Confluent Hypergeometric Functions - 13.4 Integral Representations

From testwiki
Revision as of 11:32, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision β†’ (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.4.E1 𝐌 ⁑ ( a , b , z ) = 1 Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b - a ) ⁒ ∫ 0 1 e z ⁒ t ⁒ t a - 1 ⁒ ( 1 - t ) b - a - 1 ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 1 Euler-Gamma π‘Ž Euler-Gamma 𝑏 π‘Ž superscript subscript 0 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 1 𝑑 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(a\right)\Gamma\left(b-a\right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}% \mathrm{d}t}}
\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{a}\EulerGamma@{b-a}}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\diff{t}		 β„œ ⁑ b > β„œ ⁑ a , β„œ ⁑ a > 0 , β„œ ⁑ ( b - a ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence 𝑏 π‘Ž formulae-sequence π‘Ž 0 formulae-sequence 𝑏 π‘Ž 0 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re b>\Re a,\Re a>0,\Re(b-a)>0,\Re(b+s)>0}} 
KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(a)*GAMMA(b - a))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = 0..1)

Hypergeometric1F1Regularized[a, b, z] == Divide[1,Gamma[a]*Gamma[b - a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, 0, 1}, GenerateConditions->None]
Successful Successful - Successful [Tested: 21]
13.4.E2 𝐌 ⁑ ( a , b , z ) = 1 Ξ“ ⁑ ( b - c ) ⁒ ∫ 0 1 𝐌 ⁑ ( a , c , z ⁒ t ) ⁒ t c - 1 ⁒ ( 1 - t ) b - c - 1 ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 1 Euler-Gamma 𝑏 𝑐 superscript subscript 0 1 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑐 𝑧 𝑑 superscript 𝑑 𝑐 1 superscript 1 𝑑 𝑏 𝑐 1 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(b-c\right)}\int_{0}^{1}{\mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-% 1}\mathrm{d}t}}
\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{b-c}}\int_{0}^{1}\OlverconfhyperM@{a}{c}{zt}t^{c-1}(1-t)^{b-c-1}\diff{t}		 β„œ ⁑ b > β„œ ⁑ c , β„œ ⁑ c > 0 , β„œ ⁑ ( b - c ) > 0 , β„œ ⁑ ( b + s ) > 0 , β„œ ⁑ ( c + s ) > 0 formulae-sequence 𝑏 𝑐 formulae-sequence 𝑐 0 formulae-sequence 𝑏 𝑐 0 formulae-sequence 𝑏 𝑠 0 𝑐 𝑠 0 {\displaystyle{\displaystyle\Re b>\Re c,\Re c>0,\Re(b-c)>0,\Re(b+s)>0,\Re(c+s)% >0}} 
KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(b - c))*int(KummerM(a, c, z*t)/GAMMA(c)*(t)^(c - 1)*(1 - t)^(b - c - 1), t = 0..1)

Hypergeometric1F1Regularized[a, b, z] == Divide[1,Gamma[b - c]]*Integrate[Hypergeometric1F1Regularized[a, c, z*t]*(t)^(c - 1)*(1 - t)^(b - c - 1), {t, 0, 1}, GenerateConditions->None]
Successful Successful - Successful [Tested: 126]
13.4.E3 𝐌 ⁑ ( a , b , - z ) = z 1 2 - 1 2 ⁒ b Ξ“ ⁑ ( a ) ⁒ ∫ 0 ∞ e - t ⁒ t a - 1 2 ⁒ b - 1 2 ⁒ J b - 1 ⁑ ( 2 ⁒ z ⁒ t ) ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑧 1 2 1 2 𝑏 Euler-Gamma π‘Ž superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 π‘Ž 1 2 𝑏 1 2 Bessel-J 𝑏 1 2 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{% 2}-\frac{1}{2}b}}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}% b-\frac{1}{2}}J_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}
\OlverconfhyperM@{a}{b}{-z} = \frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\BesselJ{b-1}@{2\sqrt{zt}}\diff{t}		 β„œ ⁑ a > 0 , β„œ ⁑ ( ( b - 1 ) + k + 1 ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 1 π‘˜ 1 0 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re a>0,\Re((b-1)+k+1)>0,\Re(b+s)>0}} 
KummerM(a, b, - z)/GAMMA(b) = ((z)^((1)/(2)-(1)/(2)*b))/(GAMMA(a))*int(exp(- t)*(t)^(a -(1)/(2)*b -(1)/(2))* BesselJ(b - 1, 2*sqrt(z*t)), t = 0..infinity)

Hypergeometric1F1Regularized[a, b, - z] == Divide[(z)^(Divide[1,2]-Divide[1,2]*b),Gamma[a]]*Integrate[Exp[- t]*(t)^(a -Divide[1,2]*b -Divide[1,2])* BesselJ[b - 1, 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Error Skipped - Because timed out
13.4.E4 U ⁑ ( a , b , z ) = 1 Ξ“ ⁑ ( a ) ⁒ ∫ 0 ∞ e - z ⁒ t ⁒ t a - 1 ⁒ ( 1 + t ) b - a - 1 ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 1 Euler-Gamma π‘Ž superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 1 𝑑 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)% }\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\mathrm{d}t}}
\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}		 β„œ ⁑ a > 0 , | ph ⁑ z | < 1 2 ⁒ Ο€ formulae-sequence π‘Ž 0 phase 𝑧 1 2 πœ‹ {\displaystyle{\displaystyle\Re a>0,|\operatorname{ph}{z}|<\frac{1}{2}\pi}} 
KummerU(a, b, z) = (1)/(GAMMA(a))*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = 0..infinity)

HypergeometricU[a, b, z] == Divide[1,Gamma[a]]*Integrate[Exp[- z*t]*(t)^(a - 1)*(1 + t)^(b - a - 1), {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 90]
13.4.E5 U ⁑ ( a , b , z ) = z 1 - a Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( 1 + a - b ) ⁒ ∫ 0 ∞ U ⁑ ( b - a , b , t ) ⁒ e - t ⁒ t a - 1 t + z ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 superscript 𝑧 1 π‘Ž Euler-Gamma π‘Ž Euler-Gamma 1 π‘Ž 𝑏 superscript subscript 0 Kummer-confluent-hypergeometric-U 𝑏 π‘Ž 𝑏 𝑑 superscript 𝑒 𝑑 superscript 𝑑 π‘Ž 1 𝑑 𝑧 𝑑 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a% \right)\Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^% {-t}t^{a-1}}{t+z}\mathrm{d}t}}
\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-a}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}\int_{0}^{\infty}\frac{\KummerconfhyperU@{b-a}{b}{t}e^{-t}t^{a-1}}{t+z}\diff{t}		 | ph z | < Ο€ , β„œ ⁑ a > max ( β„œ ⁑ b - 1 , β„œ ⁑ a > 0 , β„œ ⁑ ( 1 + a - b ) > 0 fragments | phase z | Ο€ , π‘Ž fragments ( 𝑏 1 , π‘Ž 0 , 1 π‘Ž 𝑏 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\pi,\Re a>\max\left(\Re b-1% ,\Re a>0,\Re(1+a-b)>0}\)\@add@PDF@RDFa@triples\end{document}} 
KummerU(a, b, z) = ((z)^(1 - a))/(GAMMA(a)*GAMMA(1 + a - b))*int((KummerU(b - a, b, t)*exp(- t)*(t)^(a - 1))/(t + z), t = 0..infinity)

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

HypergeometricU[a, b, z] == Divide[(- 1)^(n)* (z)^(1 - b - n),Gamma[1 + a - b]]*Integrate[Divide[Hypergeometric1F1Regularized[b - a, b, t]*Exp[- t]*(t)^(b + n - 1),t + z], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.4.E7 U ⁑ ( a , b , z ) = 2 ⁒ z 1 2 - 1 2 ⁒ b Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( a - b + 1 ) ⁒ ∫ 0 ∞ e - t ⁒ t a - 1 2 ⁒ b - 1 2 ⁒ K b - 1 ⁑ ( 2 ⁒ z ⁒ t ) ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 2 superscript 𝑧 1 2 1 2 𝑏 Euler-Gamma π‘Ž Euler-Gamma π‘Ž 𝑏 1 superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 π‘Ž 1 2 𝑏 1 2 modified-Bessel-second-kind 𝑏 1 2 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1% }{2}b}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}\*\int_{0}^{\infty}e^{-t}% t^{a-\frac{1}{2}b-\frac{1}{2}}K_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}
\KummerconfhyperU@{a}{b}{z} = \frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\*\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\modBesselK{b-1}@{2\sqrt{zt}}\diff{t}		 β„œ ⁑ a > max ( β„œ ⁑ b - 1 , β„œ ⁑ a > 0 , β„œ ⁑ ( a - b + 1 ) > 0 fragments π‘Ž fragments ( 𝑏 1 , π‘Ž 0 , π‘Ž 𝑏 1 0 {\displaystyle{\displaystyle\Re a>\max\left(\Re b-1,\Re a>0,\Re(a-b+1)>0}\)\@add@PDF@RDFa@triples\end{document}} 
KummerU(a, b, z) = (2*(z)^((1)/(2)-(1)/(2)*b))/(GAMMA(a)*GAMMA(a - b + 1))* int(exp(- t)*(t)^(a -(1)/(2)*b -(1)/(2))* BesselK(b - 1, 2*sqrt(z*t)), t = 0..infinity)

HypergeometricU[a, b, z] == Divide[2*(z)^(Divide[1,2]-Divide[1,2]*b),Gamma[a]*Gamma[a - b + 1]]* Integrate[Exp[- t]*(t)^(a -Divide[1,2]*b -Divide[1,2])* BesselK[b - 1, 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
Successful Aborted - Skipped - Because timed out
13.4.E8 U ⁑ ( a , b , z ) = z c - a ⁒ ∫ 0 ∞ e - z ⁒ t ⁒ t c - 1 ⁒ 𝐅 1 2 ⁑ ( a , a - b + 1 ; c ; - t ) ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 superscript 𝑧 𝑐 π‘Ž superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑐 1 hypergeometric-bold-pFq 2 1 π‘Ž π‘Ž 𝑏 1 𝑐 𝑑 𝑑 {\displaystyle{\displaystyle U\left(a,b,z\right)=z^{c-a}\*\int_{0}^{\infty}e^{% -zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left(a,a-b+1;c;-t\right)\mathrm{d}t}}
\KummerconfhyperU@{a}{b}{z} = z^{c-a}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@{a,a-b+1}{c}{-t}\diff{t}		 | ph ⁑ z | < 1 2 ⁒ Ο€ phase 𝑧 1 2 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\frac{1}{2}\pi}} 
KummerU(a, b, z) = (z)^(c - a)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([a , a - b + 1], [c], - t), t = 0..infinity)

HypergeometricU[a, b, z] == (z)^(c - a)* Integrate[Exp[- z*t]*(t)^(c - 1)* HypergeometricPFQRegularized[{a , a - b + 1}, {c}, - t], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [294 / 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}


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

... skip entries to safe data
 Skipped - Because timed out
13.4.E9 𝐌 ⁑ ( a , b , z ) = Ξ“ ⁑ ( 1 + a - b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ ∫ 0 ( 1 + ) e z ⁒ t ⁒ t a - 1 ⁒ ( t - 1 ) b - a - 1 ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 Euler-Gamma 1 π‘Ž 𝑏 2 πœ‹ imaginary-unit Euler-Gamma π‘Ž superscript subscript 0 limit-from 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 𝑑 1 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1% +a-b\right)}{2\pi\mathrm{i}\Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(% t-1)^{b-a-1}}\mathrm{d}t}}
\OlverconfhyperM@{a}{b}{z} = \frac{\EulerGamma@{1+a-b}}{2\pi\iunit\EulerGamma@{a}}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\diff{t}		 β„œ ⁑ a > 0 , β„œ ⁑ ( 1 + a - b ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence π‘Ž 0 formulae-sequence 1 π‘Ž 𝑏 0 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re a>0,\Re(1+a-b)>0,\Re(b+s)>0}} 
KummerM(a, b, z)/GAMMA(b) = (GAMMA(1 + a - b))/(2*Pi*I*GAMMA(a))*int(exp(z*t)*(t)^(a - 1)*(t - 1)^(b - a - 1), t = 0..(1 +))

Hypergeometric1F1Regularized[a, b, z] == Divide[Gamma[1 + a - b],2*Pi*I*Gamma[a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(t - 1)^(b - a - 1), {t, 0, (1 +)}, GenerateConditions->None]
Error Failure - Error
13.4.E10 𝐌 ⁑ ( a , b , z ) = e - a ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - a ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( b - a ) ⁒ ∫ 1 ( 0 + ) e z ⁒ t ⁒ t a - 1 ⁒ ( 1 - t ) b - a - 1 ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑒 π‘Ž πœ‹ imaginary-unit Euler-Gamma 1 π‘Ž 2 πœ‹ imaginary-unit Euler-Gamma 𝑏 π‘Ž superscript subscript 1 limit-from 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 1 𝑑 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-a\pi\mathrm{i}}% \frac{\Gamma\left(1-a\right)}{2\pi\mathrm{i}\Gamma\left(b-a\right)}\int_{1}^{(% 0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}
\OlverconfhyperM@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit\EulerGamma@{b-a}}\int_{1}^{(0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}		 β„œ ⁑ ( b - a ) > 0 , β„œ ⁑ ( 1 - a ) > 0 , β„œ ⁑ ( b - a ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence 𝑏 π‘Ž 0 formulae-sequence 1 π‘Ž 0 formulae-sequence 𝑏 π‘Ž 0 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re\left(b-a\right)>0,\Re(1-a)>0,\Re(b-a)>0,\Re(b+% s)>0}} 
KummerM(a, b, z)/GAMMA(b) = exp(- a*Pi*I)*(GAMMA(1 - a))/(2*Pi*I*GAMMA(b - a))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = 1..(0 +))

Hypergeometric1F1Regularized[a, b, z] == Exp[- a*Pi*I]*Divide[Gamma[1 - a],2*Pi*I*Gamma[b - a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, 1, (0 +)}, GenerateConditions->None]
Error Failure - Error
13.4.E11 𝐌 ⁑ ( a , b , z ) = e - b ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - a ) ⁒ Ξ“ ⁑ ( 1 + a - b ) ⁒ 1 4 ⁒ Ο€ 2 ⁒ ∫ Ξ± ( 0 + , 1 + , 0 - , 1 - ) e z ⁒ t ⁒ t a - 1 ⁒ ( 1 - t ) b - a - 1 ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑒 𝑏 πœ‹ imaginary-unit Euler-Gamma 1 π‘Ž Euler-Gamma 1 π‘Ž 𝑏 1 4 superscript πœ‹ 2 superscript subscript 𝛼 limit-from 0 limit-from 1 limit-from 0 limit-from 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 1 𝑑 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-b\pi\mathrm{i}}% \Gamma\left(1-a\right)\Gamma\left(1+a-b\right)\*\frac{1}{4\pi^{2}}\int_{\alpha% }^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}
\OlverconfhyperM@{a}{b}{z} = e^{-b\pi\iunit}\EulerGamma@{1-a}\EulerGamma@{1+a-b}\*\frac{1}{4\pi^{2}}\int_{\alpha}^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}		 β„œ ⁑ ( 1 - a ) > 0 , β„œ ⁑ ( 1 + a - b ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence 1 π‘Ž 0 formulae-sequence 1 π‘Ž 𝑏 0 𝑏 𝑠 0 {\displaystyle{\displaystyle\Re(1-a)>0,\Re(1+a-b)>0,\Re(b+s)>0}} 
KummerM(a, b, z)/GAMMA(b) = exp(- b*Pi*I)*GAMMA(1 - a)*GAMMA(1 + a - b)*(1)/(4*(Pi)^(2))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = alpha..(0 + , 1 + , 0 - , 1 -))

Hypergeometric1F1Regularized[a, b, z] == Exp[- b*Pi*I]*Gamma[1 - a]*Gamma[1 + a - b]*Divide[1,4*(Pi)^(2)]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, \[Alpha], (0 + , 1 + , 0 - , 1 -)}, GenerateConditions->None]
Error Failure - Error
13.4.E12 𝐌 ⁑ ( a , c , z ) = Ξ“ ⁑ ( b ) 2 ⁒ Ο€ ⁒ i ⁒ z 1 - b ⁒ ∫ - ∞ ( 0 + , 1 + ) e z ⁒ t ⁒ t - b ⁒ 𝐅 1 2 ⁑ ( a , b ; c ; 1 / t ) ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑐 𝑧 Euler-Gamma 𝑏 2 πœ‹ imaginary-unit superscript 𝑧 1 𝑏 superscript subscript limit-from 0 limit-from 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑏 hypergeometric-bold-pFq 2 1 π‘Ž 𝑏 𝑐 1 𝑑 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b% \right)}{2\pi\mathrm{i}}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;c;\ifrac{1}{t}\right)\mathrm{d}t}}
\OlverconfhyperM@{a}{c}{z} = \frac{\EulerGamma@{b}}{2\pi\iunit}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{1}{t}}\diff{t}		 | ph ⁑ z | < 1 2 ⁒ Ο€ , β„œ ⁑ b > 0 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 𝑧 1 2 πœ‹ formulae-sequence 𝑏 0 𝑐 𝑠 0 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi,\Re b% >0,\Re(c+s)>0}} 
KummerM(a, c, z)/GAMMA(c) = (GAMMA(b))/(2*Pi*I)*(z)^(1 - b)* int(exp(z*t)*(t)^(- b)* hypergeom([a , b], [c], (1)/(t)), t = - infinity..(0 + , 1 +))

Hypergeometric1F1Regularized[a, c, z] == Divide[Gamma[b],2*Pi*I]*(z)^(1 - b)* Integrate[Exp[z*t]*(t)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[1,t]], {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]
Error Failure - Error
13.4.E13 𝐌 ⁑ ( a , b , z ) = z 1 - b 2 ⁒ Ο€ ⁒ i ⁒ ∫ - ∞ ( 0 + , 1 + ) e z ⁒ t ⁒ t - b ⁒ ( 1 - 1 t ) - a ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑧 1 𝑏 2 πœ‹ imaginary-unit superscript subscript limit-from 0 limit-from 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑏 superscript 1 1 𝑑 π‘Ž 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi% \mathrm{i}}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-% a}\mathrm{d}t}}
\OlverconfhyperM@{a}{b}{z} = \frac{z^{1-b}}{2\pi\iunit}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\diff{t}		 | ph ⁑ z | < 1 2 ⁒ Ο€ , β„œ ⁑ ( b + s ) > 0 formulae-sequence phase 𝑧 1 2 πœ‹ 𝑏 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\frac{1}{2}\pi,\Re(b+s)>0}} 
KummerM(a, b, z)/GAMMA(b) = ((z)^(1 - b))/(2*Pi*I)*int(exp(z*t)*(t)^(- b)*(1 -(1)/(t))^(- a), t = - infinity..(0 + , 1 +))

Hypergeometric1F1Regularized[a, b, z] == Divide[(z)^(1 - b),2*Pi*I]*Integrate[Exp[z*t]*(t)^(- b)*(1 -Divide[1,t])^(- a), {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]
Error Failure - Error
13.4.E14 U ⁑ ( a , b , z ) = e - a ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - a ) 2 ⁒ Ο€ ⁒ i ⁒ ∫ ∞ ( 0 + ) e - z ⁒ t ⁒ t a - 1 ⁒ ( 1 + t ) b - a - 1 ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 superscript 𝑒 π‘Ž πœ‹ imaginary-unit Euler-Gamma 1 π‘Ž 2 πœ‹ imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 1 𝑑 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle U\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{% \Gamma\left(1-a\right)}{2\pi\mathrm{i}}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t% )^{b-a-1}}\mathrm{d}t}}
\KummerconfhyperU@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t)^{b-a-1}}\diff{t}		 | ph ⁑ z | < 1 2 ⁒ Ο€ , β„œ ⁑ ( 1 - a ) > 0 formulae-sequence phase 𝑧 1 2 πœ‹ 1 π‘Ž 0 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi,\Re% (1-a)>0}} 
KummerU(a, b, z) = exp(- a*Pi*I)*(GAMMA(1 - a))/(2*Pi*I)*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = infinity..(0 +))

HypergeometricU[a, b, z] == Exp[- a*Pi*I]*Divide[Gamma[1 - a],2*Pi*I]*Integrate[Exp[- z*t]*(t)^(a - 1)*(1 + t)^(b - a - 1), {t, Infinity, (0 +)}, GenerateConditions->None]
Error Failure - Error
13.4.E15 U ⁑ ( a , b , z ) Ξ“ ⁑ ( c ) ⁒ Ξ“ ⁑ ( c - b + 1 ) = z 1 - c 2 ⁒ Ο€ ⁒ i ⁒ ∫ - ∞ ( 0 + ) e z ⁒ t ⁒ t - c ⁒ 𝐅 1 2 ⁑ ( a , c ; a + c - b + 1 ; 1 - 1 t ) ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 Euler-Gamma 𝑐 Euler-Gamma 𝑐 𝑏 1 superscript 𝑧 1 𝑐 2 πœ‹ imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑐 hypergeometric-bold-pFq 2 1 π‘Ž 𝑐 π‘Ž 𝑐 𝑏 1 1 1 𝑑 𝑑 {\displaystyle{\displaystyle\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)% \Gamma\left(c-b+1\right)}=\frac{z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e% ^{zt}t^{-c}{{}_{2}{\mathbf{F}}_{1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)% \mathrm{d}t}}
\frac{\KummerconfhyperU@{a}{b}{z}}{\EulerGamma@{c}\EulerGamma@{c-b+1}} = \frac{z^{1-c}}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt}t^{-c}\genhyperOlverF{2}{1}@{a,c}{a+c-b+1}{1-\frac{1}{t}}\diff{t}		 | ph ⁑ z | < 1 2 ⁒ Ο€ , β„œ ⁑ c > 0 , β„œ ⁑ ( c - b + 1 ) > 0 formulae-sequence phase 𝑧 1 2 πœ‹ formulae-sequence 𝑐 0 𝑐 𝑏 1 0 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi,\Re c% >0,\Re(c-b+1)>0}} 
(KummerU(a, b, z))/(GAMMA(c)*GAMMA(c - b + 1)) = ((z)^(1 - c))/(2*Pi*I)*int(exp(z*t)*(t)^(- c)* hypergeom([a , c], [a + c - b + 1], 1 -(1)/(t)), t = - infinity..(0 +))

Divide[HypergeometricU[a, b, z],Gamma[c]*Gamma[c - b + 1]] == Divide[(z)^(1 - c),2*Pi*I]*Integrate[Exp[z*t]*(t)^(- c)* HypergeometricPFQRegularized[{a , c}, {a + c - b + 1}, 1 -Divide[1,t]], {t, - Infinity, (0 +)}, GenerateConditions->None]
Error Failure - Error
13.4.E16 𝐌 ⁑ ( a , b , - z ) = 1 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( a + t ) ⁒ Ξ“ ⁑ ( - t ) Ξ“ ⁑ ( b + t ) ⁒ z t ⁒ d t Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 1 2 πœ‹ imaginary-unit Euler-Gamma π‘Ž superscript subscript imaginary-unit imaginary-unit Euler-Gamma π‘Ž 𝑑 Euler-Gamma 𝑑 Euler-Gamma 𝑏 𝑑 superscript 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi% \mathrm{i}\Gamma\left(a\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}% \frac{\Gamma\left(a+t\right)\Gamma\left(-t\right)}{\Gamma\left(b+t\right)}z^{t% }\mathrm{d}t}}
\OlverconfhyperM@{a}{b}{-z} = \frac{1}{2\pi\iunit\EulerGamma@{a}}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{-t}}{\EulerGamma@{b+t}}z^{t}\diff{t}		 | ph ⁑ z | < 1 2 ⁒ Ο€ , β„œ ⁑ a > 0 , β„œ ⁑ ( a + t ) > 0 , β„œ ⁑ ( - t ) > 0 , β„œ ⁑ ( b + t ) > 0 , β„œ ⁑ ( b + s ) > 0 formulae-sequence phase 𝑧 1 2 πœ‹ formulae-sequence π‘Ž 0 formulae-sequence π‘Ž 𝑑 0 formulae-sequence 𝑑 0 formulae-sequence 𝑏 𝑑 0 𝑏 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{1}{2}\pi,\Re a>0,\Re% (a+t)>0,\Re(-t)>0,\Re(b+t)>0,\Re(b+s)>0}} 
KummerM(a, b, - z)/GAMMA(b) = (1)/(2*Pi*I*GAMMA(a))*int((GAMMA(a + t)*GAMMA(- t))/(GAMMA(b + t))*(z)^(t), t = - I*infinity..I*infinity)

Hypergeometric1F1Regularized[a, b, - z] == Divide[1,2*Pi*I*Gamma[a]]*Integrate[Divide[Gamma[a + t]*Gamma[- t],Gamma[b + t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.4.E17 U ⁑ ( a , b , z ) = z - a 2 ⁒ Ο€ ⁒ i ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( a + t ) ⁒ Ξ“ ⁑ ( 1 + a - b + t ) ⁒ Ξ“ ⁑ ( - t ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( 1 + a - b ) ⁒ z - t ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 superscript 𝑧 π‘Ž 2 πœ‹ imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma π‘Ž 𝑑 Euler-Gamma 1 π‘Ž 𝑏 𝑑 Euler-Gamma 𝑑 Euler-Gamma π‘Ž Euler-Gamma 1 π‘Ž 𝑏 superscript 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{-a}}{2\pi\mathrm{i}}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(1+a-b+t\right)\Gamma\left(-t\right)}{\Gamma\left(a\right)\Gamma\left(1+a% -b\right)}z^{-t}\mathrm{d}t}}
\KummerconfhyperU@{a}{b}{z} = \frac{z^{-a}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{1+a-b+t}\EulerGamma@{-t}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}z^{-t}\diff{t}		 | ph ⁑ z | < 3 2 ⁒ Ο€ , β„œ ⁑ ( a + t ) > 0 , β„œ ⁑ ( 1 + a - b + t ) > 0 , β„œ ⁑ ( - t ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( 1 + a - b ) > 0 formulae-sequence phase 𝑧 3 2 πœ‹ formulae-sequence π‘Ž 𝑑 0 formulae-sequence 1 π‘Ž 𝑏 𝑑 0 formulae-sequence 𝑑 0 formulae-sequence π‘Ž 0 1 π‘Ž 𝑏 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{3}{2}\pi,\Re(a+t)>0,% \Re(1+a-b+t)>0,\Re(-t)>0,\Re a>0,\Re(1+a-b)>0}} 
KummerU(a, b, z) = ((z)^(- a))/(2*Pi*I)*int((GAMMA(a + t)*GAMMA(1 + a - b + t)*GAMMA(- t))/(GAMMA(a)*GAMMA(1 + a - b))*(z)^(- t), t = - I*infinity..I*infinity)

HypergeometricU[a, b, z] == Divide[(z)^(- a),2*Pi*I]*Integrate[Divide[Gamma[a + t]*Gamma[1 + a - b + t]*Gamma[- t],Gamma[a]*Gamma[1 + a - b]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
13.4.E18 U ⁑ ( a , b , z ) = z 1 - b ⁒ e z 2 ⁒ Ο€ ⁒ i ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( b - 1 + t ) ⁒ Ξ“ ⁑ ( t ) Ξ“ ⁑ ( a + t ) ⁒ z - t ⁒ d t Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 superscript 𝑧 1 𝑏 superscript 𝑒 𝑧 2 πœ‹ imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑏 1 𝑑 Euler-Gamma 𝑑 Euler-Gamma π‘Ž 𝑑 superscript 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-b}e^{z}}{2\pi% \mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(b-1+t% \right)\Gamma\left(t\right)}{\Gamma\left(a+t\right)}z^{-t}\mathrm{d}t}}
\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-b}e^{z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{b-1+t}\EulerGamma@{t}}{\EulerGamma@{a+t}}z^{-t}\diff{t}		 | ph ⁑ z | < 1 2 ⁒ Ο€ , β„œ ⁑ ( b - 1 + t ) > 0 , β„œ ⁑ t > 0 , β„œ ⁑ ( a + t ) > 0 formulae-sequence phase 𝑧 1 2 πœ‹ formulae-sequence 𝑏 1 𝑑 0 formulae-sequence 𝑑 0 π‘Ž 𝑑 0 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\tfrac{1}{2}\pi,\Re(b-1+t)>% 0,\Re t>0,\Re(a+t)>0}} 
KummerU(a, b, z) = ((z)^(1 - b)* exp(z))/(2*Pi*I)*int((GAMMA(b - 1 + t)*GAMMA(t))/(GAMMA(a + t))*(z)^(- t), t = - I*infinity..I*infinity)

HypergeometricU[a, b, z] == Divide[(z)^(1 - b)* Exp[z],2*Pi*I]*Integrate[Divide[Gamma[b - 1 + t]*Gamma[t],Gamma[a + t]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
Retrieved from "https://lct.wmflabs.org/w/index.php?title=13.4&oldid=58254"