8.11.E2
Ξ
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incomplete-Gamma
π
π§
superscript
π§
π
1
superscript
π
π§
superscript
subscript
π
0
π
1
subscript
π’
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superscript
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{\displaystyle{\displaystyle\Gamma\left(a,z\right)=z^{a-1}e^{-z}\left(\sum_{k=%
0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)}}
\incGamma @{ a}{ z} = z^{ a-1} e^{ -z} \left (\sum _{ k=0}^{ n-1} \frac { u_{ k}}{ z^{ k}} +R_{ n} (a,z)\right )GAMMA ( a , z ) = ( z ) ^ ( a - 1 ) * exp ( - z ) * ( sum ((( - 1 ) ^ ( k ) * pochhammer ( 1 - a , k )) / (( z ) ^ ( k )), k = 0. . n - 1 ) + R [ n ]( a , z ))
Gamma [ a , z ] == ( z ) ^ ( a - 1 ) * Exp [ - z ] * ( Sum [ Divide [( - 1 ) ^ ( k ) * Pochhammer [ 1 - a , k ],( z ) ^ ( k )], { k , 0 , n - 1 }, GenerateConditions -> None ] + Subscript [ R , n ][ a , z ])
Failure
Failure
Expand Failed [300 / 300]
Result : .1072320848 e -1 -.1480251451 * I + ( .9924715785 e -1 + .4087434498 * I ) * ( 1.000000000 + 0. * I + ( .8660254040 + .5000000000 * I ) * ( -1.5 , .8660254040 + .5000000000 * I ))
Test Values : { a = -1.5 , z = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , R [ n ] = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , n = 1 , n = 3 }
Result : .1072320848 e -1 -.1480251451 * I + ( .9924715785 e -1 + .4087434498 * I ) * ( -1.165063509 + 1.250000000 * I + ( .8660254040 + .5000000000 * I ) * ( -1.5 , .8660254040 + .5000000000 * I ))
Test Values : { a = -1.5 , z = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , R [ n ] = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , n = 2 , n = 3 }
... skip entries to safe data
Error
8.11.E4
Ξ³
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incomplete-gamma
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π§
superscript
π§
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superscript
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superscript
subscript
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superscript
π§
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Pochhammer
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1
{\displaystyle{\displaystyle\gamma\left(a,z\right)=z^{a}e^{-z}\sum_{k=0}^{%
\infty}\frac{z^{k}}{{\left(a\right)_{k+1}}}}}
\incgamma @{ a}{ z} = z^{ a} e^{ -z} \sum _{ k=0}^{ \infty } \frac { z^{ k}}{ \Pochhammersym { a}{ k+1}}
β
β‘
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0
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0
{\displaystyle{\displaystyle\Re a>0}}
GAMMA ( a ) - GAMMA ( a , z ) = ( z ) ^ ( a ) * exp ( - z ) * sum ((( z ) ^ ( k )) / ( pochhammer ( a , k + 1 )), k = 0. . infinity )
Gamma [ a , 0 , z ] == ( z ) ^ ( a ) * Exp [ - z ] * Sum [ Divide [( z ) ^ ( k ), Pochhammer [ a , k + 1 ]], { k , 0 , Infinity }, GenerateConditions -> None ]
Successful
Successful
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Successful [Tested: 7]
8.11#Ex1
b
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subscript
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1
{\displaystyle{\displaystyle b_{0}(\lambda)=1}}
b_{ 0} (\lambda ) = 1Subscript[b, 0][\[Lambda]] == 1 Skipped - no semantic math
Skipped - no semantic math
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8.11#Ex2
b
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subscript
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π
{\displaystyle{\displaystyle b_{1}(\lambda)=\lambda}}
b_{ 1} (\lambda ) = \lambda Subscript[b, 1][\[Lambda]] == \[Lambda] Skipped - no semantic math
Skipped - no semantic math
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8.11#Ex3
b
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subscript
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2
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2
π
1
{\displaystyle{\displaystyle b_{2}(\lambda)=\lambda(2\lambda+1)}}
b_{ 2} (\lambda ) = \lambda (2\lambda +1)b[2](lambda) = lambda*(2*lambda + 1) Subscript[b, 2][\[Lambda]] == \[Lambda]*(2*\[Lambda]+ 1) Skipped - no semantic math
Skipped - no semantic math
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8.11.E15
S
n
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subscript
π
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π₯
incomplete-gamma
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1
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π₯
superscript
π
π₯
π
superscript
π
π
π₯
{\displaystyle{\displaystyle S_{n}(x)=\frac{\gamma\left(n+1,nx\right)}{(nx)^{n%
}e^{-nx}}}}
S_{ n} (x) = \frac { \incgamma @{ n+1}{ nx}}{ (nx)^{ n} e^{ -nx}}
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{\displaystyle{\displaystyle\Re(n+1)>0}}
S [ n ]( x ) = ( GAMMA ( n + 1 ) - GAMMA ( n + 1 , n * x )) / (( n * x ) ^ ( n ) * exp ( - n * x ))
Subscript [ S , n ][ x ] == Divide [ Gamma [ n + 1 , 0 , n * x ],( n * x ) ^ ( n ) * Exp [ - n * x ]]
Failure
Failure
Expand Failed [90 / 90]
Result : -.22087941 e -1 + .7500000000 * I
Test Values : { x = 1.5 , S [ n ] = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , n = 1 }
Result : -1.275525655 + .7500000000 * I
Test Values : { x = 1.5 , S [ n ] = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , n = 2 }
... skip entries to safe data
Expand Failed [90 / 90]
Result : Complex [ -0.02208794121538471 , 0.7499999999999999 ]
Test Values : { Rule [ n , 1 ], Rule [ x , 1.5 ], Rule [ Subscript [ S , n ], Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]]}
Result : Complex [ -1.2755256550317124 , 0.7499999999999999 ]
Test Values : { Rule [ n , 2 ], Rule [ x , 1.5 ], Rule [ Subscript [ S , n ], Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]]}
... skip entries to safe data
8.11.E19
d
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subscript
π
π
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superscript
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subscript
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superscript
1
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2
π
1
{\displaystyle{\displaystyle d_{k}(x)=\frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}}}
d_{ k} (x) = \frac { (-1)^{ k} b_{ k} (x)}{ (1-x)^{ 2k+1}} d[k](x) = ((- 1)^(k)* b[k](x))/((1 - x)^(2*k + 1)) Subscript[d, k][x] == Divide[(- 1)^(k)* Subscript[b, k][x],(1 - x)^(2*k + 1)] Skipped - no semantic math
Skipped - no semantic math
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