Numerical Methods - 3.5 Quadrature

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
3.5.E14 ∫ 0 ∞ e - p ⁢ t ⁢ J 0 ⁡ ( t ) ⁢ d t = 1 p 2 + 1 superscript subscript 0 superscript 𝑒 𝑝 𝑡 Bessel-J 0 𝑡 𝑡 1 superscript 𝑝 2 1 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-pt}J_{0}\left(t\right)\mathrm% {d}t=\frac{1}{\sqrt{p^{2}+1}}}}
\int_{0}^{\infty}e^{-pt}\BesselJ{0}@{t}\diff{t} = \frac{1}{\sqrt{p^{2}+1}}		 

int(exp(- p*t)*BesselJ(0, t), t = 0..infinity) = (1)/(sqrt((p)^(2)+ 1))

Integrate[Exp[- p*t]*BesselJ[0, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[(p)^(2)+ 1]]
Successful Aborted -
Failed [5 / 10]
Result: Complex[-1.732050807568877, -1.0]
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-1.4678898250138706, 0.39331989319032856]
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
3.5.E16 w k = g k n ⁢ ( 1 - ∑ j = 1 ⌊ n / 2 ⌋ b j 4 ⁢ j 2 - 1 ⁢ cos ⁡ ( 2 ⁢ j ⁢ k ⁢ π / n ) ) subscript 𝑤 𝑘 subscript 𝑔 𝑘 𝑛 1 superscript subscript 𝑗 1 𝑛 2 subscript 𝑏 𝑗 4 superscript 𝑗 2 1 2 𝑗 𝑘 𝑛 {\displaystyle{\displaystyle w_{k}=\frac{g_{k}}{n}\left(1-\sum_{j=1}^{\left% \lfloor n/2\right\rfloor}\frac{b_{j}}{4j^{2}-1}\cos\left(2jk\pi/n\right)\right% )}}
w_{k} = \frac{g_{k}}{n}\left(1-\sum_{j=1}^{\floor{n/2}}\frac{b_{j}}{4j^{2}-1}\cos@{2jk\cpi/n}\right)		 

w[k] = (g[k])/(n)*(1 - sum((b[j])/(4*(j)^(2)- 1)*cos(2*j*k*Pi/n), j = 1..floor(n/2)))

Subscript[w, k] == Divide[Subscript[g, k],n]*(1 - Sum[Divide[Subscript[b, j],4*(j)^(2)- 1]*Cos[2*j*k*Pi/n], {j, 1, Floor[n/2]}, GenerateConditions->None])
Failure Failure
Failed [290 / 300]
Result: .3496793685+.1056624326*I
Test Values: {b[j] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: .5495724917+.2852208110*I
Test Values: {b[j] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: .5163460354+.3943375674*I
Test Values: {b[j] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, w[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 2}


Result: .5495724917+.2852208110*I
Test Values: {b[j] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, w[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 3}

... skip entries to safe data
 Skipped - Because timed out
3.5.E19 E n ⁢ ( f ) = γ n ⁢ f ( 2 ⁢ n ) ⁢ ( ξ ) / ( 2 ⁢ n ) ! subscript 𝐸 𝑛 𝑓 subscript 𝛾 𝑛 superscript 𝑓 2 𝑛 𝜉 2 𝑛 {\displaystyle{\displaystyle E_{n}(f)=\gamma_{n}f^{(2n)}(\xi)/(2n)!}}
E_{n}(f) = \gamma_{n}f^{(2n)}(\xi)/(2n)!		 a < ξ , ξ < b formulae-sequence 𝑎 𝜉 𝜉 𝑏 {\displaystyle{\displaystyle a<\xi,\xi<b}} 
(-(b - a)/(180)*(h)^(4)* (f(xi))^(4)) = (int((p[n])^(2)(x)* w(x), x = a..b))*(f(xi))^(2*n)/factorial(2*n)
 
(-Divide[b - a,180]*(h)^(4)* (f[\[Xi]])^(4)) == (Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None])*(f[\[Xi]])^(2*n)/(2*n)!
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex1 [ a , b ] = [ - 1 , 1 ] 𝑎 𝑏 1 1 {\displaystyle{\displaystyle[a,b]=[-1,1]}}
[a,b] = [-1,1]		 

[a , b] = [- 1 , 1]
 
[a , b] == [- 1 , 1]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex2 w ⁢ ( x ) = 1 𝑤 𝑥 1 {\displaystyle{\displaystyle w(x)=1}}
w(x) = 1		 

w(x) = 1
 
w[x] == 1
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex3 γ n = 2 2 ⁢ n + 1 2 ⁢ n + 1 ⁢ ( n ! ) 4 ( ( 2 ⁢ n ) ! ) 2 subscript 𝛾 𝑛 superscript 2 2 𝑛 1 2 𝑛 1 superscript 𝑛 4 superscript 2 𝑛 2 {\displaystyle{\displaystyle\gamma_{n}=\frac{2^{2n+1}}{2n+1}\,\frac{(n!)^{4}}{% ((2n)!)^{2}}}}
\gamma_{n} = \frac{2^{2n+1}}{2n+1}\,\frac{(n!)^{4}}{((2n)!)^{2}}		 

(int((p[n])^(2)(x)* w(x), x = a..b)) = ((2)^(2*n + 1))/(2*n + 1)*((factorial(n))^(4))/((factorial(2*n))^(2))
 
(Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None]) == Divide[(2)^(2*n + 1),2*n + 1]*Divide[((n)!)^(4),((2*n)!)^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex4 [ a , b ] = [ - 1 , 1 ] 𝑎 𝑏 1 1 {\displaystyle{\displaystyle[a,b]=[-1,1]}}
[a,b] = [-1,1]		 

[a , b] = [- 1 , 1]
 
[a , b] == [- 1 , 1]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex5 w ⁢ ( x ) = ( 1 - x 2 ) - 1 / 2 𝑤 𝑥 superscript 1 superscript 𝑥 2 1 2 {\displaystyle{\displaystyle w(x)=(1-x^{2})^{-1/2}}}
w(x) = (1-x^{2})^{-1/2}		 

w(x) = (1 - (x)^(2))^(- 1/2)
 
w[x] == (1 - (x)^(2))^(- 1/2)
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex6 γ n = π 2 2 ⁢ n - 1 subscript 𝛾 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle\gamma_{n}=\frac{\pi}{2^{2n-1}}}}
\gamma_{n} = \frac{\cpi}{2^{2n-1}}		 

(int((p[n])^(2)(x)* w(x), x = a..b)) = (Pi)/((2)^(2*n - 1))

(Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None]) == Divide[Pi,(2)^(2*n - 1)]
Failure Failure
Failed [300 / 300]
Result: -1.570796327
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.3926990818
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: -.9817477044e-1
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -1.570796327
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: -1.5707963267948966
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: -0.39269908169872414
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex7 x k = cos ⁡ ( 2 ⁢ k - 1 2 ⁢ n ⁢ π ) subscript 𝑥 𝑘 2 𝑘 1 2 𝑛 {\displaystyle{\displaystyle x_{k}=\cos\left(\frac{2k-1}{2n}\pi\right)}}
x_{k} = \cos@{\frac{2k-1}{2n}\cpi}		 

x[k] = cos((2*k - 1)/(2*n)*Pi)

Subscript[x, k] == Cos[Divide[2*k - 1,2*n]*Pi]
Failure Failure
Failed [90 / 90]
Result: .8660254042+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: .1589186229+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: .2e-9+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: .8660254034+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 1}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.15891862259789125, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex8 w k = π n subscript 𝑤 𝑘 𝑛 {\displaystyle{\displaystyle w_{k}=\frac{\pi}{n}}}
w_{k} = \frac{\cpi}{n}		 k = 1 𝑘 1 {\displaystyle{\displaystyle k=1}} 
w[k] = (Pi)/(n)

Subscript[w, k] == Divide[Pi,n]
Failure Failure
Failed [30 / 30]
Result: -2.275567250+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: -.7047709230+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: -.1811721470+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: -3.641592654+.8660254040*I
Test Values: {w[k] = -1/2+1/2*I*3^(1/2), k = 1, n = 1}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[-2.2755672498053543, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7047709230104579, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex9 x k = cos ⁡ ( k n + 1 ⁢ π ) subscript 𝑥 𝑘 𝑘 𝑛 1 {\displaystyle{\displaystyle x_{k}=\cos\left(\frac{k}{n+1}\pi\right)}}
x_{k} = \cos@{\frac{k}{n+1}\cpi}		 

x[k] = cos((k)/(n + 1)*Pi)

Subscript[x, k] == Cos[Divide[k,n + 1]*Pi]
Failure Failure
Failed [88 / 90]
Result: .8660254042+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: .3660254038+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: .1589186229+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: 1.866025404+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 1}

... skip entries to safe data
Failed [88 / 90]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.3660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex10 w k = π n + 1 ⁢ sin 2 ⁡ ( k n + 1 ⁢ π ) subscript 𝑤 𝑘 𝑛 1 2 𝑘 𝑛 1 {\displaystyle{\displaystyle w_{k}=\frac{\pi}{n+1}{\sin^{2}}\left(\frac{k}{n+1% }\pi\right)}}
w_{k} = \frac{\cpi}{n+1}\sin^{2}@{\frac{k}{n+1}\cpi}		 

w[k] = (Pi)/(n + 1)*(sin((k)/(n + 1)*Pi))^(2)

Subscript[w, k] == Divide[Pi,n + 1]*(Sin[Divide[k,n + 1]*Pi])^(2)
Failure Failure
Failed [90 / 90]
Result: -.7047709230+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: .806272408e-1+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: .4733263222+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: .8660254040+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 1}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[-0.7047709230104579, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.08062724038699043, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex11 γ n = π 2 2 ⁢ n + 1 subscript 𝛾 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle\gamma_{n}=\frac{\pi}{2^{2n+1}}}}
\gamma_{n} = \frac{\cpi}{2^{2n+1}}		 α = β , β = 1 2 formulae-sequence 𝛼 𝛽 𝛽 1 2 {\displaystyle{\displaystyle\alpha=\beta,\beta=\tfrac{1}{2}}} 
(int((p[n])^(2)(x)* w(x), x = a..b)) = (Pi)/((2)^(2*n + 1))

(Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None]) == Divide[Pi,(2)^(2*n + 1)]
Failure Failure
Failed [300 / 300]
Result: -.3926990818
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.9817477044e-1
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: -.2454369261e-1
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -.3926990818
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: -0.39269908169872414
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: -0.09817477042468103
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex12 x k = + cos ⁡ ( 2 ⁢ k 2 ⁢ n + 1 ⁢ π ) subscript 𝑥 𝑘 2 𝑘 2 𝑛 1 {\displaystyle{\displaystyle x_{k}=+\cos\left(\frac{2k}{2n+1}\pi\right)}}
x_{k} = +\cos@{\frac{2k}{2n+1}\cpi}		 

x[k] = + cos((2*k)/(2*n + 1)*Pi)

Subscript[x, k] == + Cos[Divide[2*k,2*n + 1]*Pi]
Failure Failure
Failed [88 / 90]
Result: 1.366025404+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: .5570084102+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: .2425356024+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: 1.366025403+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 1}

... skip entries to safe data
Failed [88 / 90]
Result: Complex[1.3660254037844388, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.5570084094094913, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex12 x k = - cos ⁡ ( 2 ⁢ k 2 ⁢ n + 1 ⁢ π ) subscript 𝑥 𝑘 2 𝑘 2 𝑛 1 {\displaystyle{\displaystyle x_{k}=-\cos\left(\frac{2k}{2n+1}\pi\right)}}
x_{k} = -\cos@{\frac{2k}{2n+1}\cpi}		 

x[k] = - cos((2*k)/(2*n + 1)*Pi)

Subscript[x, k] == - Cos[Divide[2*k,2*n + 1]*Pi]
Failure Failure
Failed [88 / 90]
Result: .3660254043+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: 1.175042398+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: 1.489515206+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: .3660254051+.5000000000*I
Test Values: {x[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 1}

... skip entries to safe data
Failed [88 / 90]
Result: Complex[0.3660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1750423981593863, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[Subscript[x, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex13 w k = 4 ⁢ π 2 ⁢ n + 1 ⁢ sin 2 ⁡ ( k 2 ⁢ n + 1 ⁢ π ) subscript 𝑤 𝑘 4 2 𝑛 1 2 𝑘 2 𝑛 1 {\displaystyle{\displaystyle w_{k}=\frac{4\pi}{2n+1}{\sin^{2}}\left(\frac{k}{2% n+1}\pi\right)}}
w_{k} = \frac{4\cpi}{2n+1}\sin^{2}@{\frac{k}{2n+1}\cpi}		 

w[k] = (4*Pi)/(2*n + 1)*(sin((k)/(2*n + 1)*Pi))^(2)

Subscript[w, k] == Divide[4*Pi,2*n + 1]*(Sin[Divide[k,2*n + 1]*Pi])^(2)
Failure Failure
Failed [90 / 90]
Result: -2.275567250+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: -.22894503e-2+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}


Result: .5280706399+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 3}


Result: -2.275567248+.5000000000*I
Test Values: {w[k] = 1/2*3^(1/2)+1/2*I, k = 2, n = 1}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[-2.2755672498053543, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.0022894499063851326, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex14 γ n = π 2 2 ⁢ n subscript 𝛾 𝑛 superscript 2 2 𝑛 {\displaystyle{\displaystyle\gamma_{n}=\frac{\pi}{2^{2n}}}}
\gamma_{n} = \frac{\cpi}{2^{2n}}		 

(int((p[n])^(2)(x)* w(x), x = a..b)) = (Pi)/((2)^(2*n))
 
(Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None]) == Divide[Pi,(2)^(2*n)]
 Failure Failure
Failed [300 / 300]
Result: -.7853981635
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 1, alpha = 1/2, beta = -1/2}

Result: -.1963495409
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 2, alpha = 1/2, beta = -1/2}

Result: -.4908738522e-1
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 3, alpha = 1/2, beta = -1/2}

Result: -.7853981635
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = -1/2+1/2*I*3^(1/2), n = 1, alpha = 1/2, beta = -1/2}

... skip entries to safe data
Failed [300 / 300]
Result: -0.7853981633974483
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, Rational[1, 2]], Rule[β, Rational[-1, 2]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: -0.19634954084936207
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, Rational[1, 2]], Rule[β, Rational[-1, 2]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex15 [ a , b ] = [ - 1 , 1 ] 𝑎 𝑏 1 1 {\displaystyle{\displaystyle[a,b]=[-1,1]}}
[a,b] = [-1,1]		 

[a , b] = [- 1 , 1]
 
[a , b] == [- 1 , 1]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex16 w ⁢ ( x ) = ( 1 - x ) α ⁢ ( 1 + x ) β 𝑤 𝑥 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 {\displaystyle{\displaystyle w(x)=(1-x)^{\alpha}(1+x)^{\beta}}}
w(x) = (1-x)^{\alpha}(1+x)^{\beta}		 

w(x) = (1 - x)^(alpha)*(1 + x)^(beta)
 
w[x] == (1 - x)^\[Alpha]*(1 + x)^\[Beta]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex17 γ n = Γ ⁡ ( n + α + 1 ) ⁢ Γ ⁡ ( n + β + 1 ) ⁢ Γ ⁡ ( n + α + β + 1 ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( Γ ⁡ ( 2 ⁢ n + α + β + 1 ) ) 2 ⁢ 2 2 ⁢ n + α + β + 1 ⁢ n ! subscript 𝛾 𝑛 Euler-Gamma 𝑛 𝛼 1 Euler-Gamma 𝑛 𝛽 1 Euler-Gamma 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 superscript Euler-Gamma 2 𝑛 𝛼 𝛽 1 2 superscript 2 2 𝑛 𝛼 𝛽 1 𝑛 {\displaystyle{\displaystyle\gamma_{n}=\dfrac{\Gamma\left(n+\alpha+1\right)% \Gamma\left(n+\beta+1\right)\Gamma\left(n+\alpha+\beta+1\right)}{(2n+\alpha+% \beta+1)(\Gamma\left(2n+\alpha+\beta+1\right))^{2}}2^{2n+\alpha+\beta+1}n!}}
\gamma_{n} = \dfrac{\EulerGamma@{n+\alpha+1}\EulerGamma@{n+\beta+1}\EulerGamma@{n+\alpha+\beta+1}}{(2n+\alpha+\beta+1)(\EulerGamma@{2n+\alpha+\beta+1})^{2}}2^{2n+\alpha+\beta+1}n!		 α > - 1 , β > - 1 , ℜ ⁡ ( n + α + 1 ) > 0 , ℜ ⁡ ( n + β + 1 ) > 0 , ℜ ⁡ ( n + α + β + 1 ) > 0 , ℜ ⁡ ( 2 ⁢ n + α + β + 1 ) > 0 formulae-sequence 𝛼 1 formulae-sequence 𝛽 1 formulae-sequence 𝑛 𝛼 1 0 formulae-sequence 𝑛 𝛽 1 0 formulae-sequence 𝑛 𝛼 𝛽 1 0 2 𝑛 𝛼 𝛽 1 0 {\displaystyle{\displaystyle\alpha>-1,\beta>-1,\Re(n+\alpha+1)>0,\Re(n+\beta+1% )>0,\Re(n+\alpha+\beta+1)>0,\Re(2n+\alpha+\beta+1)>0}} 
(int((p[n])^(2)(x)* w(x), x = a..b)) = (GAMMA(n + alpha + 1)*GAMMA(n + beta + 1)*GAMMA(n + alpha + beta + 1))/((2*n + alpha + beta + 1)*(GAMMA(2*n + alpha + beta + 1))^(2))*(2)^(2*n + alpha + beta + 1)* factorial(n)
 
(Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None]) == Divide[Gamma[n + \[Alpha]+ 1]*Gamma[n + \[Beta]+ 1]*Gamma[n + \[Alpha]+ \[Beta]+ 1],(2*n + \[Alpha]+ \[Beta]+ 1)*(Gamma[2*n + \[Alpha]+ \[Beta]+ 1])^(2)]*(2)^(2*n + \[Alpha]+ \[Beta]+ 1)* (n)!
 Failure Failure
Failed [300 / 300]
Result: -.1963495408
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, beta = 1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 1}

Result: -.4090615438e-1
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, beta = 1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 2}

Result: -.9203884728e-2
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, beta = 1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 3}

Result: -.1963495408
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, beta = 1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
 Skipped - Because timed out
3.5#Ex19 w ⁢ ( x ) = x α ⁢ e - x 𝑤 𝑥 superscript 𝑥 𝛼 superscript 𝑒 𝑥 {\displaystyle{\displaystyle w(x)=x^{\alpha}e^{-x}}}
w(x) = x^{\alpha}e^{-x}		 

w(x) = (x)^(alpha)* exp(- x)
 
w[x] == (x)^\[Alpha]* Exp[- x]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex20 γ n = n ! ⁢ Γ ⁡ ( n + α + 1 ) subscript 𝛾 𝑛 𝑛 Euler-Gamma 𝑛 𝛼 1 {\displaystyle{\displaystyle\gamma_{n}=n!\,\Gamma\left(n+\alpha+1\right)}}
\gamma_{n} = n!\,\EulerGamma@{n+\alpha+1}		 α > - 1 , ℜ ⁡ ( n + α + 1 ) > 0 formulae-sequence 𝛼 1 𝑛 𝛼 1 0 {\displaystyle{\displaystyle\alpha>-1,\Re(n+\alpha+1)>0}} 
(int((p[n])^(2)(x)* w(x), x = a..b)) = factorial(n)*GAMMA(n + alpha + 1)
 
(Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None]) == (n)!*Gamma[n + \[Alpha]+ 1]
 Failure Failure
Failed [300 / 300]
Result: -3.323350970
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 1}

Result: -23.26345680
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 2}

Result: -314.0566667
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 3}

Result: -3.323350970
Test Values: {a = -1.5, alpha = 1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: -3.3233509704478426
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: -23.2634567931349
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex21 ( a , b ) = ( - ∞ , ∞ ) 𝑎 𝑏 {\displaystyle{\displaystyle(a,b)=(-\infty,\infty)}}
(a,b) = (-\infty,\infty)		 

(a , b) = (- infinity , infinity)
 
(a , b) == (- Infinity , Infinity)
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex22 w ⁢ ( x ) = e - x 2 𝑤 𝑥 superscript 𝑒 superscript 𝑥 2 {\displaystyle{\displaystyle w(x)=e^{-x^{2}}}}
w(x) = e^{-x^{2}}		 

w(x) = exp(- (x)^(2))
 
w[x] == Exp[- (x)^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex23 γ n = π ⁢ n ! 2 n subscript 𝛾 𝑛 𝑛 superscript 2 𝑛 {\displaystyle{\displaystyle\gamma_{n}=\sqrt{\pi}\,\frac{n!}{2^{n}}}}
\gamma_{n} = \sqrt{\cpi}\,\frac{n!}{2^{n}}		 

(int((p[n])^(2)(x)* w(x), x = a..b)) = sqrt(Pi)*(factorial(n))/((2)^(n))
 
(Integrate[(Subscript[p, n])^(2)[x]* w[x], {x, a, b}, GenerateConditions->None]) == Sqrt[Pi]*Divide[(n)!,(2)^(n)]
 Failure Failure
Failed [300 / 300]
Result: -.8862269255
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 1}

Result: -.8862269255
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 2}

Result: -1.329340388
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, n = 3}

Result: -.8862269255
Test Values: {a = -1.5, b = -1.5, w = 1/2*3^(1/2)+1/2*I, p[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: -0.8862269254527579
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: -0.8862269254527579
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5#Ex24 [ a , b ] = [ 0 , 1 ] 𝑎 𝑏 0 1 {\displaystyle{\displaystyle[a,b]=[0,1]}}
[a,b] = [0,1]		 

[a , b] = [0 , 1]
 
[a , b] == [0 , 1]
 Skipped - no semantic math Skipped - no semantic math - -
3.5#Ex25 w ⁢ ( x ) = ln ⁡ ( 1 / x ) 𝑤 𝑥 1 𝑥 {\displaystyle{\displaystyle w(x)=\ln\left(1/x\right)}}
w(x) = \ln@{1/x}		 

w(x) = ln(1/x)
 
w[x] == Log[1/x]
 Failure Failure
Failed [30 / 30]
Result: 1.704503214+.7500000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 1.5}

Result: -.2601344786+.2500000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, x = .5}

Result: 2.425197989+1.*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 2}

Result: -.3445348919+1.299038106*I
Test Values: {w = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[1.7045032137848224, 0.7499999999999999]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}

Result: Complex[-0.26013447866772593, 0.24999999999999997]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
3.5.E30 p k + 1 ⁢ ( x ) = ( x - α k ) ⁢ p k ⁢ ( x ) - β k ⁢ p k - 1 ⁢ ( x ) subscript 𝑝 𝑘 1 𝑥 𝑥 subscript 𝛼 𝑘 subscript 𝑝 𝑘 𝑥 subscript 𝛽 𝑘 subscript 𝑝 𝑘 1 𝑥 {\displaystyle{\displaystyle p_{k+1}(x)=(x-\alpha_{k})p_{k}(x)-\beta_{k}p_{k-1% }(x)}}
p_{k+1}(x) = (x-\alpha_{k})p_{k}(x)-\beta_{k}p_{k-1}(x)		 

p[k + 1](x) = (x - alpha[k])*p[k](x)- beta[k]*p[k - 1](x)
 
Subscript[p, k + 1][x] == (x - Subscript[\[Alpha], k])*Subscript[p, k][x]- Subscript[\[Beta], k]*Subscript[p, k - 1][x]
 Skipped - no semantic math Skipped - no semantic math - -
3.5.E32 w k = β 0 ⁢ v k , 1 2 subscript 𝑤 𝑘 subscript 𝛽 0 superscript subscript 𝑣 𝑘 1 2 {\displaystyle{\displaystyle w_{k}=\beta_{0}v_{k,1}^{2}}}
w_{k} = \beta_{0}v_{k,1}^{2}		 k = 1 𝑘 1 {\displaystyle{\displaystyle k=1}} 
w[k] = beta[0]*(v[k , 1])^(2)
 
Subscript[w, k] == Subscript[\[Beta], 0]*(Subscript[v, k , 1])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
3.5.E37 ∫ c - i ⁢ ∞ c + i ⁢ ∞ e ζ ⁢ ζ - s ⁢ p k ⁢ ( 1 / ζ ) ⁢ p ℓ ⁢ ( 1 / ζ ) ⁢ d ζ = 0 superscript subscript 𝑐 imaginary-unit 𝑐 imaginary-unit superscript 𝑒 𝜁 superscript 𝜁 𝑠 subscript 𝑝 𝑘 1 𝜁 subscript 𝑝 ℓ 1 𝜁 𝜁 0 {\displaystyle{\displaystyle\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}e^{% \zeta}\zeta^{-s}p_{k}(1/\zeta)p_{\ell}(1/\zeta)\mathrm{d}\zeta=0}}
\int_{c-\iunit\infty}^{c+\iunit\infty}e^{\zeta}\zeta^{-s}p_{k}(1/\zeta)p_{\ell}(1/\zeta)\diff{\zeta} = 0		 k ≠ ℓ 𝑘 ℓ {\displaystyle{\displaystyle k\neq\ell}} 
int(exp(zeta)*(zeta)^(- s)* p[k]*(1/zeta)*p[ell]*(1/zeta), zeta = c - I*infinity..c + I*infinity) = 0
 
Integrate[Exp[\[Zeta]]*\[Zeta]^(- s)* Subscript[p, k]*(1/\[Zeta])*Subscript[p, \[ScriptL]]*(1/\[Zeta]), {\[Zeta], c - I*Infinity, c + I*Infinity}, GenerateConditions->None] == 0
 Successful Aborted -
Failed [300 / 300]
Result: Complex[-3.0699801238394655, 1.7724538509055168]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[s, -1.5], Rule[Subscript[p, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-3.0699801238394655, 1.7724538509055168]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[s, -1.5], Rule[Subscript[p, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.5.E42 erfc ⁡ λ = 1 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ e ζ - 2 ⁢ λ ⁢ ζ ⁢ d ζ ζ complementary-error-function 𝜆 1 2 imaginary-unit superscript subscript 𝑐 imaginary-unit 𝑐 imaginary-unit superscript 𝑒 𝜁 2 𝜆 𝜁 𝜁 𝜁 {\displaystyle{\displaystyle\operatorname{erfc}\lambda=\frac{1}{2\pi\mathrm{i}% }\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}e^{\zeta-2\lambda\sqrt{\zeta}}% \frac{\mathrm{d}\zeta}{\zeta}}}
\erfc@@{\lambda} = \frac{1}{2\cpi\iunit}\int_{c-\iunit\infty}^{c+\iunit\infty}e^{\zeta-2\lambda\sqrt{\zeta}}\frac{\diff{\zeta}}{\zeta}		 c > 0 𝑐 0 {\displaystyle{\displaystyle c>0}} 
erfc(lambda) = (1)/(2*Pi*I)*int(exp(zeta - 2*lambda*sqrt(zeta))*(1)/(zeta), zeta = c - I*infinity..c + I*infinity)
 
Erfc[\[Lambda]] == Divide[1,2*Pi*I]*Integrate[Exp[\[Zeta]- 2*\[Lambda]*Sqrt[\[Zeta]]]*Divide[1,\[Zeta]], {\[Zeta], c - I*Infinity, c + I*Infinity}, GenerateConditions->None]
 Failure Aborted
Failed [30 / 30]
Result: .9788588170e-1-.2531649186*I
Test Values: {c = 1.5, lambda = 1/2*3^(1/2)+1/2*I}

Result: 1.977726380-.8570608782*I
Test Values: {c = 1.5, lambda = -1/2+1/2*I*3^(1/2)}

Result: .2227361984e-1+.8570608782*I
Test Values: {c = 1.5, lambda = 1/2-1/2*I*3^(1/2)}

Result: 1.902114118+.2531649186*I
Test Values: {c = 1.5, lambda = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
 Skipped - Because timed out
3.5.E44 erfc ⁡ λ = 1 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ e λ 2 ⁢ ( t - 2 ⁢ t ) ⁢ d t t complementary-error-function 𝜆 1 2 imaginary-unit superscript subscript 𝑐 imaginary-unit 𝑐 imaginary-unit superscript 𝑒 superscript 𝜆 2 𝑡 2 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\operatorname{erfc}\lambda=\frac{1}{2\pi\mathrm{i}% }\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}e^{\lambda^{2}(t-2\sqrt{t})}% \frac{\mathrm{d}t}{t}}}
\erfc@@{\lambda} = \frac{1}{2\cpi\iunit}\int_{c-\iunit\infty}^{c+\iunit\infty}e^{\lambda^{2}(t-2\sqrt{t})}\frac{\diff{t}}{t}		 c > 0 𝑐 0 {\displaystyle{\displaystyle c>0}} 
erfc(lambda) = (1)/(2*Pi*I)*int(exp((lambda)^(2)*(t - 2*sqrt(t)))*(1)/(t), t = c - I*infinity..c + I*infinity)
 
Erfc[\[Lambda]] == Divide[1,2*Pi*I]*Integrate[Exp[\[Lambda]^(2)*(t - 2*Sqrt[t])]*Divide[1,t], {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]
 Failure Aborted
Failed [30 / 30]
Result: .9788588170e-1-.2531649186*I
Test Values: {c = 1.5, lambda = 1/2*3^(1/2)+1/2*I}

Result: 1.977726380-.8570608782*I
Test Values: {c = 1.5, lambda = -1/2+1/2*I*3^(1/2)}

Result: .2227361984e-1+.8570608782*I
Test Values: {c = 1.5, lambda = 1/2-1/2*I*3^(1/2)}

Result: 1.902114118+.2531649186*I
Test Values: {c = 1.5, lambda = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
 Skipped - Because timed out
3.5.E45 erfc ⁡ λ = e - λ 2 2 ⁢ π ⁢ ∫ - π π e - λ 2 ⁢ tan 2 ⁡ ( 1 2 ⁢ θ ) ⁢ d θ complementary-error-function 𝜆 superscript 𝑒 superscript 𝜆 2 2 superscript subscript superscript 𝑒 superscript 𝜆 2 2 1 2 𝜃 𝜃 {\displaystyle{\displaystyle\operatorname{erfc}\lambda=\frac{e^{-\lambda^{2}}}% {2\pi}\int_{-\pi}^{\pi}e^{-\lambda^{2}{\tan^{2}}\left(\frac{1}{2}\theta\right)% }\mathrm{d}\theta}}
\erfc@@{\lambda} = \frac{e^{-\lambda^{2}}}{2\cpi}\int_{-\cpi}^{\cpi}e^{-\lambda^{2}\tan^{2}@{\frac{1}{2}\theta}}\diff{\theta}		 

erfc(lambda) = (exp(- (lambda)^(2)))/(2*Pi)*int(exp(- (lambda)^(2)* (tan((1)/(2)*theta))^(2)), theta = - Pi..Pi)
 
Erfc[\[Lambda]] == Divide[Exp[- \[Lambda]^(2)],2*Pi]*Integrate[Exp[- \[Lambda]^(2)* (Tan[Divide[1,2]*\[Theta]])^(2)], {\[Theta], - Pi, Pi}, GenerateConditions->None]
 Failure Failure
Failed [6 / 10]
Result: Float(infinity)+Float(infinity)*I
Test Values: {lambda = -1/2+1/2*I*3^(1/2)}

Result: Float(infinity)+Float(infinity)*I
Test Values: {lambda = 1/2-1/2*I*3^(1/2)}

Result: 1.804228236+.5063298371*I
Test Values: {lambda = -1/2*3^(1/2)-1/2*I}

Result: 1.932210292
Test Values: {lambda = -1.5}

... skip entries to safe data
Failed [5 / 10]
Result: Complex[1.9554527597185267, -1.7141217559576072]
Test Values: {Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[1.80422823640912, 0.5063298374329107]
Test Values: {Rule[λ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
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