DLMF Results

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DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
1.2.E1 binomial(n,k)=(factorial(n))/(factorial(n - k)*factorial(k)) Binomial[n,k]=Divide[(n)!,(n - k)!*(k)!] Successful Successful - -
1.2.E1 (factorial(n))/(factorial(n - k)*factorial(k))=binomial(n,n - k) Divide[(n)!,(n - k)!*(k)!]=Binomial[n,n - k] Successful Successful - -
1.2.E6 binomial(z,k)=(z*(z - 1)..(z - k + 1))/(factorial(k)) Binomial[z,k]=Divide[z*(z - 1) ... (z - k + 1),(k)!] Successful Successful - -
1.2.E7 binomial(z + 1,k)=binomial(z,k)+binomial(z,k - 1) Binomial[z + 1,k]=Binomial[z,k]+Binomial[z,k - 1] Successful Successful - -
1.2.E8 sum(binomial(z + k,k), k = 0..m)=binomial(z + m + 1,m) Sum[Binomial[z + k,k], {k, 0, m}]=Binomial[z + m + 1,m] Successful Successful - -
1.4.E8 (f)^(2)*(x)= diff(f, [x$(2)]) (f)^(2)*(x)= D[f, {x, 2}] Failure Failure
Fail
0.+3.999999998*I <- {f = 2^(1/2)+I*2^(1/2), x = 1}
0.+7.999999996*I <- {f = 2^(1/2)+I*2^(1/2), x = 2}
0.+11.99999999*I <- {f = 2^(1/2)+I*2^(1/2), x = 3}
0.-3.999999998*I <- {f = 2^(1/2)-I*2^(1/2), x = 1}
0.-7.999999996*I <- {f = 2^(1/2)-I*2^(1/2), x = 2}
0.-11.99999999*I <- {f = 2^(1/2)-I*2^(1/2), x = 3}
0.+3.999999998*I <- {f = -2^(1/2)-I*2^(1/2), x = 1}
0.+7.999999996*I <- {f = -2^(1/2)-I*2^(1/2), x = 2}
0.+11.99999999*I <- {f = -2^(1/2)-I*2^(1/2), x = 3}
0.-3.999999998*I <- {f = -2^(1/2)+I*2^(1/2), x = 1}
0.-7.999999996*I <- {f = -2^(1/2)+I*2^(1/2), x = 2}
0.-11.99999999*I <- {f = -2^(1/2)+I*2^(1/2), x = 3}
Fail
Complex[0.0, 4.0] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[0.0, 8.0] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[0.0, 12.0] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[0.0, -4.0] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[0.0, -8.0] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[0.0, -12.0] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[0.0, 4.0] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[0.0, 8.0] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[0.0, 12.0] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[0.0, -4.0] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[0.0, -8.0] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[0.0, -12.0] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
1.4.E8 diff(f, [x$(2)])= diff(diff(f, x), x) D[f, {x, 2}]= D[D[f, x], x] Successful Successful - -
1.4.E9 (f)^(n)= (f)^(n)*(x) (f)^(n)= (f)^(n)*(x) Failure Failure
Fail
-1.414213562-1.414213562*I <- {f = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
-2.828427124-2.828427124*I <- {f = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
-0.-3.999999998*I <- {f = 2^(1/2)+I*2^(1/2), n = 2, x = 2}
-0.-7.999999996*I <- {f = 2^(1/2)+I*2^(1/2), n = 2, x = 3}
5.656854245-5.656854245*I <- {f = 2^(1/2)+I*2^(1/2), n = 3, x = 2}
11.31370849-11.31370849*I <- {f = 2^(1/2)+I*2^(1/2), n = 3, x = 3}
-1.414213562+1.414213562*I <- {f = 2^(1/2)-I*2^(1/2), n = 1, x = 2}
-2.828427124+2.828427124*I <- {f = 2^(1/2)-I*2^(1/2), n = 1, x = 3}
-0.+3.999999998*I <- {f = 2^(1/2)-I*2^(1/2), n = 2, x = 2}
-0.+7.999999996*I <- {f = 2^(1/2)-I*2^(1/2), n = 2, x = 3}
5.656854245+5.656854245*I <- {f = 2^(1/2)-I*2^(1/2), n = 3, x = 2}
11.31370849+11.31370849*I <- {f = 2^(1/2)-I*2^(1/2), n = 3, x = 3}
1.414213562+1.414213562*I <- {f = -2^(1/2)-I*2^(1/2), n = 1, x = 2}
2.828427124+2.828427124*I <- {f = -2^(1/2)-I*2^(1/2), n = 1, x = 3}
-0.-3.999999998*I <- {f = -2^(1/2)-I*2^(1/2), n = 2, x = 2}
-0.-7.999999996*I <- {f = -2^(1/2)-I*2^(1/2), n = 2, x = 3}
-5.656854245+5.656854245*I <- {f = -2^(1/2)-I*2^(1/2), n = 3, x = 2}
-11.31370849+11.31370849*I <- {f = -2^(1/2)-I*2^(1/2), n = 3, x = 3}
1.414213562-1.414213562*I <- {f = -2^(1/2)+I*2^(1/2), n = 1, x = 2}
2.828427124-2.828427124*I <- {f = -2^(1/2)+I*2^(1/2), n = 1, x = 3}
-0.+3.999999998*I <- {f = -2^(1/2)+I*2^(1/2), n = 2, x = 2}
-0.+7.999999996*I <- {f = -2^(1/2)+I*2^(1/2), n = 2, x = 3}
-5.656854245-5.656854245*I <- {f = -2^(1/2)+I*2^(1/2), n = 3, x = 2}
-11.31370849-11.31370849*I <- {f = -2^(1/2)+I*2^(1/2), n = 3, x = 3}
Fail
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[0.0, -4.0] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[0.0, -8.0] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[5.656854249492381, -5.656854249492381] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[11.313708498984761, -11.313708498984761] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[-2.8284271247461903, 2.8284271247461903] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[0.0, 4.0] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[0.0, 8.0] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[5.656854249492381, 5.656854249492381] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[11.313708498984761, 11.313708498984761] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[0.0, -4.0] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[0.0, -8.0] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[-5.656854249492381, 5.656854249492381] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[-11.313708498984761, 11.313708498984761] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[2.8284271247461903, -2.8284271247461903] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[0.0, 4.0] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[0.0, 8.0] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[-5.656854249492381, -5.656854249492381] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[-11.313708498984761, -11.313708498984761] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
1.4.E9 (f)^(n)*(x)= diff((f)^(n - 1)*(x), x) (f)^(n)*(x)= D[(f)^(n - 1)*(x), x] Failure Failure
Fail
.414213562+1.414213562*I <- {f = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
1.828427124+2.828427124*I <- {f = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
3.242640686+4.242640686*I <- {f = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
-1.414213562+2.585786436*I <- {f = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
-1.414213562+6.585786434*I <- {f = 2^(1/2)+I*2^(1/2), n = 2, x = 2}
-1.414213562+10.58578643*I <- {f = 2^(1/2)+I*2^(1/2), n = 2, x = 3}
-5.656854245+1.656854247*I <- {f = 2^(1/2)+I*2^(1/2), n = 3, x = 1}
-11.31370849+7.313708492*I <- {f = 2^(1/2)+I*2^(1/2), n = 3, x = 2}
-16.97056274+12.97056274*I <- {f = 2^(1/2)+I*2^(1/2), n = 3, x = 3}
.414213562-1.414213562*I <- {f = 2^(1/2)-I*2^(1/2), n = 1, x = 1}
1.828427124-2.828427124*I <- {f = 2^(1/2)-I*2^(1/2), n = 1, x = 2}
3.242640686-4.242640686*I <- {f = 2^(1/2)-I*2^(1/2), n = 1, x = 3}
-1.414213562-2.585786436*I <- {f = 2^(1/2)-I*2^(1/2), n = 2, x = 1}
-1.414213562-6.585786434*I <- {f = 2^(1/2)-I*2^(1/2), n = 2, x = 2}
-1.414213562-10.58578643*I <- {f = 2^(1/2)-I*2^(1/2), n = 2, x = 3}
-5.656854245-1.656854247*I <- {f = 2^(1/2)-I*2^(1/2), n = 3, x = 1}
-11.31370849-7.313708492*I <- {f = 2^(1/2)-I*2^(1/2), n = 3, x = 2}
-16.97056274-12.97056274*I <- {f = 2^(1/2)-I*2^(1/2), n = 3, x = 3}
-2.414213562-1.414213562*I <- {f = -2^(1/2)-I*2^(1/2), n = 1, x = 1}
-3.828427124-2.828427124*I <- {f = -2^(1/2)-I*2^(1/2), n = 1, x = 2}
-5.242640686-4.242640686*I <- {f = -2^(1/2)-I*2^(1/2), n = 1, x = 3}
1.414213562+5.414213560*I <- {f = -2^(1/2)-I*2^(1/2), n = 2, x = 1}
1.414213562+9.414213558*I <- {f = -2^(1/2)-I*2^(1/2), n = 2, x = 2}
1.414213562+13.41421355*I <- {f = -2^(1/2)-I*2^(1/2), n = 2, x = 3}
5.656854245-9.656854243*I <- {f = -2^(1/2)-I*2^(1/2), n = 3, x = 1}
11.31370849-15.31370849*I <- {f = -2^(1/2)-I*2^(1/2), n = 3, x = 2}
16.97056274-20.97056274*I <- {f = -2^(1/2)-I*2^(1/2), n = 3, x = 3}
-2.414213562+1.414213562*I <- {f = -2^(1/2)+I*2^(1/2), n = 1, x = 1}
-3.828427124+2.828427124*I <- {f = -2^(1/2)+I*2^(1/2), n = 1, x = 2}
-5.242640686+4.242640686*I <- {f = -2^(1/2)+I*2^(1/2), n = 1, x = 3}
1.414213562-5.414213560*I <- {f = -2^(1/2)+I*2^(1/2), n = 2, x = 1}
1.414213562-9.414213558*I <- {f = -2^(1/2)+I*2^(1/2), n = 2, x = 2}
1.414213562-13.41421355*I <- {f = -2^(1/2)+I*2^(1/2), n = 2, x = 3}
5.656854245+9.656854243*I <- {f = -2^(1/2)+I*2^(1/2), n = 3, x = 1}
11.31370849+15.31370849*I <- {f = -2^(1/2)+I*2^(1/2), n = 3, x = 2}
16.97056274+20.97056274*I <- {f = -2^(1/2)+I*2^(1/2), n = 3, x = 3}
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}
Complex[1.8284271247461903, 2.8284271247461903] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[3.2426406871192857, 4.242640687119286] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[-1.4142135623730951, 2.585786437626905] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}
Complex[-1.4142135623730951, 6.585786437626905] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[-1.4142135623730951, 10.585786437626904] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[-5.656854249492381, 1.6568542494923806] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 1]}
Complex[-11.313708498984761, 7.313708498984761] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[-16.970562748477143, 12.970562748477143] <- {Rule[f, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}
Complex[1.8284271247461903, -2.8284271247461903] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[3.2426406871192857, -4.242640687119286] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[-1.4142135623730951, -2.585786437626905] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}
Complex[-1.4142135623730951, -6.585786437626905] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[-1.4142135623730951, -10.585786437626904] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[-5.656854249492381, -1.6568542494923806] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 1]}
Complex[-11.313708498984761, -7.313708498984761] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[-16.970562748477143, -12.970562748477143] <- {Rule[f, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}
Complex[-3.8284271247461903, -2.8284271247461903] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[-5.242640687119286, -4.242640687119286] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[1.4142135623730951, 5.414213562373095] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}
Complex[1.4142135623730951, 9.414213562373096] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[1.4142135623730951, 13.414213562373096] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[5.656854249492381, -9.65685424949238] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 1]}
Complex[11.313708498984761, -15.313708498984761] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[16.970562748477143, -20.970562748477143] <- {Rule[f, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}
Complex[-3.8284271247461903, 2.8284271247461903] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[-5.242640687119286, 4.242640687119286] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[1.4142135623730951, -5.414213562373095] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}
Complex[1.4142135623730951, -9.414213562373096] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 2]}
Complex[1.4142135623730951, -13.414213562373096] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 3]}
Complex[5.656854249492381, 9.65685424949238] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 1]}
Complex[11.313708498984761, 15.313708498984761] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 2]}
Complex[16.970562748477143, 20.970562748477143] <- {Rule[f, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3], Rule[x, 3]}
1.4.E16 int(f*g, x)=(int(f, x))* g - int((int(f, x))* diff(g, x), x) Integrate[f*g, x]=(Integrate[f, x])* g - Integrate[(Integrate[f, x])* D[g, x], x] Successful Successful - -