Gamma Function - 5.18 -Gamma and -Beta Functions
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 5.18.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qPochhammer{a}{q}{n} = \prod_{k=0}^{n-1}(1-aq^{k})}
\qPochhammer{a}{q}{n} = \prod_{k=0}^{n-1}(1-aq^{k}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QPochhammer(a, q, n) = product(1 - a*(q)^(k), k = 0..n - 1)
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QPochhammer[a, q, n] == Product[1 - a*(q)^(k), {k, 0, n - 1}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 60] |
| 5.18.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qPochhammer{a}{q}{\infty} = \prod_{k=0}^{\infty}(1-aq^{k})}
\qPochhammer{a}{q}{\infty} = \prod_{k=0}^{\infty}(1-aq^{k}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QPochhammer(a, q, infinity) = product(1 - a*(q)^(k), k = 0..infinity)
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QPochhammer[a, q, Infinity] == Product[1 - a*(q)^(k), {k, 0, Infinity}, GenerateConditions->None]
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Failure | Failure | Error | Failed [48 / 60]
Result: Plus[Times[-1.0, QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994]]], QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 5.18.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qGamma{q}@{z} = \qPochhammer{q}{q}{\infty}(1-q)^{1-z}/\qPochhammer{q^{z}}{q}{\infty}}
\qGamma{q}@{z} = \qPochhammer{q}{q}{\infty}(1-q)^{1-z}/\qPochhammer{q^{z}}{q}{\infty} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QGAMMA(q, z) = QPochhammer(q, q, infinity)*(1 - q)^(1 - z)/QPochhammer((q)^(z), q, infinity)
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QGamma[z,q] == QPochhammer[q, q, Infinity]*(1 - q)^(1 - z)/QPochhammer[(q)^(z), q, Infinity]
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Error | Failure | - | Failed [56 / 70]
Result: Plus[QGamma[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Times[Complex[-0.4701974928403924, -0.07292434984262404], Power[QPochhammer[Complex[0.6918839380246471, 0.3371668184918191], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[QGamma[Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], Times[Complex[-0.021172596861766507, 0.11798586945608598], Power[QPochhammer[Complex[0.6137803977754971, -0.16446196191399762], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 5.18.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qGamma{q}@{1} = \qGamma{q}@{2}}
\qGamma{q}@{1} = \qGamma{q}@{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QGAMMA(q, 1) = QGAMMA(q, 2)
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QGamma[1,q] == QGamma[2,q]
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Error | Successful | - | Successful [Tested: 10] |
| 5.18.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qGamma{q}@{2} = 1}
\qGamma{q}@{2} = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QGAMMA(q, 2) = 1
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QGamma[2,q] == 1
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Error | Successful | - | Successful [Tested: 10] |
| 5.18.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qfactorial{n}{q} = \qGamma{q}@{n+1}}
\qfactorial{n}{q} = \qGamma{q}@{n+1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QFactorial(n, q) = QGAMMA(q, n + 1)
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QFactorial[n,q] == QGamma[n + 1,q]
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Error | Successful | - | Successful [Tested: 30] |
| 5.18.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qGamma{q}@{z+1} = \frac{1-q^{z}}{1-q}\qGamma{q}@{z}}
\qGamma{q}@{z+1} = \frac{1-q^{z}}{1-q}\qGamma{q}@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QGAMMA(q, z + 1) = (1 - (q)^(z))/(1 - q)*QGAMMA(q, z)
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QGamma[z + 1,q] == Divide[1 - (q)^(z),1 - q]*QGamma[z,q]
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Error | Failure | - | Failed [17 / 70]
Result: Indeterminate
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Indeterminate
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 5.18.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qGamma{q}@{x} < \qGamma{r}@{x}}
\qGamma{q}@{x} < \qGamma{r}@{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QGAMMA(q, x) < QGAMMA(r, x)
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QGamma[x,q] < QGamma[x,r]
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Failure | Failure | Error | Failed [174 / 180]
Result: Less[QGamma[1.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[0.4591522571856908, -0.3749002120921232]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 1.5]}
Result: Less[QGamma[0.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[4.854857756142472*^-6, 0.9372445842681697]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 0.5]}
... skip entries to safe data |
| 5.18.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \qGamma{q}@{x} > \qGamma{r}@{x}}
\qGamma{q}@{x} > \qGamma{r}@{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | QGAMMA(q, x) > QGAMMA(r, x)
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QGamma[x,q] > QGamma[x,r]
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Failure | Failure | Error | Failed [174 / 180]
Result: Greater[QGamma[1.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[0.4591522571856908, -0.3749002120921232]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 1.5]}
Result: Greater[QGamma[0.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[4.854857756142472*^-6, 0.9372445842681697]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 0.5]}
... skip entries to safe data |
| 5.18.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{q\to 1-}\qGamma{q}@{z} = \EulerGamma@{z}}
\lim_{q\to 1-}\qGamma{q}@{z} = \EulerGamma@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0} | limit(QGAMMA(q, z), q = 1, left) = GAMMA(z)
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Limit[QGamma[z,q], q -> 1, Direction -> "FromBelow", GenerateConditions->None] == Gamma[z]
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Failure | Aborted | Error | Skipped - Because timed out |