Gamma Function - 5.19 Mathematical Applications

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.19#Ex1 S = βˆ‘ k = 0 ∞ a k 𝑆 superscript subscript π‘˜ 0 subscript π‘Ž π‘˜ {\displaystyle{\displaystyle S=\sum_{k=0}^{\infty}a_{k}}}
S = \sum_{k=0}^{\infty}a_{k}		 

S = sum(a[k], k = 0..infinity)
 
S == Sum[Subscript[a, k], {k, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
5.19#Ex2 a k = k ( 3 ⁒ k + 2 ) ⁒ ( 2 ⁒ k + 1 ) ⁒ ( k + 1 ) subscript π‘Ž π‘˜ π‘˜ 3 π‘˜ 2 2 π‘˜ 1 π‘˜ 1 {\displaystyle{\displaystyle a_{k}=\frac{k}{(3k+2)(2k+1)(k+1)}}}
a_{k} = \frac{k}{(3k+2)(2k+1)(k+1)}		 

a[k] = (k)/((3*k + 2)*(2*k + 1)*(k + 1))
 
Subscript[a, k] == Divide[k,(3*k + 2)*(2*k + 1)*(k + 1)]
 Skipped - no semantic math Skipped - no semantic math - -
5.19.E2 a k = 2 k + 2 3 - 1 k + 1 2 - 1 k + 1 subscript π‘Ž π‘˜ 2 π‘˜ 2 3 1 π‘˜ 1 2 1 π‘˜ 1 {\displaystyle{\displaystyle a_{k}=\frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}% {2}}-\frac{1}{k+1}}}
a_{k} = \frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}		 

a[k] = (2)/(k +(2)/(3))-(1)/(k +(1)/(2))-(1)/(k + 1)
 
Subscript[a, k] == Divide[2,k +Divide[2,3]]-Divide[1,k +Divide[1,2]]-Divide[1,k + 1]
 Skipped - no semantic math Skipped - no semantic math - -
5.19.E3 S = ψ ⁑ ( 1 2 ) - 2 ⁒ ψ ⁑ ( 2 3 ) - Ξ³ 𝑆 digamma 1 2 2 digamma 2 3 {\displaystyle{\displaystyle S=\psi\left(\tfrac{1}{2}\right)-2\psi\left(\tfrac% {2}{3}\right)-\gamma}}
S = \digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant		 

S = Psi((1)/(2))- 2*Psi((2)/(3))- gamma

S == PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma
Failure Failure
Failed [10 / 10]
Result: .7702822630+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I}


Result: -.5957431410+.8660254040*I
Test Values: {S = -1/2+1/2*I*3^(1/2)}


Result: .4042568590-.8660254040*I
Test Values: {S = 1/2-1/2*I*3^(1/2)}


Result: -.9617685450-.5000000000*I
Test Values: {S = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[0.7702822631342183, 0.49999999999999994]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.5957431406502202, 0.8660254037844387]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
5.19.E3 ψ ⁑ ( 1 2 ) - 2 ⁒ ψ ⁑ ( 2 3 ) - Ξ³ = 3 ⁒ ln ⁑ 3 - 2 ⁒ ln ⁑ 2 - 1 3 ⁒ Ο€ ⁒ 3 digamma 1 2 2 digamma 2 3 3 3 2 2 1 3 πœ‹ 3 {\displaystyle{\displaystyle\psi\left(\tfrac{1}{2}\right)-2\psi\left(\tfrac{2}% {3}\right)-\gamma=3\ln 3-2\ln 2-\tfrac{1}{3}\pi\sqrt{3}}}
\digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant = 3\ln@@{3}-2\ln@@{2}-\tfrac{1}{3}\pi\sqrt{3}		 

Psi((1)/(2))- 2*Psi((2)/(3))- gamma = 3*ln(3)- 2*ln(2)-(1)/(3)*Pi*sqrt(3)

PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma == 3*Log[3]- 2*Log[2]-Divide[1,3]*Pi*Sqrt[3]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 1]
5.19#Ex3 V = Ο€ 1 2 ⁒ n ⁒ r n Ξ“ ⁑ ( 1 2 ⁒ n + 1 ) 𝑉 superscript πœ‹ 1 2 𝑛 superscript π‘Ÿ 𝑛 Euler-Gamma 1 2 𝑛 1 {\displaystyle{\displaystyle V=\frac{\pi^{\frac{1}{2}n}r^{n}}{\Gamma\left(% \frac{1}{2}n+1\right)}}}
V = \frac{\pi^{\frac{1}{2}n}r^{n}}{\EulerGamma@{\frac{1}{2}n+1}}		 β„œ ⁑ ( 1 2 ⁒ n + 1 ) > 0 1 2 𝑛 1 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}n+1)>0}} 
V = ((Pi)^((1)/(2)*n)* (r)^(n))/(GAMMA((1)/(2)*n + 1))

V == Divide[(Pi)^(Divide[1,2]*n)* (r)^(n),Gamma[Divide[1,2]*n + 1]]
Failure Failure
Failed [180 / 180]
Result: 3.866025403+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 1}


Result: -6.202558068+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 2}


Result: 15.00319234+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 3}


Result: -2.133974595+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = 1.5, n = 1}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[3.866025403784439, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-6.202558066792596, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
5.19#Ex4 S = 2 ⁒ Ο€ 1 2 ⁒ n ⁒ r n - 1 Ξ“ ⁑ ( 1 2 ⁒ n ) 𝑆 2 superscript πœ‹ 1 2 𝑛 superscript π‘Ÿ 𝑛 1 Euler-Gamma 1 2 𝑛 {\displaystyle{\displaystyle S=\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\Gamma\left(% \frac{1}{2}n\right)}}}
S = \frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}}		 β„œ ⁑ ( 1 2 ⁒ n ) > 0 1 2 𝑛 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}n)>0}} 
S = (2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))

S == Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]
Failure Failure
Failed [174 / 180]
Result: -1.133974596+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 1}


Result: 10.29080337+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 2}


Result: -27.40830850+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 3}


Result: -1.133974596+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = 1.5, n = 1}

... skip entries to safe data
Failed [174 / 180]
Result: Complex[-1.1339745962155612, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[r, -1.5], Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[10.290803364553819, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[r, -1.5], Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
5.19#Ex4 2 ⁒ Ο€ 1 2 ⁒ n ⁒ r n - 1 Ξ“ ⁑ ( 1 2 ⁒ n ) = n r ⁒ V 2 superscript πœ‹ 1 2 𝑛 superscript π‘Ÿ 𝑛 1 Euler-Gamma 1 2 𝑛 𝑛 π‘Ÿ 𝑉 {\displaystyle{\displaystyle\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\Gamma\left(% \frac{1}{2}n\right)}=\frac{n}{r}V}}
\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}} = \frac{n}{r}V		 β„œ ⁑ ( 1 2 ⁒ n ) > 0 1 2 𝑛 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}n)>0}} 
(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n)) = (n)/(r)*V

Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]] == Divide[n,r]*V
Failure Failure
Failed [180 / 180]
Result: 2.577350269+.3333333334*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 1}


Result: -8.270077424+.6666666665*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 2}


Result: 30.00638471+1.000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 3}


Result: 1.422649731-.3333333334*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = 1.5, n = 1}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[2.5773502691896253, 0.33333333333333326]
Test Values: {Rule[n, 1], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-8.270077422390127, 0.6666666666666665]
Test Values: {Rule[n, 2], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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