Gamma Function - 5.11 Asymptotic Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.11#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{0} = 1}
g_{0} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[0] = 1
 
Subscript[g, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{1} = \tfrac{1}{12}}
g_{1} = \tfrac{1}{12}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[1] = (1)/(12)
 
Subscript[g, 1] == Divide[1,12]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{2} = \tfrac{1}{288}}
g_{2} = \tfrac{1}{288}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[2] = (1)/(288)
 
Subscript[g, 2] == Divide[1,288]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{3} = -\tfrac{139}{51840}}
g_{3} = -\tfrac{139}{51840}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[3] = -(139)/(51840)
 
Subscript[g, 3] == -Divide[139,51840]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{4} = -\tfrac{571}{24\;88320}}
g_{4} = -\tfrac{571}{24\;88320}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[4] = -(571)/(2488320)
 
Subscript[g, 4] == -Divide[571,2488320]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{5} = \tfrac{1\;63879}{2090\;18880}}
g_{5} = \tfrac{1\;63879}{2090\;18880}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[5] = (163879)/(209018880)
 
Subscript[g, 5] == Divide[163879,209018880]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{6} = \tfrac{52\;46819}{7\;52467\;96800}}
g_{6} = \tfrac{52\;46819}{7\;52467\;96800}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[6] = (5246819)/(75246796800)
 
Subscript[g, 6] == Divide[5246819,75246796800]
 Skipped - no semantic math Skipped - no semantic math - -
5.11.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}}
g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
g[k] = sqrt(2)*pochhammer((1)/(2), k)*a[2*k]

Subscript[g, k] == Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k]
Failure Failure
Failed [300 / 300]
Result: .2536529683+.1464466095*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.5253324962e-1-.303300858e-1*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 2}


Result: -1.430371229-.8258252140*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 3}


Result: -1.112372436+.5124720135*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), k = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.25365296808864424, 0.14644660940672627]
Test Values: {Rule[k, 1], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.05253324975925311, -0.03033008588991054]
Test Values: {Rule[k, 2], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
5.11.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)}
\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0} 
GAMMA(z) = exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K](z))

Gamma[z] == Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}, GenerateConditions->None]+ Subscript[R, K][z])
Failure Failure
Failed [300 / 300]
Result: -1.892613380-.1706947928*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, K = 3}


Result: -.4529896033-2.955992714*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), K = 3}


Result: .8926845412+1.268928985*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2-1/2*I*3^(1/2), K = 3}


Result: 2.332308320-1.516368937*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2*3^(1/2)-1/2*I, K = 3}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.8926133813331316, -0.17069479199840365]
Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7462398809799414, -0.22409723911500246]
Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
5.11#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{0}(a,b) = 1}
G_{0}(a,b) = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
G[0](a , b) = 1
 
Subscript[G, 0][a , b] == 1
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)}
G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
G[1](a , b) = (1)/(2)*(a - b)*(a + b - 1)
 
Subscript[G, 1][a , b] == Divide[1,2]*(a - b)*(a + b - 1)
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))}
G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
G[2](a , b) = (1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1))

Subscript[G, 2][a , b] == Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1))
Failure Failure Error Error
5.11#Ex11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle H_{0}(a,b) = 1}
H_{0}(a,b) = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
H[0](a , b) = 1
 
Subscript[H, 0][a , b] == 1
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)}
H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
H[1](a , b) = -(1)/(12)*binomial(a - b,2)*(a - b + 1)

Subscript[H, 1][a , b] == -Divide[1,12]*Binomial[a - b,2]*(a - b + 1)
Failure Failure Error Error
5.11#Ex13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})}
H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
H[2](a , b) = (1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2))

Subscript[H, 2][a , b] == Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2))
Failure Failure Error Error
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