Confluent Hypergeometric Functions - 13.10 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.10.E1 ∫ 𝐌 ⁑ ( a , b , z ) ⁒ d z = 1 a - 1 ⁒ 𝐌 ⁑ ( a - 1 , b - 1 , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 𝑧 1 π‘Ž 1 Kummer-confluent-hypergeometric-bold-M π‘Ž 1 𝑏 1 𝑧 {\displaystyle{\displaystyle\int{\mathbf{M}}\left(a,b,z\right)\mathrm{d}z=% \frac{1}{a-1}{\mathbf{M}}\left(a-1,b-1,z\right)}}
\int\OlverconfhyperM@{a}{b}{z}\diff{z} = \frac{1}{a-1}\OlverconfhyperM@{a-1}{b-1}{z}		 β„œ ⁑ ( b + s ) > 0 , β„œ ⁑ ( ( b - 1 ) + s ) > 0 formulae-sequence 𝑏 𝑠 0 𝑏 1 𝑠 0 {\displaystyle{\displaystyle\Re(b+s)>0,\Re((b-1)+s)>0}} 
int(KummerM(a, b, z)/GAMMA(b), z) = (1)/(a - 1)*KummerM(a - 1, b - 1, z)/GAMMA(b - 1)

Integrate[Hypergeometric1F1Regularized[a, b, z], z, GenerateConditions->None] == Divide[1,a - 1]*Hypergeometric1F1Regularized[a - 1, b - 1, z]
Successful Failure -
Failed [252 / 252]
Result: Complex[-0.4231421876608173, 0.0]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.42314218766081735, 0.0]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
13.10.E2 ∫ U ⁑ ( a , b , z ) ⁒ d z = - 1 a - 1 ⁒ U ⁑ ( a - 1 , b - 1 , z ) Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 𝑧 1 π‘Ž 1 Kummer-confluent-hypergeometric-U π‘Ž 1 𝑏 1 𝑧 {\displaystyle{\displaystyle\int U\left(a,b,z\right)\mathrm{d}z=-\frac{1}{a-1}% U\left(a-1,b-1,z\right)}}
\int\KummerconfhyperU@{a}{b}{z}\diff{z} = -\frac{1}{a-1}\KummerconfhyperU@{a-1}{b-1}{z}		 

int(KummerU(a, b, z), z) = -(1)/(a - 1)*KummerU(a - 1, b - 1, z)

Integrate[HypergeometricU[a, b, z], z, GenerateConditions->None] == -Divide[1,a - 1]*HypergeometricU[a - 1, b - 1, z]
Successful Successful - Successful [Tested: 252]
13.10.E3 ∫ 0 ∞ e - z ⁒ t ⁒ t b - 1 ⁒ 𝐌 ⁑ ( a , c , k ⁒ t ) ⁒ d t = Ξ“ ⁑ ( b ) ⁒ z - b ⁒ 𝐅 1 2 ⁑ ( a , b ; c ; k / z ) superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑏 1 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑐 π‘˜ 𝑑 𝑑 Euler-Gamma 𝑏 superscript 𝑧 𝑏 hypergeometric-bold-pFq 2 1 π‘Ž 𝑏 𝑐 π‘˜ 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a% ,c,kt\right)\mathrm{d}t=\Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}% \left(a,b;c;\ifrac{k}{z}\right)}}
\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{c}{kt}\diff{t} = \EulerGamma@{b}z^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{k}{z}}		 β„œ ⁑ b > 0 , β„œ ⁑ z > max ( β„œ ⁑ k , β„œ ⁑ ( c + s ) > 0 fragments 𝑏 0 , 𝑧 fragments ( π‘˜ , 𝑐 𝑠 0 {\displaystyle{\displaystyle\Re b>0,\Re z>\max\left(\Re k,\Re(c+s)>0}\)\@add@PDF@RDFa@triples\end{document}} 
int(exp(- z*t)*(t)^(b - 1)* KummerM(a, c, k*t)/GAMMA(c), t = 0..infinity) = GAMMA(b)*(z)^(- b)* hypergeom([a , b], [c], (k)/(z))

Integrate[Exp[- z*t]*(t)^(b - 1)* Hypergeometric1F1Regularized[a, c, k*t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[b]*(z)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[k,z]]
Failure Aborted
Failed [300 / 300]
Result: Float(undefined)+Float(undefined)*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: Float(undefined)+Float(undefined)*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
 Skipped - Because timed out
13.10.E4 ∫ 0 ∞ e - z ⁒ t ⁒ t b - 1 ⁒ 𝐌 ⁑ ( a , b , t ) ⁒ d t = z - b ⁒ ( 1 - 1 z ) - a superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑏 1 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑑 𝑑 superscript 𝑧 𝑏 superscript 1 1 𝑧 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a% ,b,t\right)\mathrm{d}t=z^{-b}\left(1-\frac{1}{z}\right)^{-a}}}
\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{b}{t}\diff{t} = z^{-b}\left(1-\frac{1}{z}\right)^{-a}		 β„œ ⁑ b > 0 , β„œ ⁑ z > 1 , β„œ ⁑ ( b + s ) > 0 formulae-sequence 𝑏 0 formulae-sequence 𝑧 1 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re b>0,\Re z>1,\Re(b+s)>0}} 
int(exp(- z*t)*(t)^(b - 1)* KummerM(a, b, t)/GAMMA(b), t = 0..infinity) = (z)^(- b)*(1 -(1)/(z))^(- a)

Integrate[Exp[- z*t]*(t)^(b - 1)* Hypergeometric1F1Regularized[a, b, t], {t, 0, Infinity}, GenerateConditions->None] == (z)^(- b)*(1 -Divide[1,z])^(- a)
Failure Aborted
Failed [24 / 36]
Result: -.2095131204
Test Values: {a = -3/2, b = 3/2, z = 3/2}


Result: -.2500000000
Test Values: {a = -3/2, b = 3/2, z = 2}

... skip entries to safe data
 Skipped - Because timed out
13.10.E5 ∫ 0 ∞ e - t ⁒ t b - 1 ⁒ 𝐌 ⁑ ( a , c , t ) ⁒ d t = Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c - a - b ) Ξ“ ⁑ ( c - a ) ⁒ Ξ“ ⁑ ( c - b ) superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 𝑏 1 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑐 𝑑 𝑑 Euler-Gamma 𝑏 Euler-Gamma 𝑐 π‘Ž 𝑏 Euler-Gamma 𝑐 π‘Ž Euler-Gamma 𝑐 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{b-1}{\mathbf{M}}\left(a,% c,t\right)\mathrm{d}t=\frac{\Gamma\left(b\right)\Gamma\left(c-a-b\right)}{% \Gamma\left(c-a\right)\Gamma\left(c-b\right)}}}
\int_{0}^{\infty}e^{-t}t^{b-1}\OlverconfhyperM@{a}{c}{t}\diff{t} = \frac{\EulerGamma@{b}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}		 β„œ ⁑ ( c - a ) > β„œ ⁑ b , β„œ ⁑ b > 0 , β„œ ⁑ ( c - a - b ) > 0 , β„œ ⁑ ( c - a ) > 0 , β„œ ⁑ ( c - b ) > 0 , β„œ ⁑ ( c + s ) > 0 formulae-sequence 𝑐 π‘Ž 𝑏 formulae-sequence 𝑏 0 formulae-sequence 𝑐 π‘Ž 𝑏 0 formulae-sequence 𝑐 π‘Ž 0 formulae-sequence 𝑐 𝑏 0 𝑐 𝑠 0 {\displaystyle{\displaystyle\Re\left(c-a\right)>\Re b,\Re b>0,\Re(c-a-b)>0,\Re% (c-a)>0,\Re(c-b)>0,\Re(c+s)>0}} 
int(exp(- t)*(t)^(b - 1)* KummerM(a, c, t)/GAMMA(c), t = 0..infinity) = (GAMMA(b)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))

Integrate[Exp[- t]*(t)^(b - 1)* Hypergeometric1F1Regularized[a, c, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[b]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]
Successful Aborted - Skipped - Because timed out
13.10.E6 ∫ 0 ∞ e - z ⁒ t - t 2 ⁒ t 2 ⁒ b - 2 ⁒ 𝐌 ⁑ ( a , b , t 2 ) ⁒ d t = 1 2 ⁒ Ο€ - 1 2 ⁒ Ξ“ ⁑ ( b - 1 2 ) ⁒ U ⁑ ( b - 1 2 , a + 1 2 , 1 4 ⁒ z 2 ) superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 2 superscript 𝑑 2 𝑏 2 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 superscript 𝑑 2 𝑑 1 2 superscript πœ‹ 1 2 Euler-Gamma 𝑏 1 2 Kummer-confluent-hypergeometric-U 𝑏 1 2 π‘Ž 1 2 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}% \left(a,b,t^{2}\right)\mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-% \tfrac{1}{2}\right)U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}% \right)}}
\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}\OlverconfhyperM@{a}{b}{t^{2}}\diff{t} = \tfrac{1}{2}\pi^{-\frac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{\tfrac{1}{4}z^{2}}		 β„œ ⁑ b > 1 2 , β„œ ⁑ z > 0 , β„œ ⁑ ( b - 1 2 ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence 𝑏 1 2 formulae-sequence 𝑧 0 formulae-sequence 𝑏 1 2 0 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re b>\tfrac{1}{2},\Re z>0,\Re(b-\tfrac{1}{2})>0,% \Re(b+s)>0}} 
int(exp(- z*t - (t)^(2))*(t)^(2*b - 2)* KummerM(a, b, (t)^(2))/GAMMA(b), t = 0..infinity) = (1)/(2)*(Pi)^(-(1)/(2))* GAMMA(b -(1)/(2))*KummerU(b -(1)/(2), a +(1)/(2), (1)/(4)*(z)^(2))

Integrate[Exp[- z*t - (t)^(2)]*(t)^(2*b - 2)* Hypergeometric1F1Regularized[a, b, (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*(Pi)^(-Divide[1,2])* Gamma[b -Divide[1,2]]*HypergeometricU[b -Divide[1,2], a +Divide[1,2], Divide[1,4]*(z)^(2)]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.10.E7 ∫ 0 ∞ e - z ⁒ t ⁒ t b - 1 ⁒ U ⁑ ( a , c , t ) ⁒ d t = Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( b - c + 1 ) ⁒ z - b ⁒ 𝐅 1 2 ⁑ ( a , b ; a + b - c + 1 ; 1 - 1 z ) superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑏 1 Kummer-confluent-hypergeometric-U π‘Ž 𝑐 𝑑 𝑑 Euler-Gamma 𝑏 Euler-Gamma 𝑏 𝑐 1 superscript 𝑧 𝑏 hypergeometric-bold-pFq 2 1 π‘Ž 𝑏 π‘Ž 𝑏 𝑐 1 1 1 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}U\left(a,c,t\right)% \mathrm{d}t=\Gamma\left(b\right)\Gamma\left(b-c+1\right)\*z^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;a+b-c+1;1-\frac{1}{z}\right)}}
\int_{0}^{\infty}e^{-zt}t^{b-1}\KummerconfhyperU@{a}{c}{t}\diff{t} = \EulerGamma@{b}\EulerGamma@{b-c+1}\*z^{-b}\genhyperOlverF{2}{1}@{a,b}{a+b-c+1}{1-\frac{1}{z}}		 β„œ ⁑ b > max ( β„œ ⁑ c - 1 , β„œ ⁑ z > 0 , β„œ ⁑ b > 0 , β„œ ⁑ ( b - c + 1 ) > 0 fragments 𝑏 fragments ( 𝑐 1 , 𝑧 0 , 𝑏 0 , 𝑏 𝑐 1 0 {\displaystyle{\displaystyle\Re b>\max\left(\Re c-1,\Re z>0,\Re b>0,\Re(b-c+1)% >0}\)\@add@PDF@RDFa@triples\end{document}} 
int(exp(- z*t)*(t)^(b - 1)* KummerU(a, c, t), t = 0..infinity) = GAMMA(b)*GAMMA(b - c + 1)* (z)^(- b)* hypergeom([a , b], [a + b - c + 1], 1 -(1)/(z))

Integrate[Exp[- z*t]*(t)^(b - 1)* HypergeometricU[a, c, t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[b]*Gamma[b - c + 1]* (z)^(- b)* HypergeometricPFQRegularized[{a , b}, {a + b - c + 1}, 1 -Divide[1,z]]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.10.E8 1 2 ⁒ Ο€ ⁒ i ⁒ ∫ - ∞ ( 0 + ) e t ⁒ z ⁒ t - a ⁒ 𝐌 ⁑ ( a , b , y / t ) ⁒ d t = 1 Ξ“ ⁑ ( a ) ⁒ z 1 2 ⁒ ( 2 ⁒ a - b - 1 ) ⁒ y 1 2 ⁒ ( 1 - b ) ⁒ I b - 1 ⁑ ( 2 ⁒ z ⁒ y ) 1 2 πœ‹ imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑑 𝑧 superscript 𝑑 π‘Ž Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑦 𝑑 𝑑 1 Euler-Gamma π‘Ž superscript 𝑧 1 2 2 π‘Ž 𝑏 1 superscript 𝑦 1 2 1 𝑏 modified-Bessel-first-kind 𝑏 1 2 𝑧 𝑦 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz% }t^{-a}{\mathbf{M}}\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{1}{\Gamma% \left(a\right)}z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}I_{b-1}\left(2\sqrt{% zy}\right)}}
\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\OlverconfhyperM@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{1}{\EulerGamma@{a}}z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}\modBesselI{b-1}@{2\sqrt{zy}}		 β„œ ⁑ z > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( b + s ) > 0 , β„œ ⁑ ( ( b - 1 ) + k + 1 ) > 0 formulae-sequence 𝑧 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 𝑠 0 𝑏 1 π‘˜ 1 0 {\displaystyle{\displaystyle\Re z>0,\Re a>0,\Re(b+s)>0,\Re((b-1)+k+1)>0}} 
(1)/(2*Pi*I)*int(exp(t*(x + y*I))*(t)^(- a)* KummerM(a, b, (y)/(t))/GAMMA(b), t = - infinity..(0 +)) = (1)/(GAMMA(a))*(x + y*I)^((1)/(2)*(2*a - b - 1))* (y)^((1)/(2)*(1 - b))* BesselI(b - 1, 2*sqrt((x + y*I)*y))

Divide[1,2*Pi*I]*Integrate[Exp[t*(x + y*I)]*(t)^(- a)* Hypergeometric1F1Regularized[a, b, Divide[y,t]], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[1,Gamma[a]]*(x + y*I)^(Divide[1,2]*(2*a - b - 1))* (y)^(Divide[1,2]*(1 - b))* BesselI[b - 1, 2*Sqrt[(x + y*I)*y]]
Error Failure - Error
13.10.E9 1 2 ⁒ Ο€ ⁒ i ⁒ ∫ - ∞ ( 0 + ) e t ⁒ z ⁒ t - a ⁒ U ⁑ ( a , b , y / t ) ⁒ d t = 2 ⁒ z 1 2 ⁒ ( 2 ⁒ a - b - 1 ) ⁒ y 1 2 ⁒ ( 1 - b ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( a - b + 1 ) ⁒ K b - 1 ⁑ ( 2 ⁒ z ⁒ y ) 1 2 πœ‹ imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑑 𝑧 superscript 𝑑 π‘Ž Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑦 𝑑 𝑑 2 superscript 𝑧 1 2 2 π‘Ž 𝑏 1 superscript 𝑦 1 2 1 𝑏 Euler-Gamma π‘Ž Euler-Gamma π‘Ž 𝑏 1 modified-Bessel-second-kind 𝑏 1 2 𝑧 𝑦 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz% }t^{-a}U\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)% }y^{\frac{1}{2}(1-b)}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}% \left(2\sqrt{zy}\right)}}
\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\KummerconfhyperU@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{2z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\modBesselK{b-1}@{2\sqrt{zy}}		 β„œ ⁑ z > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( a - b + 1 ) > 0 formulae-sequence 𝑧 0 formulae-sequence π‘Ž 0 π‘Ž 𝑏 1 0 {\displaystyle{\displaystyle\Re z>0,\Re a>0,\Re(a-b+1)>0}} 
(1)/(2*Pi*I)*int(exp(t*(x + y*I))*(t)^(- a)* KummerU(a, b, (y)/(t)), t = - infinity..(0 +)) = (2*(x + y*I)^((1)/(2)*(2*a - b - 1))* (y)^((1)/(2)*(1 - b)))/(GAMMA(a)*GAMMA(a - b + 1))*BesselK(b - 1, 2*sqrt((x + y*I)*y))

Divide[1,2*Pi*I]*Integrate[Exp[t*(x + y*I)]*(t)^(- a)* HypergeometricU[a, b, Divide[y,t]], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[2*(x + y*I)^(Divide[1,2]*(2*a - b - 1))* (y)^(Divide[1,2]*(1 - b)),Gamma[a]*Gamma[a - b + 1]]*BesselK[b - 1, 2*Sqrt[(x + y*I)*y]]
Error Failure - Error
13.10.E10 ∫ 0 ∞ t Ξ» - 1 ⁒ 𝐌 ⁑ ( a , b , - t ) ⁒ d t = Ξ“ ⁑ ( Ξ» ) ⁒ Ξ“ ⁑ ( a - Ξ» ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b - Ξ» ) superscript subscript 0 superscript 𝑑 πœ† 1 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑑 𝑑 Euler-Gamma πœ† Euler-Gamma π‘Ž πœ† Euler-Gamma π‘Ž Euler-Gamma 𝑏 πœ† {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,% b,-t\right)\mathrm{d}t=\frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda% \right)}{\Gamma\left(a\right)\Gamma\left(b-\lambda\right)}}}
\int_{0}^{\infty}t^{\lambda-1}\OlverconfhyperM@{a}{b}{-t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}}{\EulerGamma@{a}\EulerGamma@{b-\lambda}}		 0 < β„œ ⁑ Ξ» , β„œ ⁑ Ξ» < β„œ ⁑ a , β„œ ⁑ ( Ξ» ) > 0 , β„œ ⁑ ( a - Ξ» ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( b - Ξ» ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence 0 πœ† formulae-sequence πœ† π‘Ž formulae-sequence πœ† 0 formulae-sequence π‘Ž πœ† 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 πœ† 0 𝑏 𝑠 0 {\displaystyle{\displaystyle 0<\Re\lambda,\Re\lambda<\Re a,\Re(\lambda)>0,\Re(% a-\lambda)>0,\Re a>0,\Re(b-\lambda)>0,\Re(b+s)>0}} 
int((t)^(lambda - 1)* KummerM(a, b, - t)/GAMMA(b), t = 0..infinity) = (GAMMA(lambda)*GAMMA(a - lambda))/(GAMMA(a)*GAMMA(b - lambda))

Integrate[(t)^(\[Lambda]- 1)* Hypergeometric1F1Regularized[a, b, - t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Lambda]]*Gamma[a - \[Lambda]],Gamma[a]*Gamma[b - \[Lambda]]]
Successful Aborted - Skipped - Because timed out
13.10.E11 ∫ 0 ∞ t Ξ» - 1 ⁒ U ⁑ ( a , b , t ) ⁒ d t = Ξ“ ⁑ ( Ξ» ) ⁒ Ξ“ ⁑ ( a - Ξ» ) ⁒ Ξ“ ⁑ ( Ξ» - b + 1 ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( a - b + 1 ) superscript subscript 0 superscript 𝑑 πœ† 1 Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑑 𝑑 Euler-Gamma πœ† Euler-Gamma π‘Ž πœ† Euler-Gamma πœ† 𝑏 1 Euler-Gamma π‘Ž Euler-Gamma π‘Ž 𝑏 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)% \mathrm{d}t=\frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)\Gamma% \left(\lambda-b+1\right)}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}}}
\int_{0}^{\infty}t^{\lambda-1}\KummerconfhyperU@{a}{b}{t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}\EulerGamma@{\lambda-b+1}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}		 max ⁑ ( β„œ ⁑ b - 1 < β„œ ⁑ Ξ» , 0 ) < β„œ ⁑ Ξ» , β„œ ⁑ Ξ» < β„œ ⁑ a , β„œ ⁑ ( Ξ» ) > 0 , β„œ ⁑ ( a - Ξ» ) > 0 , β„œ ⁑ ( Ξ» - b + 1 ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( a - b + 1 ) > 0 formulae-sequence 𝑏 1 πœ† 0 πœ† formulae-sequence πœ† π‘Ž formulae-sequence πœ† 0 formulae-sequence π‘Ž πœ† 0 formulae-sequence πœ† 𝑏 1 0 formulae-sequence π‘Ž 0 π‘Ž 𝑏 1 0 {\displaystyle{\displaystyle\max\left(\Re b-1<\Re\lambda,0\right)<\Re\lambda,% \Re\lambda<\Re a,\Re(\lambda)>0,\Re(a-\lambda)>0,\Re(\lambda-b+1)>0,\Re a>0,% \Re(a-b+1)>0}} 
int((t)^(lambda - 1)* KummerU(a, b, t), t = 0..infinity) = (GAMMA(lambda)*GAMMA(a - lambda)*GAMMA(lambda - b + 1))/(GAMMA(a)*GAMMA(a - b + 1))

Integrate[(t)^(\[Lambda]- 1)* HypergeometricU[a, b, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Lambda]]*Gamma[a - \[Lambda]]*Gamma[\[Lambda]- b + 1],Gamma[a]*Gamma[a - b + 1]]
Successful Successful - Successful [Tested: 300]
13.10.E12 ∫ 0 ∞ cos ⁑ ( 2 ⁒ x ⁒ t ) ⁒ 𝐌 ⁑ ( a , b , - t 2 ) ⁒ d t = Ο€ 2 ⁒ Ξ“ ⁑ ( a ) ⁒ x 2 ⁒ a - 1 ⁒ e - x 2 ⁒ U ⁑ ( b - 1 2 , a + 1 2 , x 2 ) superscript subscript 0 2 π‘₯ 𝑑 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 superscript 𝑑 2 𝑑 πœ‹ 2 Euler-Gamma π‘Ž superscript π‘₯ 2 π‘Ž 1 superscript 𝑒 superscript π‘₯ 2 Kummer-confluent-hypergeometric-U 𝑏 1 2 π‘Ž 1 2 superscript π‘₯ 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(2xt\right){\mathbf{M}}% \left(a,b,-t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2\Gamma\left(a\right)}x^{% 2a-1}e^{-x^{2}}U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right)}}
\int_{0}^{\infty}\cos@{2xt}\OlverconfhyperM@{a}{b}{-t^{2}}\diff{t} = \frac{\sqrt{\pi}}{2\EulerGamma@{a}}x^{2a-1}e^{-x^{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{x^{2}}		 β„œ ⁑ a > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence π‘Ž 0 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re a>0,\Re(b+s)>0}} 
int(cos(2*x*t)*KummerM(a, b, - (t)^(2))/GAMMA(b), t = 0..infinity) = (sqrt(Pi))/(2*GAMMA(a))*(x)^(2*a - 1)* exp(- (x)^(2))*KummerU(b -(1)/(2), a +(1)/(2), (x)^(2))

Integrate[Cos[2*x*t]*Hypergeometric1F1Regularized[a, b, - (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],2*Gamma[a]]*(x)^(2*a - 1)* Exp[- (x)^(2)]*HypergeometricU[b -Divide[1,2], a +Divide[1,2], (x)^(2)]
Failure Aborted
Failed [51 / 54]
Result: Float(undefined)+Float(undefined)*I
Test Values: {a = 3/2, b = -3/2, x = 3/2}


Result: Float(undefined)+Float(undefined)*I
Test Values: {a = 3/2, b = -3/2, x = 1/2}

... skip entries to safe data
 Skipped - Because timed out
13.10.E13 ∫ 0 ∞ e - t ⁒ t b - 1 - 1 2 ⁒ Ξ½ ⁒ 𝐌 ⁑ ( a , b , t ) ⁒ J Ξ½ ⁑ ( 2 ⁒ x ⁒ t ) ⁒ d t = x - a + 1 2 ⁒ Ξ½ ⁒ e - x ⁒ 𝐌 ⁑ ( Ξ½ - b + 1 , Ξ½ - a + 1 , x ) superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 𝑏 1 1 2 𝜈 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑑 Bessel-J 𝜈 2 π‘₯ 𝑑 𝑑 superscript π‘₯ π‘Ž 1 2 𝜈 superscript 𝑒 π‘₯ Kummer-confluent-hypergeometric-bold-M 𝜈 𝑏 1 𝜈 π‘Ž 1 π‘₯ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{% \mathbf{M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=x^{-a+% \frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(\nu-b+1,\nu-a+1,x\right)}}
\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = x^{-a+\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{\nu-b+1}{\nu-a+1}{x}		 x > 0 , 2 ⁒ β„œ ⁑ a < β„œ ⁑ Ξ½ + 5 2 , β„œ ⁑ b > 0 , β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( b + s ) > 0 , β„œ ⁑ ( ( Ξ½ - a + 1 ) + s ) > 0 formulae-sequence π‘₯ 0 formulae-sequence 2 π‘Ž 𝜈 5 2 formulae-sequence 𝑏 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝑏 𝑠 0 𝜈 π‘Ž 1 𝑠 0 {\displaystyle{\displaystyle x>0,2\Re a<\Re\nu+\tfrac{5}{2},\Re b>0,\Re(\nu+k+% 1)>0,\Re(b+s)>0,\Re((\nu-a+1)+s)>0}} 
int(exp(- t)*(t)^(b - 1 -(1)/(2)*nu)* KummerM(a, b, t)/GAMMA(b)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (x)^(- a +(1)/(2)*nu)* exp(- x)*KummerM(nu - b + 1, nu - a + 1, x)/GAMMA(nu - a + 1)

Integrate[Exp[- t]*(t)^(b - 1 -Divide[1,2]*\[Nu])* Hypergeometric1F1Regularized[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == (x)^(- a +Divide[1,2]*\[Nu])* Exp[- x]*Hypergeometric1F1Regularized[\[Nu]- b + 1, \[Nu]- a + 1, x]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.10.E14 ∫ 0 ∞ e - t ⁒ t 1 2 ⁒ Ξ½ ⁒ 𝐌 ⁑ ( a , b , t ) ⁒ J Ξ½ ⁑ ( 2 ⁒ x ⁒ t ) ⁒ d t = x 1 2 ⁒ Ξ½ ⁒ e - x Ξ“ ⁑ ( b - a ) ⁒ U ⁑ ( a , a - b + Ξ½ + 2 , x ) superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 1 2 𝜈 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑑 Bessel-J 𝜈 2 π‘₯ 𝑑 𝑑 superscript π‘₯ 1 2 𝜈 superscript 𝑒 π‘₯ Euler-Gamma 𝑏 π‘Ž Kummer-confluent-hypergeometric-U π‘Ž π‘Ž 𝑏 𝜈 2 π‘₯ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{% M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{x^{\frac{% 1}{2}\nu}e^{-x}}{\Gamma\left(b-a\right)}U\left(a,a-b+\nu+2,x\right)}}
\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{x^{\frac{1}{2}\nu}e^{-x}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{a-b+\nu+2}{x}		 x > 0 , - 1 < β„œ ⁑ Ξ½ , β„œ ⁑ Ξ½ < 2 ⁒ β„œ ⁑ ( b - a ) - 1 2 , β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( b - a ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence π‘₯ 0 formulae-sequence 1 𝜈 formulae-sequence 𝜈 2 𝑏 π‘Ž 1 2 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝑏 π‘Ž 0 𝑏 𝑠 0 {\displaystyle{\displaystyle x>0,-1<\Re\nu,\Re\nu<2\Re\left(b-a\right)-\tfrac{% 1}{2},\Re(\nu+k+1)>0,\Re(b-a)>0,\Re(b+s)>0}} 
int(exp(- t)*(t)^((1)/(2)*nu)* KummerM(a, b, t)/GAMMA(b)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = ((x)^((1)/(2)*nu)* exp(- x))/(GAMMA(b - a))*KummerU(a, a - b + nu + 2, x)

Integrate[Exp[- t]*(t)^(Divide[1,2]*\[Nu])* Hypergeometric1F1Regularized[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(x)^(Divide[1,2]*\[Nu])* Exp[- x],Gamma[b - a]]*HypergeometricU[a, a - b + \[Nu]+ 2, x]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.10.E15 ∫ 0 ∞ t 1 2 ⁒ Ξ½ ⁒ U ⁑ ( a , b , t ) ⁒ J Ξ½ ⁑ ( 2 ⁒ x ⁒ t ) ⁒ d t = Ξ“ ⁑ ( Ξ½ - b + 2 ) Ξ“ ⁑ ( a ) ⁒ x 1 2 ⁒ Ξ½ ⁒ U ⁑ ( Ξ½ - b + 2 , Ξ½ - a + 2 , x ) superscript subscript 0 superscript 𝑑 1 2 𝜈 Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑑 Bessel-J 𝜈 2 π‘₯ 𝑑 𝑑 Euler-Gamma 𝜈 𝑏 2 Euler-Gamma π‘Ž superscript π‘₯ 1 2 𝜈 Kummer-confluent-hypergeometric-U 𝜈 𝑏 2 𝜈 π‘Ž 2 π‘₯ {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\frac{1}{2}\nu}U\left(a,b,t% \right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-b+2% \right)}{\Gamma\left(a\right)}x^{\frac{1}{2}\nu}U\left(\nu-b+2,\nu-a+2,x\right% )}}
\int_{0}^{\infty}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-b+2}}{\EulerGamma@{a}}x^{\frac{1}{2}\nu}\KummerconfhyperU@{\nu-b+2}{\nu-a+2}{x}		 x > 0 , max ⁑ ( β„œ ⁑ b - 2 < β„œ ⁑ Ξ½ , - 1 ) < β„œ ⁑ Ξ½ , β„œ ⁑ Ξ½ < 2 ⁒ β„œ ⁑ a + 1 2 , β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ - b + 2 ) > 0 , β„œ ⁑ a > 0 formulae-sequence π‘₯ 0 formulae-sequence 𝑏 2 𝜈 1 𝜈 formulae-sequence 𝜈 2 π‘Ž 1 2 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝜈 𝑏 2 0 π‘Ž 0 {\displaystyle{\displaystyle x>0,\max\left(\Re b-2<\Re\nu,-1\right)<\Re\nu,\Re% \nu<2\Re a+\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu-b+2)>0,\Re a>0}} 
int((t)^((1)/(2)*nu)* KummerU(a, b, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(nu - b + 2))/(GAMMA(a))*(x)^((1)/(2)*nu)* KummerU(nu - b + 2, nu - a + 2, x)

Integrate[(t)^(Divide[1,2]*\[Nu])* HypergeometricU[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Nu]- b + 2],Gamma[a]]*(x)^(Divide[1,2]*\[Nu])* HypergeometricU[\[Nu]- b + 2, \[Nu]- a + 2, x]
Successful Aborted - Skipped - Because timed out
13.10.E16 ∫ 0 ∞ e - t ⁒ t 1 2 ⁒ Ξ½ ⁒ U ⁑ ( a , b , t ) ⁒ J Ξ½ ⁑ ( 2 ⁒ x ⁒ t ) ⁒ d t = Ξ“ ⁑ ( Ξ½ - b + 2 ) ⁒ x 1 2 ⁒ Ξ½ ⁒ e - x ⁒ 𝐌 ⁑ ( a , a - b + Ξ½ + 2 , x ) superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 1 2 𝜈 Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑑 Bessel-J 𝜈 2 π‘₯ 𝑑 𝑑 Euler-Gamma 𝜈 𝑏 2 superscript π‘₯ 1 2 𝜈 superscript 𝑒 π‘₯ Kummer-confluent-hypergeometric-bold-M π‘Ž π‘Ž 𝑏 𝜈 2 π‘₯ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}U\left(a,% b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\Gamma\left(\nu-b+2\right)% x^{\frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(a,a-b+\nu+2,x\right)}}
\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \EulerGamma@{\nu-b+2}x^{\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{a}{a-b+\nu+2}{x}		 x > 0 , max ⁑ ( β„œ ⁑ b - 2 < β„œ ⁑ Ξ½ , - 1 ) < β„œ ⁑ Ξ½ , β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ - b + 2 ) > 0 , β„œ ⁑ ( ( a - b + Ξ½ + 2 ) + s ) > 0 formulae-sequence π‘₯ 0 formulae-sequence 𝑏 2 𝜈 1 𝜈 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝜈 𝑏 2 0 π‘Ž 𝑏 𝜈 2 𝑠 0 {\displaystyle{\displaystyle x>0,\max\left(\Re b-2<\Re\nu,-1\right)<\Re\nu,\Re% (\nu+k+1)>0,\Re(\nu-b+2)>0,\Re((a-b+\nu+2)+s)>0}} 
int(exp(- t)*(t)^((1)/(2)*nu)* KummerU(a, b, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = GAMMA(nu - b + 2)*(x)^((1)/(2)*nu)* exp(- x)*KummerM(a, a - b + nu + 2, x)/GAMMA(a - b + nu + 2)

Integrate[Exp[- t]*(t)^(Divide[1,2]*\[Nu])* HypergeometricU[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Gamma[\[Nu]- b + 2]*(x)^(Divide[1,2]*\[Nu])* Exp[- x]*Hypergeometric1F1Regularized[a, a - b + \[Nu]+ 2, x]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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