Incomplete Gamma and Related Functions - 8.14 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
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8.14.E1 ∫ 0 ∞ e - a ⁒ x ⁒ Ξ³ ⁑ ( b , x ) Ξ“ ⁑ ( b ) ⁒ d x = ( 1 + a ) - b a superscript subscript 0 superscript 𝑒 π‘Ž π‘₯ incomplete-gamma 𝑏 π‘₯ Euler-Gamma 𝑏 π‘₯ superscript 1 π‘Ž 𝑏 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x% \right)}{\Gamma\left(b\right)}\mathrm{d}x=\frac{(1+a)^{-b}}{a}}}
\int_{0}^{\infty}e^{-ax}\frac{\incgamma@{b}{x}}{\EulerGamma@{b}}\diff{x} = \frac{(1+a)^{-b}}{a}		 β„œ ⁑ a > 0 , β„œ ⁑ b > - 1 , β„œ ⁑ b > 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 1 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re b>-1,\Re b>0}} 
int(exp(- a*x)*(GAMMA(b)-GAMMA(b, x))/(GAMMA(b)), x = 0..infinity) = ((1 + a)^(- b))/(a)

Integrate[Exp[- a*x]*Divide[Gamma[b, 0, x],Gamma[b]], {x, 0, Infinity}, GenerateConditions->None] == Divide[(1 + a)^(- b),a]
Successful Aborted - Skipped - Because timed out
8.14.E2 ∫ 0 ∞ e - a ⁒ x ⁒ Ξ“ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( b ) ⁒ 1 - ( 1 + a ) - b a superscript subscript 0 superscript 𝑒 π‘Ž π‘₯ incomplete-Gamma 𝑏 π‘₯ π‘₯ Euler-Gamma 𝑏 1 superscript 1 π‘Ž 𝑏 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\Gamma\left(b,x\right)% \mathrm{d}x=\Gamma\left(b\right)\frac{1-(1+a)^{-b}}{a}}}
\int_{0}^{\infty}e^{-ax}\incGamma@{b}{x}\diff{x} = \EulerGamma@{b}\frac{1-(1+a)^{-b}}{a}		 β„œ ⁑ a > - 1 , β„œ ⁑ b > - 1 , β„œ ⁑ b > 0 formulae-sequence π‘Ž 1 formulae-sequence 𝑏 1 𝑏 0 {\displaystyle{\displaystyle\Re a>-1,\Re b>-1,\Re b>0}} 
int(exp(- a*x)*GAMMA(b, x), x = 0..infinity) = GAMMA(b)*(1 -(1 + a)^(- b))/(a)

Integrate[Exp[- a*x]*Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Gamma[b]*Divide[1 -(1 + a)^(- b),a]
Failure Aborted Successful [Tested: 12] Skipped - Because timed out
8.14.E3 ∫ 0 ∞ x a - 1 ⁒ Ξ³ ⁑ ( b , x ) ⁒ d x = - Ξ“ ⁑ ( a + b ) a superscript subscript 0 superscript π‘₯ π‘Ž 1 incomplete-gamma 𝑏 π‘₯ π‘₯ Euler-Gamma π‘Ž 𝑏 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}\gamma\left(b,x\right)% \mathrm{d}x=-\frac{\Gamma\left(a+b\right)}{a}}}
\int_{0}^{\infty}x^{a-1}\incgamma@{b}{x}\diff{x} = -\frac{\EulerGamma@{a+b}}{a}		 β„œ ⁑ a < 0 , β„œ ⁑ ( a + b ) > 0 , β„œ ⁑ ( a + b ) > 0 , β„œ ⁑ b > 0 formulae-sequence π‘Ž 0 formulae-sequence π‘Ž 𝑏 0 formulae-sequence π‘Ž 𝑏 0 𝑏 0 {\displaystyle{\displaystyle\Re a<0,\Re\left(a+b\right)>0,\Re(a+b)>0,\Re b>0}} 
int((x)^(a - 1)* GAMMA(b)-GAMMA(b, x), x = 0..infinity) = -(GAMMA(a + b))/(a)

Integrate[(x)^(a - 1)* Gamma[b, 0, x], {x, 0, Infinity}, GenerateConditions->None] == -Divide[Gamma[a + b],a]
Failure Aborted
Failed [3 / 3]
Result: Float(infinity)
Test Values: {a = -1.5, b = 2}


Result: Float(infinity)
Test Values: {a = -.5, b = 1.5}

... skip entries to safe data
 Skip - No test values generated
8.14.E4 ∫ 0 ∞ x a - 1 ⁒ Ξ“ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( a + b ) a superscript subscript 0 superscript π‘₯ π‘Ž 1 incomplete-Gamma 𝑏 π‘₯ π‘₯ Euler-Gamma π‘Ž 𝑏 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)% \mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a}}}
\int_{0}^{\infty}x^{a-1}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a}		 β„œ ⁑ a > 0 , β„œ ⁑ ( a + b ) > 0 , β„œ ⁑ ( a + b ) > 0 formulae-sequence π‘Ž 0 formulae-sequence π‘Ž 𝑏 0 π‘Ž 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re\left(a+b\right)>0,\Re(a+b)>0}} 
int((x)^(a - 1)* GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(a)

Integrate[(x)^(a - 1)* Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],a]
Successful Successful - Successful [Tested: 12]
8.14.E5 ∫ 0 ∞ x a - 1 ⁒ e - s ⁒ x ⁒ Ξ³ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( a + b ) b ⁒ ( 1 + s ) a + b ⁒ F ⁑ ( 1 , a + b ; 1 + b ; 1 / ( 1 + s ) ) superscript subscript 0 superscript π‘₯ π‘Ž 1 superscript 𝑒 𝑠 π‘₯ incomplete-gamma 𝑏 π‘₯ π‘₯ Euler-Gamma π‘Ž 𝑏 𝑏 superscript 1 𝑠 π‘Ž 𝑏 Gauss-hypergeometric-F 1 π‘Ž 𝑏 1 𝑏 1 1 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\gamma\left(b,x% \right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{b(1+s)^{a+b}}\*F\left(1,a+b;1% +b;1/(1+s)\right)}}
\int_{0}^{\infty}x^{a-1}e^{-sx}\incgamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{b(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+b}{1/(1+s)}		 β„œ ⁑ s > 0 , β„œ ⁑ ( a + b ) > 0 , β„œ ⁑ ( a + b ) > 0 , β„œ ⁑ b > 0 formulae-sequence 𝑠 0 formulae-sequence π‘Ž 𝑏 0 formulae-sequence π‘Ž 𝑏 0 𝑏 0 {\displaystyle{\displaystyle\Re s>0,\Re\left(a+b\right)>0,\Re(a+b)>0,\Re b>0}} 
int((x)^(a - 1)* exp(- s*x)*GAMMA(b)-GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(b*(1 + s)^(a + b))* hypergeom([1, a + b], [1 + b], 1/(1 + s))

Integrate[(x)^(a - 1)* Exp[- s*x]*Gamma[b, 0, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],b*(1 + s)^(a + b)]* Hypergeometric2F1[1, a + b, 1 + b, 1/(1 + s)]
Failure Aborted
Failed [36 / 36]
Result: Float(infinity)
Test Values: {a = -1.5, b = 2, s = 1.5}


Result: Float(infinity)
Test Values: {a = -1.5, b = 2, s = .5}

... skip entries to safe data
 Skipped - Because timed out
8.14.E6 ∫ 0 ∞ x a - 1 ⁒ e - s ⁒ x ⁒ Ξ“ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( a + b ) a ⁒ ( 1 + s ) a + b ⁒ F ⁑ ( 1 , a + b ; 1 + a ; s / ( 1 + s ) ) superscript subscript 0 superscript π‘₯ π‘Ž 1 superscript 𝑒 𝑠 π‘₯ incomplete-Gamma 𝑏 π‘₯ π‘₯ Euler-Gamma π‘Ž 𝑏 π‘Ž superscript 1 𝑠 π‘Ž 𝑏 Gauss-hypergeometric-F 1 π‘Ž 𝑏 1 π‘Ž 𝑠 1 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x% \right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1% +a;s/(1+s)\right)}}
\int_{0}^{\infty}x^{a-1}e^{-sx}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+a}{s/(1+s)}		 β„œ ⁑ s > - 1 , β„œ ⁑ ( a + b ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( a + b ) > 0 formulae-sequence 𝑠 1 formulae-sequence π‘Ž 𝑏 0 formulae-sequence π‘Ž 0 π‘Ž 𝑏 0 {\displaystyle{\displaystyle\Re s>-1,\Re\left(a+b\right)>0,\Re a>0,\Re(a+b)>0}} 
int((x)^(a - 1)* exp(- s*x)*GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(a*(1 + s)^(a + b))* hypergeom([1, a + b], [1 + a], s/(1 + s))

Integrate[(x)^(a - 1)* Exp[- s*x]*Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],a*(1 + s)^(a + b)]* Hypergeometric2F1[1, a + b, 1 + a, s/(1 + s)]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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