calc101
Step-by-Step Derivatives
(E?)(L?) http://www.calc101.com/webMathematica/derivatives.jsp#topdoitErstellt: 2011-10
The Wolfram Mathematica Online Integrator uses webMathematica technology to bring the computational capabilities of Mathematica to the web.
The heart of The Integrator is Mathematica's built-in Integrate function - the same one you use when accessing Mathematica directly. When you type in an integral, the Integrator uses webMathematica to send the integral to the Mathematica kernel, which computes the result using the Integrate function. The result is then embedded into the web page as typeset output.
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HISTORY OF INTEGRATION
Mathematica combines the recent decades of the computer revolution with the previous few centuries of mathematical research, fulfilling the original goal of the early computer pioneers: to do mathematics by computer...
Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the paraboloid. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers.
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Functions Categories
- Elementary Functions
- Constants
- Bessel-Type Functions
- Integer Functions
- Polynomials
- Gamma, Beta, Erf
- Hypergeometric Functions
- Elliptic Integrals
- Elliptic Functions
- Zeta Functions and Polylogarithms
- Mathieu and Spheroidal Functions
- Complex Components
- Number Theory Functions
- Generalized Functions
Elementary Functions (61,455 formulas)
Power FunctionsExponential Function
- Sqrt[z] (220 formulas)
- Power[z,a] (579 formulas)
Logarithmic Functions
- Exp[z] (1523 formulas)
Trigonometric Functions
- Log[z] (565 formulas)
- Log[a,z] (110 formulas)
Inverse Trigonometric Functions
- Sin[z] (2304 formulas)
- Cos[z] (3435 formulas)
- Tan[z] (570 formulas)
- Cot[z] (596 formulas)
- Csc[z] (582 formulas)
- Sec[z] (553 formulas)
Hyperbolic Functions
- ArcSin[z] (2716 formulas)
- ArcCos[z] (2732 formulas)
- ArcTan[z] (3274 formulas)
- ArcTan[x,y] (90 formulas)
- ArcCot[z] (3277 formulas)
- ArcCsc[z] (2807 formulas)
- ArcSec[z] (2795 formulas)
Inverse Hyperbolic Functions
- Sinh[z] (4084 formulas)
- Cosh[z] (5580 formulas)
- Tanh[z] (762 formulas)
- Coth[z] (840 formulas)
- Csch[z] (1015 formulas)
- Sech[z] (1012 formulas)
Product Logarithms
- ArcSinh[z] (2442 formulas)
- ArcCosh[z] (2952 formulas)
- ArcTanh[z] (3911 formulas)
- ArcCoth[z] (3983 formulas)
- ArcCsch[z] (2774 formulas)
- ArcSech[z] (3184 formulas)
Roots
- ProductLog[z] (59 formulas)
- ProductLog[k,z] (41 formulas)
Max and Min
- Root[polynomialInzk] (16 formulas)
Cardinal Functions
- Max[x1 ,x2,...] (34 formulas)
- Min[x1 ,x2,...] (35 formulas)
- Sinc[z] (3 formulas)
Constants (735 formulas)
Imaginary UnitClassical Constants
- I (26 formulas)
Symbols
- GoldenRatio (49 formulas)
- Pi (158 formulas)
- Degree (40 formulas)
- E (94 formulas)
- EulerGamma (81 formulas)
- Catalan (99 formulas)
- Glaisher (31 formulas)
- Khinchin (33 formulas)
- Indeterminate (31 formulas)
- Infinity (27 formulas)
- ComplexInfinity (28 formulas)
- DirectedInfinity[z] (38 formulas)
Bessel-Type Functions (5,583 formulas)
Bessel FunctionsHankel Functions
- BesselJ[nu,z] (545 formulas)
- BesselI[nu,z] (530 formulas)
- BesselY[nu,z] (557 formulas)
- BesselK[nu,z] (467 formulas)
Airy Functions
- HankelH1[nu,z] (1 formula)
- HankelH2[nu,z] (1 formula)
Struve Functions
- AiryAi[z] (236 formulas)
- AiryBi[z] (235 formulas)
- AiryAiPrime[z] (232 formulas)
- AiryBiPrime[z] (233 formulas)
Kelvin Functions
- StruveH[nu,z] (213 formulas)
- StruveL[nu,z] (198 formulas)
Spherical Bessel-Type Functions
- KelvinBei[z] (143 formulas)
- KelvinBer[z] (143 formulas)
- KelvinKei[z] (119 formulas)
- KelvinKer[z] (132 formulas)
- KelvinBei[nu,z] (289 formulas)
- KelvinBer[nu,z] (292 formulas)
- KelvinKei[nu,z] (270 formulas)
- KelvinKer[nu,z] (277 formulas)
Zeros
- SphericalBesselJ[nu,z] (463 formulas)
- SphericalBesselY[nu,z] (1 formula)
- SphericalHankelH1[nu,z] (1 formula)
- SphericalHankelH2[nu,z] (1 formula)
- BesselJZero[nu,k] (1 formula)
- BesselYZero[nu,k] (1 formula)
- AiryAiZero[k] (1 formula)
- AiryBiZero[k] (1 formula)
Integer Functions (1,966 formulas)
Rounding and Congruence FunctionsGCD and LCM
- Floor[z] (129 formulas)
- Ceiling[z] (107 formulas)
- Round[z] (106 formulas)
- IntegerPart[z] (114 formulas)
- FractionalPart[z] (118 formulas)
- Mod[m,n] (139 formulas)
- Quotient[m,n] (136 formulas)
Fibonaci and Lucas Numbers
- GCD[n1,n2,...,nm] (67 formulas)
- ExtendedGCD[n1,n2,...,nm] (30 formulas)
- LCM[n1,n2,...,nm] (52 formulas)
Euler and Bernoulli Numbers
- LucasL[nu] (113 formulas)
Stirling and Bell Numbers
- EulerE[n] (52 formulas)
- BernoulliB[n] (103 formulas)
Partitions
- StirlingS1[n,m] (92 formulas)
- StirlingS2[n,m] (109 formulas)
- BellB[n] (38 formulas)
Tensorial Functions
- PartitionsP[n] (78 formulas)
- PartitionsQ[n] (68 formulas)
- DiscreteDelta[n] (20 formulas)
- DiscreteDelta[n1,n2,...] (31 formulas)
- KroneckerDelta[n] (18 formulas)
- KroneckerDelta[n1,n2,...] (38 formulas)
- Signature[{n1,n2,...,nd}] (30 formulas)
Polynomials (2,372 formulas)
Classical Orthogonal PolynomialsAssociated Legendre Polynomials and Spherical Harmonics
- HermiteH[n,z] (165 formulas)
- LaguerreL[n,z] (108 formulas)
- LaguerreL[n,lambda,z] (158 formulas)
- LegendreP[n,z] (167 formulas)
- ChebyshevT[n,z] (165 formulas)
- ChebyshevU[n,z] (168 formulas)
- GegenbauerC[n,lambda,z] (205 formulas)
- JacobiP[n,a,b,z] (189 formulas)
Other Polynomials
- LegendreP[n,mu,2,z] (221 formulas)
- SphericalHarmonicY[n,m,theta,phi] (223 formulas)
- Cyclotomic[n,z] (42 formulas)
- EulerE[n,z] (101 formulas)
- BernoulliB[n,z] (106 formulas)
- BellB[n,z] (50 formulas)
- NorlundB[n,\[Alpha]] (50 formulas)
- NorlundB[n,\[Alpha],z] (63 formulas)
- ZernikeR[n,m,z] (87 formulas)
Gamma, Beta, Erf (5,032 formulas)
FactorialsBinomials
- Factorial[n] (153 formulas)
- Factorial2[n] (77 formulas)
- Pochhammer[a,n] (103 formulas)
Gamma Functions and Inverses
- Binomial[n,k] (147 formulas)
- Multinomial[n1,n2,...,nm] (32 formulas)
- CatalanNumber[n] (102 formulas)
Differentiated Gamma Functions
- Gamma[z] (193 formulas)
- Gamma[a,z] (153 formulas)
- Gamma[a,z1,z2] (90 formulas)
- GammaRegularized[a,z] (142 formulas)
- GammaRegularized[a,z1,z2] (85 formulas)
- Subfactorial[z] (91 formulas)
- LogGamma[z] (130 formulas)
- InverseGammaRegularized[a,z] (29 formulas)
- InverseGammaRegularized[a,z1,z2] (17 formulas)
Beta Functions and Inverses
- PolyGamma[z] (117 formulas)
- PolyGamma[nu,z] (214 formulas)
- HarmonicNumber[z] (127 formulas)
- HarmonicNumber[z,r] (92 formulas)
Probability Integrals and Inverses
- Beta[a,b] (72 formulas)
- Beta[z,a,b] (193 formulas)
- Beta[z1,z2,a,b] (106 formulas)
- BetaRegularized[z,a,b] (173 formulas)
- BetaRegularized[z1,z2,a,b] (98 formulas)
- InverseBetaRegularized[z,a,b] (31 formulas)
- InverseBetaRegularized[z1,z2,a,b] (15 formulas)
Fresnel Integrals
- Erf[z] (239 formulas)
- Erf[z1,z2] (54 formulas)
- Erfc[z] (235 formulas)
- Erfi[z] (246 formulas)
- InverseErf[z] (23 formulas)
- InverseErf[z1,z2] (20 formulas)
- InverseErfc[z] (24 formulas)
Exponential Integrals
- FresnelS[z] (108 formulas)
- FresnelC[z] (226 formulas)
- ExpIntegralE[nu,z] (133 formulas)
- ExpIntegralEi[z] (183 formulas)
- LogIntegral[z] (90 formulas)
- SinIntegral[z] (166 formulas)
- CosIntegral[z] (174 formulas)
- SinhIntegral[z] (159 formulas)
- CoshIntegral[z] (170 formulas)
Hypergeometric Functions (218,254 formulas)
Hermite, Parabolic Cylinder, and Laguerre FunctionsLegendre Functions
- HermiteH[nu,z] (229 formulas)
- ParabolicCylinderD[nu,z] (235 formulas)
- LaguerreL[nu,z] (138 formulas)
- LaguerreL[nu,lambda,z] (183 formulas)
Gegenbauer and Jacobi Functions
- LegendreP[nu,z] (188 formulas)
- LegendreP[nu,mu,z] (1 formula)
- LegendreP[nu,mu,2,z] (284 formulas)
- LegendreP[nu,mu,3,z] (315 formulas)
- LegendreQ[nu,z] (103 formulas)
- LegendreQ[nu,mu,z] (1 formula)
- LegendreQ[nu,mu,2,z] (313 formulas)
- LegendreQ[nu,mu,3,z] (366 formulas)
Confluent Hypergeometric Functions
- GegenbauerC[nu,z] (205 formulas)
- GegenbauerC[nu,lambda,z] (237 formulas)
- JacobiP[nu,a,b,z] (185 formulas)
Hypergeometric Functions
- Hypergeometric1F1[a,b,z] (750 formulas)
- Hypergeometric1F1Regularized[a,b,z] (777 formulas)
- HypergeometricU[a,b,z] (1017 formulas)
- WhittakerM[nu,mu,z] (289 formulas)
- WhittakerW[nu,mu,z] (181 formulas)
Meijer G Functions
- HypergeometricPFQ[{},{},z] (18 formulas)
- Hypergeometric0F1[b,z] (416 formulas)
- Hypergeometric0F1Regularized[b,z] (409 formulas)
- HypergeometricPFQ[{a},{},z] (42 formulas)
- HypergeometricPFQ[{a1},{b1,b2},z] (21885 formulas)
- Hypergeometric2F1[a,b,c,z] (111951 formulas)
- Hypergeometric2F1Regularized[a,b,c,z] (865 formulas)
- HypergeometricPFQ[{a1,a2},{b1,b2},z] (31337 formulas)
- HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] (145 formulas)
- HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] (40964 formulas)
- HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] (360 formulas)
- HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] (78 formulas)
- HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] (15 formulas)
- HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] (333 formulas)
- HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] (310 formulas)
Hypergeometric Functions of Several Variables
- MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z] (1363 formulas)
- MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] (198 formulas)
Angular Momentum Functions
- AppellF1[a,b1,b2,c,z1,z2] (102 formulas)
- SphericalHarmonicY[lambda,mu,theta,phi] (266 formulas)
- ClebschGordan[{j1,m1},{j2,m2},{j,m}] (184 formulas)
- ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}] (120 formulas)
- SixJSymbol[{j1,j2,j3},{j4,j5,j6}] (254 formulas)
Elliptic Integrals (1,236 formulas)
Complete Elliptic IntegralsIncomplete Elliptic Integrals
- EllipticE[z] (182 formulas)
- EllipticK[z] (269 formulas)
- EllipticPi[n,m] (116 formulas)
- EllipticE[z,m] (177 formulas)
- EllipticF[z,m] (190 formulas)
- EllipticPi[n,z,m] (215 formulas)
- JacobiZeta[z,m] (87 formulas)
Elliptic Functions (7,248 formulas)
Jacobi Theta FunctionsSiegel Theta Functions
- EllipticTheta[1,z,q] (291 formulas)
- EllipticTheta[2,z,q] (105 formulas)
- EllipticTheta[3,z,q] (104 formulas)
- EllipticTheta[4,z,q] (100 formulas)
- EllipticThetaPrime[1,z,q] (87 formulas)
- EllipticThetaPrime[2,z,q] (82 formulas)
- EllipticThetaPrime[3,z,q] (73 formulas)
- EllipticThetaPrime[4,z,q] (73 formulas)
Neville Theta Functions
- SiegelTheta[{{m1,1,...,m1,r},...,{mr,1,...,mr,r}},{s1,...,sr}] (1 formula)
- SiegelTheta[{{u1,...,ur},{v1,...,vr}},{{m1,1,...,m1,r},...,{mr,1,...,mr,r}},{s1,...,sr}] (1 formula)
Weierstrass Functions and Inverses
- NevilleThetaC[z,m] (31 formulas)
- NevilleThetaD[z,m] (29 formulas)
- NevilleThetaN[z,m] (29 formulas)
- NevilleThetaS[z,m] (28 formulas)
Weierstrass Utility Functions
- WeierstrassP[z,{g2,g3}] (142 formulas)
- WeierstrassPPrime[z,{g2,g3}] (82 formulas)
- WeierstrassSigma[z,{g2,g3}] (96 formulas)
- WeierstrassSigma[n,z,{g2,g3}] (77 formulas)
- WeierstrassZeta[z,{g2,g3}] (91 formulas)
- InverseWeierstrassP[z,{g2,g3}] (30 formulas)
- InverseWeierstrassP[{z1,z2},{g2,g3}] (14 formulas)
Jacobi Functions
- WeierstrassHalfPeriods[{g2,g3}] (23 formulas)
- WeierstrassInvariants[{w1,w3}] (58 formulas)
- WeierstrassPHalfPeriodValues[{g2,g3}] (41 formulas)
- WeierstrassZetaHalfPeriodValues[{g2,g3}] (44 formulas)
Inverse Jacobi Functions
- JacobiAmplitude[z,m] (91 formulas)
- JacobiCD[z,m] (300 formulas)
- JacobiCN[z,m] (445 formulas)
- JacobiCS[z,m] (255 formulas)
- JacobiDC[z,m] (286 formulas)
- JacobiDN[z,m] (642 formulas)
- JacobiDS[z,m] (232 formulas)
- JacobiNC[z,m] (306 formulas)
- JacobiND[z,m] (295 formulas)
- JacobiNS[z,m] (261 formulas)
- JacobiSC[z,m] (299 formulas)
- JacobiSD[z,m] (282 formulas)
- JacobiSN[z,m] (472 formulas)
Modular Functions and Inverses
- InverseJacobiCD[z,m] (92 formulas)
- InverseJacobiCN[z,m] (100 formulas)
- InverseJacobiCS[z,m] (80 formulas)
- InverseJacobiDC[z,m] (82 formulas)
- InverseJacobiDN[z,m] (86 formulas)
- InverseJacobiDS[z,m] (82 formulas)
- InverseJacobiNC[z,m] (82 formulas)
- InverseJacobiND[z,m] (84 formulas)
- InverseJacobiNS[z,m] (85 formulas)
- InverseJacobiSC[z,m] (98 formulas)
- InverseJacobiSD[z,m] (87 formulas)
- InverseJacobiSN[z,m] (86 formulas)
Arithmetic Geometric Mean
- DedekindEta[z] (57 formulas)
- KleinInvariantJ[z] (51 formulas)
- ModularLambda[z] (38 formulas)
- InverseEllipticNomeQ[z] (40 formulas)
- EllipticNomeQ[m] (36 formulas)
Elliptic Exp and Elliptic Log
- ArithmeticGeometricMean[a,b] (53 formulas)
- EllipticExp[z,{a,b}] (11 formulas)
- EllipticExpPrime[z,{a,b}] (5 formulas)
- EllipticLog[{z1,z2},{a,b}] (15 formulas)
Zeta Functions and Polylogarithms (1,571 formulas)
Zeta FunctionsZeta-Related Functions
- Zeta[s] (198 formulas)
- Zeta[s,a] (321 formulas)
- RiemannSiegelZ[z] (47 formulas)
- LerchPhi[z,s,a] (273 formulas)
- RamanujanTauL[z] (16 formulas)
- RamanujanTauZ[z] (13 formulas)
Polylogarithms
- RiemannSiegelTheta[z] (103 formulas)
- RamanujanTauTheta[z] (156 formulas)
Derivatives
- PolyLog[2,z] (156 formulas)
- PolyLog[nu,z] (224 formulas)
- PolyLog[nu,p,z] (48 formulas)
Zeros
- StieltjesGamma[n] (14 formulas)
- ZetaZero[k] (2 formulas)
Mathieu and Spheroidal Functions (388 formulas)
Mathieu FunctionsMathieu Characteristics
- MathieuC[a,q,z] (34 formulas)
- MathieuS[a,q,z] (32 formulas)
- MathieuCPrime[a,q,z] (32 formulas)
- MathieuSPrime[a,q,z] (31 formulas)
Spheroidal Functions
- MathieuCharacteristicA[r,q] (22 formulas)
- MathieuCharacteristicB[r,q] (21 formulas)
- MathieuCharacteristicExponent[a,q] (8 formulas)
Spheroidal Characteristics
- SpheroidalPS[nu,mu,gamma,z] (32 formulas)
- SpheroidalQS[nu,mu,gamma,z] (27 formulas)
- SpheroidalS1[nu,mu,gamma,z] (21 formulas)
- SpheroidalS2[nu,mu,gamma,z] (21 formulas)
- SpheroidalPSPrime[nu,mu,gamma,z] (28 formulas)
- SpheroidalQSPrime[nu,mu,gamma,z] (24 formulas)
- SpheroidalS1Prime[nu,mu,gamma,z] (21 formulas)
- SpheroidalS2Prime[nu,mu,gamma,z] (21 formulas)
- SpheroidalEigenvalue[nu,mu,gamma] (11 formulas)
- SpheroidalJoiningFactor[nu,mu,gamma] (1 formula)
- SpheroidalRadialFactor[nu,mu,gamma] (1 formula)
Complex Components (689 formulas)
- Abs[z] (128 formulas)
- Arg[z] (178 formulas)
- Re[z] (90 formulas)
- Im[z] (88 formulas)
- Conjugate[z] (90 formulas)
- Sign[z] (115 formulas)
Number Theory Functions (906 formulas)
Prime Factorization Related FunctionsOther Functions
- FactorInteger[n] (72 formulas)
- Divisors[n] (67 formulas)
- Prime[n] (87 formulas)
- PrimePi[x] (85 formulas)
- DivisorSigma[k,n] (135 formulas)
- EulerPhi[n] (112 formulas)
- MoebiusMu[n] (89 formulas)
- JacobiSymbol[n,m] (98 formulas)
- KroneckerSymbol[n,m] (1 formula)
- CarmichaelLambda[n] (63 formulas)
- DigitCount[n,b] (66 formulas)
- SquaresR[m,n] (1 formula)
- RamanujanTau[n] (30 formulas)
Generalized Functions (280 formulas)
Discontinuous FunctionsFunctions with Singular Support
- UnitStep[x] (66 formulas)
- UnitStep[x1,x2,...] (34 formulas)
- DiracDelta[x] (61 formulas)
- DiracDelta[x1,x2,...] (23 formulas)
- HeavisideTheta[x] (63 formulas)
- HeavisideTheta[x1,x2,...] (33 formulas)