Algebraic and Analytic Methods - 1.9 Calculus of a Complex Variable

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.9.E1 z = x + i ⁒ y 𝑧 π‘₯ 𝑖 𝑦 {\displaystyle{\displaystyle z=x+iy}}
z = x+iy		 

(x + y*I) = x + I*y

(x + y*I) == x + I*y
Successful Successful - Successful [Tested: 1]
1.9#Ex1 β„œ ⁑ z = x 𝑧 π‘₯ {\displaystyle{\displaystyle\Re z=x}}
\realpart@@{z} = x		 

Re(x + y*I) = x

Re[x + y*I] == x
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
1.9#Ex2 β„‘ ⁑ z = y 𝑧 𝑦 {\displaystyle{\displaystyle\Im z=y}}
\imagpart@@{z} = y		 

Im(x + y*I) = y

Im[x + y*I] == y
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
1.9#Ex3 x = r ⁒ cos ⁑ ΞΈ π‘₯ π‘Ÿ πœƒ {\displaystyle{\displaystyle x=r\cos\theta}}
x = r\cos@@{\theta}		 

x = r*cos(theta)

x == r*Cos[\[Theta]]
Failure Failure
Failed [180 / 180]
Result: 2.595814528-.5954243254*I
Test Values: {r = -1.5, theta = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: 1.595814528-.5954243254*I
Test Values: {r = -1.5, theta = 1/2*3^(1/2)+1/2*I, x = .5}


Result: 3.095814528-.5954243254*I
Test Values: {r = -1.5, theta = 1/2*3^(1/2)+1/2*I, x = 2}


Result: 3.341648276+.7036130646*I
Test Values: {r = -1.5, theta = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[2.595814528585838, -0.5954243253435487]
Test Values: {Rule[r, -1.5], Rule[x, 1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.3416482752961656, 0.7036130644027555]
Test Values: {Rule[r, -1.5], Rule[x, 1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9#Ex4 y = r ⁒ sin ⁑ ΞΈ 𝑦 π‘Ÿ πœƒ {\displaystyle{\displaystyle y=r\sin\theta}}
y = r\sin@@{\theta}		 

y = r*sin(theta)

y == r*Sin[\[Theta]]
Failure Failure
Failed [300 / 300]
Result: -.211529498+.5063946946*I
Test Values: {r = -1.5, theta = 1/2*3^(1/2)+1/2*I, y = -1.5}


Result: 2.788470502+.5063946946*I
Test Values: {r = -1.5, theta = 1/2*3^(1/2)+1/2*I, y = 1.5}


Result: .788470502+.5063946946*I
Test Values: {r = -1.5, theta = 1/2*3^(1/2)+1/2*I, y = -.5}


Result: 1.788470502+.5063946946*I
Test Values: {r = -1.5, theta = 1/2*3^(1/2)+1/2*I, y = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.21152949854979308, 0.506394694834305]
Test Values: {Rule[r, -1.5], Rule[y, -1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.506097038210817, 1.2879550752257174]
Test Values: {Rule[r, -1.5], Rule[y, -1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9.E4 r = ( x 2 + y 2 ) 1 / 2 π‘Ÿ superscript superscript π‘₯ 2 superscript 𝑦 2 1 2 {\displaystyle{\displaystyle r=(x^{2}+y^{2})^{1/2}}}
r = (x^{2}+y^{2})^{1/2}		 

r = ((x)^(2)+ (y)^(2))^(1/2)
 
r == ((x)^(2)+ (y)^(2))^(1/2)
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E6 Ο‰ = arctan ⁑ ( | y / x | ) ∈ [ 0 , 1 2 ⁒ Ο€ ] πœ” 𝑦 π‘₯ 0 1 2 πœ‹ {\displaystyle{\displaystyle\omega=\operatorname{arctan}\left(|y/x|\right)\in% \left[0,\tfrac{1}{2}\pi\right]}}
\omega = \atan@{|y/x|}\in\left[0,\tfrac{1}{2}\pi\right]		 

omega 0 <= arctan(abs(y/x)) <= (1)/(2)*Pi

\[Omega] 0 <= ArcTan[Abs[y/x]] <= Divide[1,2]*Pi
Error Failure -
Failed [180 / 180]
Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[-1.0, True]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ο‰, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.4999999999999998, 0.8660254037844387], Times[-1.0, True]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ο‰, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9#Ex5 | z | = r 𝑧 π‘Ÿ {\displaystyle{\displaystyle|z|=r}}
|z| = r		 

abs(z) = r

Abs[z] == r
Failure Failure
Failed [39 / 42]
Result: 2.5
Test Values: {r = -1.5, z = 1/2*3^(1/2)+1/2*I}


Result: 2.5
Test Values: {r = -1.5, z = -1/2+1/2*I*3^(1/2)}


Result: 2.5
Test Values: {r = -1.5, z = 1/2-1/2*I*3^(1/2)}


Result: 2.5
Test Values: {r = -1.5, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [39 / 42]
Result: 2.5
Test Values: {Rule[r, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: 2.5
Test Values: {Rule[r, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9#Ex6 ph ⁑ z = ΞΈ + 2 ⁒ n ⁒ Ο€ phase 𝑧 πœƒ 2 𝑛 πœ‹ {\displaystyle{\displaystyle\operatorname{ph}z=\theta+2n\pi}}
\phase@@{z} = \theta+2n\pi		 

argument(z) = theta + 2*n*Pi

Arg[z] == \[Theta]+ 2*n*Pi
Error Failure -
Failed [70 / 70]
Result: Complex[-19.191982549724898, -0.49999999999999994]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-17.82595714594046, -0.8660254037844387]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9#Ex7 | β„œ ⁑ z | ≀ | z | 𝑧 𝑧 {\displaystyle{\displaystyle|\Re z|\leq|z|}}
|\realpart@@{z}| \leq |z|		 

abs(Re(z)) <= abs(z)

Abs[Re[z]] <= Abs[z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
1.9#Ex8 | β„‘ ⁑ z | ≀ | z | 𝑧 𝑧 {\displaystyle{\displaystyle|\Im z|\leq|z|}}
|\imagpart@@{z}| \leq |z|		 

abs(Im(z)) <= abs(z)

Abs[Im[z]] <= Abs[z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
1.9.E9 z = r ⁒ e i ⁒ ΞΈ 𝑧 π‘Ÿ superscript 𝑒 𝑖 πœƒ {\displaystyle{\displaystyle z=re^{i\theta}}}
z = re^{i\theta}		 

z = r*exp(I*theta)
 
z == r*Exp[I*\[Theta]]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E10 e i ⁒ ΞΈ = cos ⁑ ΞΈ + i ⁒ sin ⁑ ΞΈ superscript 𝑒 𝑖 πœƒ πœƒ 𝑖 πœƒ {\displaystyle{\displaystyle e^{i\theta}=\cos\theta+i\sin\theta}}
e^{i\theta} = \cos@@{\theta}+i\sin@@{\theta}		 

exp(I*theta) = cos(theta)+ I*sin(theta)

Exp[I*\[Theta]] == Cos[\[Theta]]+ I*Sin[\[Theta]]
Successful Successful - Successful [Tested: 10]
1.9.E11 z Β― = x - i ⁒ y 𝑧 π‘₯ 𝑖 𝑦 {\displaystyle{\displaystyle\overline{z}=x-iy}}
\conj{z} = x-iy		 

conjugate(x + y*I) = x - I*y

Conjugate[x + y*I] == x - I*y
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
1.9.E12 | z Β― | = | z | 𝑧 𝑧 {\displaystyle{\displaystyle|\overline{z}|=|z|}}
|\conj{z}| = |z|		 

abs(conjugate(z)) = abs(z)

Abs[Conjugate[z]] == Abs[z]
Successful Successful - Successful [Tested: 7]
1.9.E13 ph ⁑ z Β― = - ph ⁑ z phase 𝑧 phase 𝑧 {\displaystyle{\displaystyle\operatorname{ph}\overline{z}=-\operatorname{ph}z}}
\phase@@{\conj{z}} = -\phase@@{z}		 

argument(conjugate(z)) = - argument(z)

Arg[Conjugate[z]] == - Arg[z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
1.9.E14 z 1 + z 2 = x 1 + x 2 + i ⁒ ( y 1 + y 2 ) subscript 𝑧 1 subscript 𝑧 2 subscript π‘₯ 1 subscript π‘₯ 2 imaginary-unit subscript 𝑦 1 subscript 𝑦 2 {\displaystyle{\displaystyle z_{1}+z_{2}=x_{1}+x_{2}+\mathrm{i}(y_{1}+y_{2})}}
z_{1}+ z_{2} = x_{1}+ x_{2}+\iunit(y_{1}+ y_{2})		 

x + y*I[1]+x + y*I[2] = x[1]+ x[2]+ I*(y[1]+ y[2])

Subscript[x + y*I, 1]+Subscript[x + y*I, 2] == Subscript[x, 1]+ Subscript[x, 2]+ I*(Subscript[y, 1]+ Subscript[y, 2])
Failure Failure Error
Failed [300 / 300]
Result: Plus[Complex[-0.7320508075688775, -2.732050807568877], Subscript[Complex[1.5, -1.5], 1], Subscript[Complex[1.5, -1.5], 2]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.3660254037844388, -1.3660254037844388], Subscript[Complex[1.5, -1.5], 1], Subscript[Complex[1.5, -1.5], 2]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9.E14 z 1 - z 2 = x 1 - x 2 + i ⁒ ( y 1 - y 2 ) subscript 𝑧 1 subscript 𝑧 2 subscript π‘₯ 1 subscript π‘₯ 2 imaginary-unit subscript 𝑦 1 subscript 𝑦 2 {\displaystyle{\displaystyle z_{1}-z_{2}=x_{1}-x_{2}+\mathrm{i}(y_{1}-y_{2})}}
z_{1}- z_{2} = x_{1}- x_{2}+\iunit(y_{1}- y_{2})		 

x + y*I[1]-x + y*I[2] = x[1]- x[2]+ I*(y[1]- y[2])

Subscript[x + y*I, 1]-Subscript[x + y*I, 2] == Subscript[x, 1]- Subscript[x, 2]+ I*(Subscript[y, 1]- Subscript[y, 2])
Failure Failure Error
Failed [300 / 300]
Result: Plus[Subscript[Complex[1.5, -1.5], 1], Times[-1.0, Subscript[Complex[1.5, -1.5], 2]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.36602540378443876, -1.3660254037844384], Subscript[Complex[1.5, -1.5], 1], Times[-1.0, Subscript[Complex[1.5, -1.5], 2]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9.E15 z 1 ⁒ z 2 = x 1 ⁒ x 2 - y 1 ⁒ y 2 + i ⁒ ( x 1 ⁒ y 2 + x 2 ⁒ y 1 ) subscript 𝑧 1 subscript 𝑧 2 subscript π‘₯ 1 subscript π‘₯ 2 subscript 𝑦 1 subscript 𝑦 2 𝑖 subscript π‘₯ 1 subscript 𝑦 2 subscript π‘₯ 2 subscript 𝑦 1 {\displaystyle{\displaystyle z_{1}z_{2}=x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+x_{% 2}y_{1})}}
z_{1}z_{2} = x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+x_{2}y_{1})		 

x + y*I[1]*x + y*I[2] = x[1]*x[2]- y[1]*y[2]+ I*(x[1]*y[2]+ x[2]*y[1])
 
Subscript[x + y*I, 1]*Subscript[x + y*I, 2] == Subscript[x, 1]*Subscript[x, 2]- Subscript[y, 1]*Subscript[y, 2]+ I*(Subscript[x, 1]*Subscript[y, 2]+ Subscript[x, 2]*Subscript[y, 1])
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E16 z 1 z 2 = z 1 ⁒ z Β― 2 | z 2 | 2 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 superscript subscript 𝑧 2 2 {\displaystyle{\displaystyle\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z}_{2}}{|% z_{2}|^{2}}}}
\frac{z_{1}}{z_{2}} = \frac{z_{1}\conj{z}_{2}}{|z_{2}|^{2}}		 

(z[1])/(z[2]) = (z[1]*conjugate(z)[2])/((abs(z[2]))^(2))

Divide[Subscript[z, 1],Subscript[z, 2]] == Divide[Subscript[z, 1]*Subscript[Conjugate[z], 2],(Abs[Subscript[z, 2]])^(2)]
Failure Failure Error
Failed [300 / 300]
Result: Plus[1.0, Times[Complex[-0.8660254037844387, -0.49999999999999994], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 2]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.0, -1.0], Times[Complex[-0.8660254037844387, -0.49999999999999994], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 2]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9.E16 z 1 ⁒ z Β― 2 | z 2 | 2 = x 1 ⁒ x 2 + y 1 ⁒ y 2 + i ⁒ ( x 2 ⁒ y 1 - x 1 ⁒ y 2 ) x 2 2 + y 2 2 subscript 𝑧 1 subscript 𝑧 2 superscript subscript 𝑧 2 2 subscript π‘₯ 1 subscript π‘₯ 2 subscript 𝑦 1 subscript 𝑦 2 𝑖 subscript π‘₯ 2 subscript 𝑦 1 subscript π‘₯ 1 subscript 𝑦 2 superscript subscript π‘₯ 2 2 superscript subscript 𝑦 2 2 {\displaystyle{\displaystyle\frac{z_{1}\overline{z}_{2}}{|z_{2}|^{2}}=\frac{x_% {1}x_{2}+y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}}}}
\frac{z_{1}\conj{z}_{2}}{|z_{2}|^{2}} = \frac{x_{1}x_{2}+y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}}		 

(x + y*I[1]*conjugate(x + y*I)[2])/((abs(x + y*I[2]))^(2)) = (x[1]*x[2]+ y[1]*y[2]+ I*(x[2]*y[1]- x[1]*y[2]))/((x[2])^(2)+ (y[2])^(2))

Divide[Subscript[x + y*I, 1]*Subscript[Conjugate[x + y*I], 2],(Abs[Subscript[x + y*I, 2]])^(2)] == Divide[Subscript[x, 1]*Subscript[x, 2]+ Subscript[y, 1]*Subscript[y, 2]+ I*(Subscript[x, 2]*Subscript[y, 1]- Subscript[x, 1]*Subscript[y, 2]),(Subscript[x, 2])^(2)+ (Subscript[y, 2])^(2)]
Failure Failure Error
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.6666666666666669, -0.6666666666666667], Times[Power[Abs[Subscript[Complex[1.5, -1.5], 2]], -2], Subscript[Complex[1.5, -1.5], 1], Subscript[Complex[1.5, 1.5], 2]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[y, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9.E17 | z 1 ⁒ z 2 | = | z 1 | ⁒ | z 2 | subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle|z_{1}z_{2}|=|z_{1}|\;|z_{2}|}}
|z_{1}z_{2}| = |z_{1}|\;|z_{2}|		 

abs(z[1]*z[2]) = abs(z[1])*abs(z[2])
 
Abs[Subscript[z, 1]*Subscript[z, 2]] == Abs[Subscript[z, 1]]*Abs[Subscript[z, 2]]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E18 ph ⁑ ( z 1 ⁒ z 2 ) = ph ⁑ z 1 + ph ⁑ z 2 phase subscript 𝑧 1 subscript 𝑧 2 phase subscript 𝑧 1 phase subscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{ph}\left(z_{1}z_{2}\right)=% \operatorname{ph}z_{1}+\operatorname{ph}z_{2}}}
\phase@{z_{1}z_{2}} = \phase@@{z_{1}}+\phase@@{z_{2}}		 

argument(z[1]*z[2]) = argument(z[1])+ argument(z[2])

Arg[Subscript[z, 1]*Subscript[z, 2]] == Arg[Subscript[z, 1]]+ Arg[Subscript[z, 2]]
Failure Failure
Failed [25 / 100]
Result: -6.283185308
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -1.5}


Result: -6.283185308
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -.5}


Result: -6.283185308
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -2}


Result: -6.283185309
Test Values: {z[1] = -1/2+1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [25 / 100]
Result: -6.283185307179587
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -1.5]}


Result: -6.283185307179587
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -0.5]}

... skip entries to safe data
1.9.E19 | z 1 z 2 | = | z 1 | | z 2 | subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\left|\frac{z_{1}}{z_{2}}\right|=\frac{|z_{1}|}{|z% _{2}|}}}
\abs{\frac{z_{1}}{z_{2}}} = \frac{|z_{1}|}{|z_{2}|}		 

abs((z[1])/(z[2])) = (abs(z[1]))/(abs(z[2]))

Abs[Divide[Subscript[z, 1],Subscript[z, 2]]] == Divide[Abs[Subscript[z, 1]],Abs[Subscript[z, 2]]]
Successful Successful - Successful [Tested: 100]
1.9.E20 ph ⁑ z 1 z 2 = ph ⁑ z 1 - ph ⁑ z 2 phase subscript 𝑧 1 subscript 𝑧 2 phase subscript 𝑧 1 phase subscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{ph}\frac{z_{1}}{z_{2}}=\operatorname% {ph}z_{1}-\operatorname{ph}z_{2}}}
\phase@@{\frac{z_{1}}{z_{2}}} = \phase@@{z_{1}}-\phase@@{z_{2}}		 

argument((z[1])/(z[2])) = argument(z[1])- argument(z[2])
 
Arg[Divide[Subscript[z, 1],Subscript[z, 2]]] == Arg[Subscript[z, 1]]- Arg[Subscript[z, 2]]
 Failure Failure
Failed [25 / 100]
Result: -6.283185308
Test Values: {z[1] = -1/2+1/2*I*3^(1/2), z[2] = -1/2*3^(1/2)-1/2*I}

Result: 6.283185308
Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)}

Result: 6.283185307
Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1.5}

Result: 6.283185307
Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -.5}

... skip entries to safe data
Failed [25 / 100]
Result: -6.283185307179586
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

Result: 6.283185307179586
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9.E22 cos ⁑ n ⁒ ΞΈ + i ⁒ sin ⁑ n ⁒ ΞΈ = ( cos ⁑ ΞΈ + i ⁒ sin ⁑ ΞΈ ) n 𝑛 πœƒ 𝑖 𝑛 πœƒ superscript πœƒ 𝑖 πœƒ 𝑛 {\displaystyle{\displaystyle\cos n\theta+i\sin n\theta=(\cos\theta+i\sin\theta% )^{n}}}
\cos@@{n\theta}+i\sin@@{n\theta} = (\cos@@{\theta}+i\sin@@{\theta})^{n}		 

cos(n*theta)+ I*sin(n*theta) = (cos(theta)+ I*sin(theta))^(n)
 
Cos[n*\[Theta]]+ I*Sin[n*\[Theta]] == (Cos[\[Theta]]+ I*Sin[\[Theta]])^(n)
 Error Successful - Successful [Tested: 10]
1.9.E23 | | z 1 | - | z 2 | | ≀ | z 1 + z 2 | subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\left|\left|z_{1}\right|-\left|z_{2}\right|\right|% \leq\left|z_{1}+z_{2}\right|}}
\abs{\abs{z_{1}}-\abs{z_{2}}} \leq \abs{z_{1}+z_{2}}		 

abs(abs(z[1])- abs(z[2])) <= abs(z[1]+ z[2])
 
Abs[Abs[Subscript[z, 1]]- Abs[Subscript[z, 2]]] <= Abs[Subscript[z, 1]+ Subscript[z, 2]]
 Failure Failure Successful [Tested: 100] Successful [Tested: 100]
1.9.E23 | z 1 + z 2 | ≀ | z 1 | + | z 2 | subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\left|z_{1}+z_{2}\right|\leq\left|z_{1}\right|+% \left|z_{2}\right|}}
\abs{z_{1}+z_{2}} \leq \abs{z_{1}}+\abs{z_{2}}		 

abs(z[1]+ z[2]) <= abs(z[1])+ abs(z[2])
 
Abs[Subscript[z, 1]+ Subscript[z, 2]] <= Abs[Subscript[z, 1]]+ Abs[Subscript[z, 2]]
 Failure Failure Successful [Tested: 100] Successful [Tested: 100]
1.9#Ex9 βˆ‚ ⁑ u βˆ‚ ⁑ x = βˆ‚ ⁑ v βˆ‚ ⁑ y partial-derivative 𝑒 π‘₯ partial-derivative 𝑣 𝑦 {\displaystyle{\displaystyle\frac{\partial u}{\partial x}=\frac{\partial v}{% \partial y}}}
\pderiv{u}{x} = \pderiv{v}{y}		 

diff(u, x) = diff(v, y)
 
D[u, x] == D[v, y]
 Successful Successful - Successful [Tested: 300]
1.9#Ex10 βˆ‚ ⁑ u βˆ‚ ⁑ y = - βˆ‚ ⁑ v βˆ‚ ⁑ x partial-derivative 𝑒 𝑦 partial-derivative 𝑣 π‘₯ {\displaystyle{\displaystyle\frac{\partial u}{\partial y}=-\frac{\partial v}{% \partial x}}}
\pderiv{u}{y} = -\pderiv{v}{x}		 

diff(u, y) = - diff(v, x)
 
D[u, y] == - D[v, x]
 Successful Successful - Successful [Tested: 300]
1.9.E26 βˆ‚ 2 ⁑ u βˆ‚ ⁑ x 2 + βˆ‚ 2 ⁑ u βˆ‚ ⁑ y 2 = βˆ‚ 2 ⁑ v βˆ‚ ⁑ x 2 + βˆ‚ 2 ⁑ v βˆ‚ ⁑ y 2 partial-derivative 𝑒 π‘₯ 2 partial-derivative 𝑒 𝑦 2 partial-derivative 𝑣 π‘₯ 2 partial-derivative 𝑣 𝑦 2 {\displaystyle{\displaystyle\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{% \partial}^{2}u}{{\partial y}^{2}}=\frac{{\partial}^{2}v}{{\partial x}^{2}}+% \frac{{\partial}^{2}v}{{\partial y}^{2}}}}
\pderiv[2]{u}{x}+\pderiv[2]{u}{y} = \pderiv[2]{v}{x}+\pderiv[2]{v}{y}		 

diff(u, [x$(2)])+ diff(u, [y$(2)]) = diff(v, [x$(2)])+ diff(v, [y$(2)])
 
D[u, {x, 2}]+ D[u, {y, 2}] == D[v, {x, 2}]+ D[v, {y, 2}]
 Successful Successful - Successful [Tested: 300]
1.9.E26 βˆ‚ 2 ⁑ v βˆ‚ ⁑ x 2 + βˆ‚ 2 ⁑ v βˆ‚ ⁑ y 2 = 0 partial-derivative 𝑣 π‘₯ 2 partial-derivative 𝑣 𝑦 2 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}v}{{\partial x}^{2}}+\frac{{% \partial}^{2}v}{{\partial y}^{2}}=0}}
\pderiv[2]{v}{x}+\pderiv[2]{v}{y} = 0		 

diff(v, [x$(2)])+ diff(v, [y$(2)]) = 0
 
D[v, {x, 2}]+ D[v, {y, 2}] == 0
 Successful Successful - Successful [Tested: 180]
1.9.E27 βˆ‚ 2 ⁑ u βˆ‚ ⁑ r 2 + 1 r ⁒ βˆ‚ ⁑ u βˆ‚ ⁑ r + 1 r 2 ⁒ βˆ‚ 2 ⁑ u βˆ‚ ⁑ ΞΈ 2 = 0 partial-derivative 𝑒 π‘Ÿ 2 1 π‘Ÿ partial-derivative 𝑒 π‘Ÿ 1 superscript π‘Ÿ 2 partial-derivative 𝑒 πœƒ 2 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}u}{{\partial r}^{2}}+\frac{1}{% r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{{\partial}^{2}u}{{% \partial\theta}^{2}}=0}}
\pderiv[2]{u}{r}+\frac{1}{r}\pderiv{u}{r}+\frac{1}{r^{2}}\pderiv[2]{u}{\theta} = 0		 

diff(u, [r$(2)])+(1)/(r)*diff(u, r)+(1)/((r)^(2))*diff(u, [theta$(2)]) = 0
 
D[u, {r, 2}]+Divide[1,r]*D[u, r]+Divide[1,(r)^(2)]*D[u, {\[Theta], 2}] == 0
 Successful Successful - Successful [Tested: 300]
1.9.E33 u ⁒ ( z ) = 1 2 ⁒ Ο€ ⁒ ∫ 0 2 ⁒ Ο€ u ⁒ ( z + r ⁒ e i ⁒ Ο• ) ⁒ d Ο• 𝑒 𝑧 1 2 πœ‹ subscript superscript 2 πœ‹ 0 𝑒 𝑧 π‘Ÿ superscript 𝑒 𝑖 italic-Ο• italic-Ο• {\displaystyle{\displaystyle u(z)=\frac{1}{2\pi}\int^{2\pi}_{0}u(z+re^{i\phi})% \mathrm{d}\phi}}
u(z) = \frac{1}{2\pi}\int^{2\pi}_{0}u(z+re^{i\phi})\diff{\phi}		 

u(z) = (1)/(2*Pi)*int(u(z + r*exp(I*phi)), phi = 0..2*Pi)
 
u[z] == Divide[1,2*Pi]*Integrate[u[z + r*Exp[I*\[Phi]]], {\[Phi], 0, 2*Pi}, GenerateConditions->None]
 Successful Successful - Successful [Tested: 300]
1.9.E34 u ⁒ ( r ⁒ e i ⁒ ΞΈ ) = 1 2 ⁒ Ο€ ⁒ ∫ 0 2 ⁒ Ο€ ( R 2 - r 2 ) ⁒ h ⁒ ( R ⁒ e i ⁒ Ο• ) ⁒ d Ο• R 2 - 2 ⁒ R ⁒ r ⁒ cos ⁑ ( Ο• - ΞΈ ) + r 2 𝑒 π‘Ÿ superscript 𝑒 𝑖 πœƒ 1 2 πœ‹ subscript superscript 2 πœ‹ 0 superscript 𝑅 2 superscript π‘Ÿ 2 β„Ž 𝑅 superscript 𝑒 𝑖 italic-Ο• italic-Ο• superscript 𝑅 2 2 𝑅 π‘Ÿ italic-Ο• πœƒ superscript π‘Ÿ 2 {\displaystyle{\displaystyle u(re^{i\theta})=\frac{1}{2\pi}\int^{2\pi}_{0}% \frac{(R^{2}-r^{2})h(Re^{i\phi})\mathrm{d}\phi}{R^{2}-2Rr\cos\left(\phi-\theta% \right)+r^{2}}}}
u(re^{i\theta}) = \frac{1}{2\pi}\int^{2\pi}_{0}\frac{(R^{2}-r^{2})h(Re^{i\phi})\diff{\phi}}{R^{2}-2Rr\cos@{\phi-\theta}+r^{2}}		 

u(r*exp(I*theta)) = (1)/(2*Pi)*int((((R)^(2)- (r)^(2))*h(R*exp(I*phi)))/((R)^(2)- 2*R*r*cos(phi - theta)+ (r)^(2)), phi = 0..2*Pi)
 
u[r*Exp[I*\[Theta]]] == Divide[1,2*Pi]*Integrate[Divide[((R)^(2)- (r)^(2))*h[R*Exp[I*\[Phi]]],(R)^(2)- 2*R*r*Cos[\[Phi]- \[Theta]]+ (r)^(2)], {\[Phi], 0, 2*Pi}, GenerateConditions->None]
 Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[-0.1639294614698989, -0.894905511379796]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[R, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.6307543640677387, -0.014887794479775784]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[R, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.9.E36 ∞ + z = z + ∞ 𝑧 𝑧 {\displaystyle{\displaystyle\infty+z=z+\infty}}
\infty+ z = z+\infty		 

infinity + z = z + infinity
 
Infinity + z == z + Infinity
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E37 ∞ β‹… z = z β‹… ∞ β‹… 𝑧 β‹… 𝑧 {\displaystyle{\displaystyle\infty\cdot z=z\cdot\infty}}
\infty\cdot z = z\cdot\infty		 

infinity * z = z * infinity
 
Infinity * z == z * Infinity
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E38 z / ∞ = 0 𝑧 0 {\displaystyle{\displaystyle z/\infty=0}}
z/\infty = 0		 

z/infinity = 0
 
z/Infinity == 0
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E39 z / 0 = ∞ 𝑧 0 {\displaystyle{\displaystyle z/0=\infty}}
z/0 = \infty		 z β‰  0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}} 
z/0 = infinity
 
z/0 == Infinity
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E44 z = d ⁒ w - b - c ⁒ w + a 𝑧 𝑑 𝑀 𝑏 𝑐 𝑀 π‘Ž {\displaystyle{\displaystyle z=\frac{dw-b}{-cw+a}}}
z = \frac{dw-b}{-cw+a}		 

z = (d*w - b)/(- c*w + a)
 
z == Divide[d*w - b,- c*w + a]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E48 a n = f ( n ) ⁒ ( z 0 ) n ! subscript π‘Ž 𝑛 superscript 𝑓 𝑛 subscript 𝑧 0 𝑛 {\displaystyle{\displaystyle a_{n}=\frac{f^{(n)}(z_{0})}{n!}}}
a_{n} = \frac{f^{(n)}(z_{0})}{n!}		 

a[n] = ((f(z[0]))^(n))/(factorial(n))
 
Subscript[a, n] == Divide[(f[Subscript[z, 0]])^(n),(n)!]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E50 βˆ‘ n = 0 ∞ ( a n + b n ) ⁒ z n = βˆ‘ n = 0 ∞ a n ⁒ z n + βˆ‘ n = 0 ∞ b n ⁒ z n subscript superscript 𝑛 0 subscript π‘Ž 𝑛 subscript 𝑏 𝑛 superscript 𝑧 𝑛 subscript superscript 𝑛 0 subscript π‘Ž 𝑛 superscript 𝑧 𝑛 subscript superscript 𝑛 0 subscript 𝑏 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\sum^{\infty}_{n=0}(a_{n}+b_{n})z^{n}=\sum^{\infty% }_{n=0}a_{n}z^{n}+\sum^{\infty}_{n=0}b_{n}z^{n}}}
\sum^{\infty}_{n=0}(a_{n}+ b_{n})z^{n} = \sum^{\infty}_{n=0}a_{n}z^{n}+\sum^{\infty}_{n=0}b_{n}z^{n}		 

sum((a[n]+ b[n])*(z)^(n), n = 0..infinity) = sum(a[n]*(z)^(n), n = 0..infinity)+ sum(b[n]*(z)^(n), n = 0..infinity)
 
Sum[(Subscript[a, n]+ Subscript[b, n])*(z)^(n), {n, 0, Infinity}, GenerateConditions->None] == Sum[Subscript[a, n]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None]+ Sum[Subscript[b, n]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E51 ( βˆ‘ n = 0 ∞ a n ⁒ z n ) ⁒ ( βˆ‘ n = 0 ∞ b n ⁒ z n ) = βˆ‘ n = 0 ∞ c n ⁒ z n subscript superscript 𝑛 0 subscript π‘Ž 𝑛 superscript 𝑧 𝑛 subscript superscript 𝑛 0 subscript 𝑏 𝑛 superscript 𝑧 𝑛 subscript superscript 𝑛 0 subscript 𝑐 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\left(\sum^{\infty}_{n=0}a_{n}z^{n}\right)\left(% \sum^{\infty}_{n=0}b_{n}z^{n}\right)=\sum^{\infty}_{n=0}c_{n}z^{n}}}
\left(\sum^{\infty}_{n=0}a_{n}z^{n}\right)\left(\sum^{\infty}_{n=0}b_{n}z^{n}\right) = \sum^{\infty}_{n=0}c_{n}z^{n}		 

(sum(a[n]*(z)^(n), n = 0..infinity))*(sum(b[n]*(z)^(n), n = 0..infinity)) = sum(c[n]*(z)^(n), n = 0..infinity)
 
(Sum[Subscript[a, n]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None])*(Sum[Subscript[b, n]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None]) == Sum[Subscript[c, n]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E52 c n = βˆ‘ k = 0 n a k ⁒ b n - k subscript 𝑐 𝑛 subscript superscript 𝑛 π‘˜ 0 subscript π‘Ž π‘˜ subscript 𝑏 𝑛 π‘˜ {\displaystyle{\displaystyle c_{n}=\sum^{n}_{k=0}a_{k}b_{n-k}}}
c_{n} = \sum^{n}_{k=0}a_{k}b_{n-k}		 

c[n] = sum(a[k]*b[n - k], k = 0..n)
 
Subscript[c, n] == Sum[Subscript[a, k]*Subscript[b, n - k], {k, 0, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex13 b 0 = 1 / a 0 subscript 𝑏 0 1 subscript π‘Ž 0 {\displaystyle{\displaystyle b_{0}=1/a_{0}}}
b_{0} = 1/a_{0}		 

b[0] = 1/a[0]
 
Subscript[b, 0] == 1/Subscript[a, 0]
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex14 b 1 = - a 1 / a 0 2 subscript 𝑏 1 subscript π‘Ž 1 superscript subscript π‘Ž 0 2 {\displaystyle{\displaystyle b_{1}=-a_{1}/a_{0}^{2}}}
b_{1} = -a_{1}/a_{0}^{2}		 

b[1] = - a[1]/(a[0])^(2)
 
Subscript[b, 1] == - Subscript[a, 1]/(Subscript[a, 0])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex15 b 2 = ( a 1 2 - a 0 ⁒ a 2 ) / a 0 3 subscript 𝑏 2 superscript subscript π‘Ž 1 2 subscript π‘Ž 0 subscript π‘Ž 2 superscript subscript π‘Ž 0 3 {\displaystyle{\displaystyle b_{2}=(a_{1}^{2}-a_{0}a_{2})/a_{0}^{3}}}
b_{2} = (a_{1}^{2}-a_{0}a_{2})/a_{0}^{3}		 

b[2] = ((a[1])^(2)- a[0]*a[2])/(a[0])^(3)
 
Subscript[b, 2] == ((Subscript[a, 1])^(2)- Subscript[a, 0]*Subscript[a, 2])/(Subscript[a, 0])^(3)
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex16 q 1 = a 1 subscript π‘ž 1 subscript π‘Ž 1 {\displaystyle{\displaystyle q_{1}=a_{1}}}
q_{1} = a_{1}		 

q[1] = a[1]
 
Subscript[q, 1] == Subscript[a, 1]
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex17 q 2 = ( 2 ⁒ a 2 - a 1 2 ) / 2 subscript π‘ž 2 2 subscript π‘Ž 2 superscript subscript π‘Ž 1 2 2 {\displaystyle{\displaystyle q_{2}=(2a_{2}-a_{1}^{2})/2}}
q_{2} = (2a_{2}-a_{1}^{2})/2		 

q[2] = (2*a[2]- (a[1])^(2))/2
 
Subscript[q, 2] == (2*Subscript[a, 2]- (Subscript[a, 1])^(2))/2
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex18 q 3 = ( 3 ⁒ a 3 - 3 ⁒ a 1 ⁒ a 2 + a 1 3 ) / 3 subscript π‘ž 3 3 subscript π‘Ž 3 3 subscript π‘Ž 1 subscript π‘Ž 2 superscript subscript π‘Ž 1 3 3 {\displaystyle{\displaystyle q_{3}=(3a_{3}-3a_{1}a_{2}+a_{1}^{3})/3}}
q_{3} = (3a_{3}-3a_{1}a_{2}+a_{1}^{3})/3		 

q[3] = (3*a[3]- 3*a[1]*a[2]+ (a[1])^(3))/3
 
Subscript[q, 3] == (3*Subscript[a, 3]- 3*Subscript[a, 1]*Subscript[a, 2]+ (Subscript[a, 1])^(3))/3
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex19 p 0 = 1 subscript 𝑝 0 1 {\displaystyle{\displaystyle p_{0}=1}}
p_{0} = 1		 

p[0] = 1
 
Subscript[p, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex20 p 1 = Ξ½ ⁒ a 1 subscript 𝑝 1 𝜈 subscript π‘Ž 1 {\displaystyle{\displaystyle p_{1}=\nu a_{1}}}
p_{1} = \nu a_{1}		 

p[1] = nu*a[1]
 
Subscript[p, 1] == \[Nu]*Subscript[a, 1]
 Skipped - no semantic math Skipped - no semantic math - -
1.9#Ex21 p 2 = Ξ½ ⁒ ( ( Ξ½ - 1 ) ⁒ a 1 2 + 2 ⁒ a 2 ) / 2 subscript 𝑝 2 𝜈 𝜈 1 superscript subscript π‘Ž 1 2 2 subscript π‘Ž 2 2 {\displaystyle{\displaystyle p_{2}=\nu((\nu-1)a_{1}^{2}+2a_{2})/2}}
p_{2} = \nu((\nu-1)a_{1}^{2}+2a_{2})/2		 

p[2] = nu*((nu - 1)*(a[1])^(2)+ 2*a[2])/2
 
Subscript[p, 2] == \[Nu]*((\[Nu]- 1)*(Subscript[a, 1])^(2)+ 2*Subscript[a, 2])/2
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E63 f ( m ) ⁒ ( z ) = βˆ‘ n = 0 ∞ ( n + 1 ) m ⁒ a n + m ⁒ ( z - z 0 ) n superscript 𝑓 π‘š 𝑧 superscript subscript 𝑛 0 Pochhammer 𝑛 1 π‘š subscript π‘Ž 𝑛 π‘š superscript 𝑧 subscript 𝑧 0 𝑛 {\displaystyle{\displaystyle f^{(m)}(z)=\sum_{n=0}^{\infty}{\left(n+1\right)_{% m}}a_{n+m}(z-z_{0})^{n}}}
f^{(m)}(z) = \sum_{n=0}^{\infty}\Pochhammersym{n+1}{m}a_{n+m}(z-z_{0})^{n}		 | z - z 0 | < R 𝑧 subscript 𝑧 0 𝑅 {\displaystyle{\displaystyle\left|z-z_{0}\right|<R}} 
(f(z))^(m) = sum(pochhammer(n + 1, m)*a[n + m]*(z - z[0])^(n), n = 0..infinity)
 
(f[z])^(m) == Sum[Pochhammer[n + 1, m]*Subscript[a, n + m]*(z - Subscript[z, 0])^(n), {n, 0, Infinity}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out Skipped - Because timed out
1.9.E64 | z m , n - z | < Ο΅ subscript 𝑧 π‘š 𝑛 𝑧 italic-Ο΅ {\displaystyle{\displaystyle|z_{m,n}-z|<\epsilon}}
|z_{m,n}-z| < \epsilon		 

abs(z[m , n]- z) < epsilon
 
Abs[Subscript[z, m , n]- z] < \[Epsilon]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E66 z p , q = βˆ‘ m = 0 p βˆ‘ n = 0 q ΞΆ m , n subscript 𝑧 𝑝 π‘ž subscript superscript 𝑝 π‘š 0 subscript superscript π‘ž 𝑛 0 subscript 𝜁 π‘š 𝑛 {\displaystyle{\displaystyle z_{p,q}=\sum^{p}_{m=0}\sum^{q}_{n=0}\zeta_{m,n}}}
z_{p,q} = \sum^{p}_{m=0}\sum^{q}_{n=0}\zeta_{m,n}		 

z[p , q] = sum(sum(zeta[m , n], n = 0..q), m = 0..p)
 
Subscript[z, p , q] == Sum[Sum[Subscript[\[Zeta], m , n], {n, 0, q}, GenerateConditions->None], {m, 0, p}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.9.E69 ∫ a b βˆ‘ n = 0 ∞ | f n ⁒ ( t ) | ⁒ d t < ∞ subscript superscript 𝑏 π‘Ž subscript superscript 𝑛 0 subscript 𝑓 𝑛 𝑑 𝑑 {\displaystyle{\displaystyle\int^{b}_{a}\sum^{\infty}_{n=0}|f_{n}(t)|\mathrm{d% }t<\infty}}
\int^{b}_{a}\sum^{\infty}_{n=0}|f_{n}(t)|\diff{t} < \infty		 

int(sum(abs(f[n](t)), n = 0..infinity), t = a..b) < infinity
 
Integrate[Sum[Abs[Subscript[f, n][t]], {n, 0, Infinity}, GenerateConditions->None], {t, a, b}, GenerateConditions->None] < Infinity
 Missing Macro Error Missing Macro Error - -
1.9.E70 βˆ‘ n = 0 ∞ ∫ a b | f n ⁒ ( t ) | ⁒ d t < ∞ subscript superscript 𝑛 0 subscript superscript 𝑏 π‘Ž subscript 𝑓 𝑛 𝑑 𝑑 {\displaystyle{\displaystyle\sum^{\infty}_{n=0}\int^{b}_{a}|f_{n}(t)|\mathrm{d% }t<\infty}}
\sum^{\infty}_{n=0}\int^{b}_{a}|f_{n}(t)|\diff{t} < \infty		 

sum(int(abs(f[n](t)), t = a..b) , n = 0..infinity)< infinity
 
Sum[Integrate[Abs[Subscript[f, n][t]], {t, a, b}, GenerateConditions->None] , {n, 0, Infinity}, GenerateConditions->None]< Infinity
 Missing Macro Error Missing Macro Error - -
1.9.E71 ∫ a b βˆ‘ n = 0 ∞ f n ⁒ ( t ) ⁒ d t = βˆ‘ n = 0 ∞ ∫ a b f n ⁒ ( t ) ⁒ d t subscript superscript 𝑏 π‘Ž subscript superscript 𝑛 0 subscript 𝑓 𝑛 𝑑 𝑑 subscript superscript 𝑛 0 subscript superscript 𝑏 π‘Ž subscript 𝑓 𝑛 𝑑 𝑑 {\displaystyle{\displaystyle\int^{b}_{a}\sum^{\infty}_{n=0}f_{n}(t)\mathrm{d}t% =\sum^{\infty}_{n=0}\int^{b}_{a}f_{n}(t)\mathrm{d}t}}
\int^{b}_{a}\sum^{\infty}_{n=0}f_{n}(t)\diff{t} = \sum^{\infty}_{n=0}\int^{b}_{a}f_{n}(t)\diff{t}		 

int(sum(f[n](t), n = 0..infinity), t = a..b) = sum(int(f[n](t), t = a..b), n = 0..infinity)
 
Integrate[Sum[Subscript[f, n][t], {n, 0, Infinity}, GenerateConditions->None], {t, a, b}, GenerateConditions->None] == Sum[Integrate[Subscript[f, n][t], {t, a, b}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None]
 Successful Aborted - Skipped - Because timed out
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