ) circle osculating implicit function u(x,y)=0 at (x0,y0) in terms of θ

IMP_PERPENDICULARAUKXVG(u,x,y,x0,y0) line perpendicular to implicit function u(x,y)=0 at (x0,y0)

IMP_TANGENTAUKXVG(u,x,y,x0,y0) line tangent to implicit function u(x,y)=0 at (x0,y0)

INCOMPLETE_BETA0FMTUG(x,z,w) incomplete beta function Bx(z,w)

INCOMPLETE_GAMMA0FMTUG(z,w) incomplete gamma function P(z,w), real part z > 0

INCOMPLETE_GAMMA_SERIES0FMTUG(z,w,m) m+1 terms of series approximation for P(z,w)

INT_DATAY5C_1X(A) antiderivative of numerical data matrix A

INT_PARTS144O15P(u,v,x) antiderivative of u(x)·v(x) using integration by parts

INT_SUBST144O15P(y,x,u) antiderivative of y(x) by substituting x for u(x)

INTEGRATING_FACTOROA7SLO(p,q,x,y,x0,y0) specific solution of p(x,y)+q(x,y)y'=0, if integrating factor exists

INTEGRATING_FACTOR_GENOA7SLO(p,q,x,y,c) general solution of p(x,y)+q(x,y)y'=0, if integrating factor exists

INVERSE144O15P(u,x) inverse of u(x) with respect to x

ISOMETRIC2WQ_YS1(v) 2D isometric projection of 3D coordinate vector v

ISOMETRICS2WQ_YS1(v,s,s0,sm,m,t,t0,tn,n) vector that 2D plots as isometric projection of 3D coordinate vector v

JACOBI19ICN1P(a,b) Jacobi symbol (a/b)

JACOBI_AM296HRX(u,m,n) Jacobi elliptic amplitude function

JACOBI_PHBCBN(n,a,b,x) nth Jacobi polynomial Pn(a,b)(x)

JACOBI_P_LISTHBCBN(n,a,b,x) vector of 1st n Jacobi (a,b) polynomials

JACOBIANI._DEZ(u,v) Jacobian matrix of the coordinate transformation x=u(v1,v2, …, vm)

KI296HRX(m,n) complete elliptic integral of the 1st kind

KRONECKERI._DEZ(i,j) Kronecker delta function

KUMMER3XSHYO3(a,b,z) Kummer's confluent hypergeometric function M(a,b,z)

KUMMER_SERIES3XSHYO3(a,b,z,m) m+1 terms of series approximation for M(a,b,z)

LAGUERRE_LHBCBN(n,x) nth Laguerre polynomial Ln(x)

LAGUERRE_L_LISTHBCBN(n,x) vector of first n Laguerre polynomials

LAPLACE2W1Y_YU(y,t,s) Laplace transform of y(t) for transform domain variable

LEFT_RIEMANN144O15P(u,x,a,b,n) left Riemann sum of n rectangles of integral of u(x) from x=a to b

LEGENDRE_PHBCBN(n,x) nth Legendre polynomial Pn(x)

LEGENDRE_P_LISTHBCBN(n,x) vector of first n Legendre polynomials

LERCH_PHI1CKZXC(z,s,a,m) m terms of series approximation for Lerch transcendent function Φ(z,s,a)

LI7RN4D(x,m) m terms of series approximation for logarithmic integral li(x), x>1

LIM2144O15P(u,x,y,x0,y0) limit of u as [x,y] -> [x0,y0] along slope of @1

LIN_FRACOA7SLO(r,a,b,c,p,q,k,x,y,x0,y0) specific solution of linear fractional equation y'=r((ax+by+c)/(px+qy+k))

LIN_FRAC_ GENOA7SLO(r,a,b,c,p,q,k,x,y,c) general solution of linear fractional equation y'=r((ax+by+c)/(px+qy+k))

LIN1_DIFFERENCEOKAKSN(p,q,x, x0,y0) specific solution of recurrence equation y(x+1)=p(x)y(x)+q(x), y(x0)=y0

LIN2_CCFOKAKSN(p,q,r,x,c1,c2) general solution of difference equation y(x+2)+p·y(x+1)+q·y(x)=r(x)

LIN2_CCF_BVOKAKSN(p,q,r,x,x0,y0,x2,y2) specific solution of difference equation y(x+2)+p·y(x+1)+q·y(x)=r(x)

LINEAR_CORRELATION_COEFFICIENT(A) linear correlation coefficient of 2-column x-y matrix A

LINEAR1OA7SLO(p,q,x,y,x0,y0) explicit solution of linear monic equation y'+p(x)y=q(x)

LINEAR1_GENOA7SLO(p,q,x,y,c) general solution of linear monic equation y'+p(x)y=q(x)

LIOUVILLEOB7SLO(p,q,x,y,c1,c2) general solution of Liouville equation y"+p(x)y'+q(y)(y')^2=0

LUCAS19ICN1P(n) nth Lucas number beginning at 1

LUCAS_LEHMER19ICN1P(p) if p is an odd prime and 2^p-1 is prime, returns true; otherwise returns false

MATPRODI._DEZ(A,B,i,j) element i,j of the dot product of A and B

MERSENNE19ICN1P(n) nth Mersenne prime 2^p - 1

MERSENNE_DEGREE19ICN1P(n) exponent p of nth Mersenne prime 2^p - 1

MERSENNE_LIST19ICN1P(n) exponents p of first n Mersenne prime 2^p - 1

MINORI._DEZ(A,i,j) delete row i and column j from A

MOEBIUS_MU19ICN1P(n) Moebius mu function of n

MONOMIAL_TESTOA7SLO(p,q,x,y) integrating factor of p(x,y)+q(x,y)y'=0, if result of the form x^m·y^n

NEWTON.S02QQ(u,x,x0,n) n iterations of Newton's method applied to equation u(x)=0 with initial guess of x=x0

NEWTONS.S02QQ(u,x,x0,n) n iterations of Newton's method applied to a system of equations

NEXT_MERSENNE_DEGREE19ICN1P(n) smallest prime p>n such that Mersenne number 2^p - 1 is prime

NORMAL_LINEAUKXVG(u,v,v0,t) line normal to surface u=0 at v=v0, using parameter t

NTH_PRIME19ICN1P(n) nth prime number

OCTAHEDRAL1338PND(n) nth octahedral number

OSCULATING_CIRCLEAUKXVG(y,x,θ) circle osculating y(x), in terms of θ

OUTERI._DEZ(v,w) outer product of vectors v and w

PADE12K9KB.(y,x,x0,n,d) approx y(x) near x=x0, n=numr deg, d=denr deg, n=d or d-1

PARA_ARC_LENGTH2W1Y_YU(v,t,t1,t2) arc length of vector v(t) from t=t1 to t2

PARA_ARC_LENGTH2W1Y_YU(v,t,t1,t2,μ) integral of μ(t) along v(t) from t=t1 to t2

PARA_CENTER_OF_CURVATUREAUKXVG(v,t) center of curvature of v=[x(t),y(t)]

PARA_CURVATUREAUKXVG(v,t) curvature of v=[x(t),y(t)]

PARA_DIFAUKXVG(v,t,n) nth derivative of v=[x(t),y(t)]

PARA_OSCULATING_CIRCLEAUKXVG(v,t,t0,Φ) circle osculating v=[x(t),y(t)] at t=t0 in terms of Φ

PARA_PERPENDICULARAUKXVG(v,t,t0,x) line perpendicular to v=[x(t),y(t)] at t=t0 in terms of x

PARA_TANGENTAUKXVG(v,t,t0,x) line tangent to v=[x(t),y(t)] at t=t0 in terms of x

PARTITIONI._DEZ(v,n,d) partition of vector v into vectors of length n with an offset delta of d

PARTS1338PND(n) number of decompositions of n into integer summands

PARTS_LIST1338PND(n) n+1 element vector of PARTS(0) through PARTS(n)

PELL19ICN1P(n) nth Pell number

PENTATOPE1338PND(n) nth pentatope number

PERFECT19ICN1P(n) nth perfect number (i.e. numbers that are equal to the sum of their divisors)

PERPENDICULARAUKXVG(y,x,x0) line perpendicular to y(x) at x=x0

PICARD1NBVQQO(r,p,x,y,x0,y0) improved series approximation of ODE, given the series p(x)

PIVOTI._DEZ(A,i,j) force column j below row i to 0 by pivoting

PlotIntPlotting_Areas_and_Integrals(u,x,a,b) plots the definite integral of the function u(x) from x = a to b

POCHHAMMER0FMTUG(a,x) Pochhammer symbol function (a)x

POISSON_DENSITY0FMTUG(k,t) Poisson probability density

POISSON_DISTRIBUTION0FMTUG(k,t) cumulative probability Poisson distribution function

POLAR2WQ_YS1(r,θ) 2D coordinate vector of point at radius r and co-longitude θ

POLAR_ARC_LENGTH2W1Y_YU(r,θ,θ1,θ2) arc length of polar r(θ) from θ1 to θ2

POLAR_ARC_LENGTH2W1Y_YU(r,θ,θ1,θ2,μ) integral of μ(θ) along arc r(θ)

POLAR_AREA2W1Y_YU(r,r1,r2,θ,θ1,θ2) area of θ=θ1 to θ2 and r=r1(θ) to r2(θ)

POLAR_AREA2W1Y_YU(r,r1,r2,θ,θ1,θ2,μ) integral of μ(θ) over region

POLAR_CENTER_OF_CURVATUREAUKXVG(r,θ) center of curvature of r(θ)

POLAR_CURVATUREAUKXVG(r,θ) curvature of r(θ)

POLAR_DIFAUKXVG(r,θ,n) nth derivative of r(θ)

POLAR_OSCULATING_CIRCLEAUKXVG(r,θ,θ0,Φ) circle osculating r(θ) at θ=θ0 in terms of Φ

POLAR_PERPENDICULARAUKXVG(r,θ,θ0,x) line perpendicular to r(θ) at θ=θ0 in terms of x

POLAR_TANGENTAUKXVG(r,θ,θ0,x) line tangent to r(θ) at θ=θ0 in terms of x

POLY_COEFF144O15P(u,x,n) coefficient of x^n term in polynomial u(x)

POLY_DEGREE144O15P(u,x) degree of polynomial u(x)

POLYGON_FILLFilling_Polygons(u) fills the 2D or 3D convex polygon u

POLY_INTERPOLATE144O15P(A,x) polynomial interpolation of data matrix A

POLY_INTERPOLATE_EXPRESSION144O15P(u,x,a) polynomial in x that interpolates u given 1D vector of points a

POLYGAMMA0FMTUG(n,z,m) m+1 terms of series approximation for the nth polygamma function Ψn(z)

POLYGONAL1338PND(n,p) nth p-sided polygonal number

POLYGONAL_PYRAMID1338PND(n,p,d) nth p-sided d-dimensional polygonal pyramid number

POLYLOG1CKZXC(n,z,m) m terms of series approximation for Jonquière's polylogarithm function Li(n)(z)

PRIMEPI19ICN1P(x,d,a) number of primes p ≤ x, such that p is of the form p = k·d + a for some k ≥ 0

PRIME_POWER?19ICN1P(n) if n is a power of a prime number, returns true; otherwise returns false

PRIMITIVE_ROOT19ICN1P(n) smallest primitive root mod n, if one exists; otherwise a ?

PROVE_SUM144O15P(t,k,a,n,s) if SUM(t,k,a,n)=s, return [0,0]

RANDOM_MATRIX144O15P(m,n,s) m by n matrix with random elements from -s to s

RANDOM_NORMAL144O15P(s,m) random normal value with a standard deviation of s and a mean value of m

RANDOM_POLY144O15P(x,d,s) poly of degree d in x with random coefficients from -s to s

RANDOM_SIGN144O15P random 1 or -1

RANDOM_VECTOR144O15P(n,s) n element vector with random elements from -s to s

RATIO_TEST144O15P(t,n) if < 1, SUM(t,n,a,inf) converges; if > 1, sum diverges

RAYSPlotting_Complex_valued_Expressions(w,z,z00,zmn,m,n) 2D plots w(z) as rays over the z-plane grid from z00 to zmn

RECURRENCE1338PND(u,v,v0,m) m recurrences of u(v) starting with vector v=v0

RECURRENCE1OKAKSN(r,x,y,x0,y0,n) n steps of y(x+1)=r(x,y(x)), y(x0)=y0

RHOMBIC_DODECAHEDRAL1338PND(n) nth rhombic dodecahedral number

RK1NBVQQO(r,v,v0,h,n) fourth order Runge-Kutta solution of system of 1st order differential equations

ROTATE_X2WQ_YS1(φ) matrix A so that A . [x,y,z] rotates angle φ about x axis

ROTATE_Y2WQ_YS1(φ) matrix A so that A . [x,y,z] rotates angle φ about y axis

ROTATE_Z2WQ_YS1(φ) matrix A so that A . [x,y,z] rotates angle φ about z axis

SCALE_ELEMENTI._DEZ(v,i,s) multiply element i of v by s

SEPARABLEOA7SLO(p,q,x,y,x0,y0) specific solution of separable equation y'=p(x)q(y)

SEPARABLE_GENOA7SLO(p,q,x,y,c) general solution of separable equation y'=p(x)q(y)

SI7RN4D(z) sine integral Si(z)

SMOOTH_COLUMNY5C_1X(A,j) matrix A with column j smoothed

SMOOTH_VECTORY5C_1X(v) smoothed copy of vector v

SOLVE_MOD19ICN1P(u,x,m) vector of solutions of the linear congruence equation u(x) mod m

SPHERE2WQ_YS1(r,θ,Φ) 3D coordinate matrix of a sphere of radius r

SPHERICAL_BESSEL_Y3JDX.V4(n,z) closed-form spherical Bessel function of 2nd kind,yn(z)

SPHERICAL_BESSEL_J3JDX.V4(n,z) spherical Bessel function of 1st kind jn(z) for integer n

SPHERICAL_BESSEL_J_LIST3JDX.V4(n,z) vector of spherical Bessel functions of 1st kind j0(z) through jn(z)

SPHERICAL_BESSEL_Y3JDX.V4(n,z) spherical Bessel function of 2nd kind yn(z) for integer n

SPHERICAL_VOLUME2W1Y_YU(r,r1,r2,θ,θ1,θ2,Φ,Φ1,Φ2) volume of region specified in spherical coordinates

SPHERICAL_VOLUME2W1Y_YU(r, r1,r2,θ,θ1,θ2,Φ,Φ1,Φ2,μ) integral of μ(r,θ,Φ) over the region

SQUARE_ROOT19ICN1P(a,p) square root of integer a mod prime p, if one exists; otherwise ?

SQUARE_WAVE144O15P(x) square wave of x

SQUAREFREE19ICN1P(n) if n is not divisible by the square of a prime, returns true; otherwise returns false

STAR1338PND(n) nth star (i.e. 12-sided centered) number

STIRLING1338PND(n,k) Stirling number

STIRLING11338PND(n,k) Stirling cycle number of the 1st kind

STIRLING21338PND(n,k) Stirling subset number of the 2nd kind

STIRLING_CYCLE1338PND(n,k) Stirling cycle number of the 1st kind

STIRLING_SUBSET1338PND(n,k) Stirling subset number of the 2nd kind

STUDENT0FMTUG(t,v) student's cumulative probability distribution A(t|v)

SUBTRACT_ELEMENTSI._DEZ(v,i,j,s) subtract element j·s from element i of v

SURFACE_AREA2W1Y_YU(z,x,x1,x2,y,y1,y2) area of surface z(x,y)

SURFACE_AREA2W1Y_YU(z,x,x1,x2,y,y1,y2,μ) integral of μ(x,y) over surface z(x,y)

SWAP_ELEMENTSI._DEZ(v,i,j) interchange elements i and j of vector v

TANGENTAUKXVG(y,x,x0) line tangent to y(x) at x=x0

TANGENT_PLANEAUKXVG(u,v,v0) plane tangent to u(x,y,z)=0 at [x,y,z]=v=v0

TAYLOR_INVERSE.S02QQ(u,x,y,x0,n) nth order series expansion of inverse of y=u(x)

TAYLOR_ODE11NBVQQO(r,x,y,x0,y0,n) nth order Taylor series solution of y'=r(x,y) with y=y0 at x=x0

TAYLOR_ODE21NBVQQO(r,x,y,v,x0,y0,v0,n) nth order Taylor series solution of y''=r(x,y,y')

TAYLOR_ODES1NBVQQO(r,x,y,x0,y0,n) vector of nth order Taylor series solutions of system of differential equations

TAYLOR_SOLVE.S02QQ(u,x,y,x0,y0,n) nth order series solution y(x) of u(x,y)=0

TETRAHEDRAL1338PND(n) nth tetrahedral number

TORUS2WQ_YS1(r,θ,Φ) 3D coordinate matrix of torus of radius r

TRIANGULAR1338PND(n) nth triangular number

U_LUCAS19ICN1P(n,p,q) nth term of the Lucas sequence L(n) where L(0)=0, L(1)=1, and L(n+2)=p·L(n+1)-q·L(n)

U_MOD19ICN1P(n,p,q,m) U_LUCAS(n, p, q) mod m, but is more efficient

V_LUCAS19ICN1P(n,p,q) nth term of the Lucas sequence L(n) where L(0)=2, L(1)=p, and L(n+2)=p·L(n+1)-q·L(n)

V_MOD19ICN1P(n,p,q,m) V_LUCAS(n,p,q) mod m, but is more efficient

VOLUME2W1Y_YU(x,x1,x2,y,y1,y2,z,z1,z2) volume y=y1(x) to y2(x), z=z1(x,y) to z2(x,y)

VOLUME2W1Y_YU(x,x1,x2,y,y1,y2,z,z1,z2,μ) integral of μ(x,y,z) over region

VOLUME_CENTROID2W1Y_YU(x,x1,x2,y,y1,y2,z,z1,z2) volumetric centroid of region

VOLUME_CENTROID2W1Y_YU(x,x1,x2,y,y1,y2,z,z1,z2,μ) centroid of density μ(x,y,z)

VOLUME_INERTIA2W1Y_YU(x,x1,x2,y,y1,y2,z,z1,z2) volumetric inertia tensor

VOLUME_INERTIA2W1Y_YU(x,x1,x2,y,y1,y2,z,z1,z2,μ) inertia tensor of μ(x,y,z)

VOLUME_OF_REVOLUTION2W1Y_YU(y,x,x1,x2) volume of y(x) revolved about x-axis

VOLUMEY_OF_REVOLUTION2W1Y_YU(y,x,x1,x2) volume of y(x) revolved about y-axis

WEBER_DHBCBN(n,x) Weber's nth parabolic cylinder function Dn(x)