13.4: Difference between revisions

Latest revision as of 11:32, 28 June 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
13.4.E1	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(a\right)\Gamma\left(b-a\right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}% \mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\Re b>\Re a,\Re a>0,\Re(b-a)>0,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(a)GAMMA(b - a))int(exp(zt)(t)^(a - 1)*(1 - t)^(b - a - 1), t = 0..1)	Hypergeometric1F1Regularized[a, b, z] == Divide[1,Gamma[a]Gamma[b - a]]Integrate[Exp[zt](t)^(a - 1)*(1 - t)^(b - a - 1), {t, 0, 1}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 21]
13.4.E2	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(b-c\right)}\int_{0}^{1}{\mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-% 1}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\Re b>\Re c,\Re c>0,\Re(b-c)>0,\Re(b+s)>0,\Re(c+s)% >0}}$	KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(b - c))int(KummerM(a, c, zt)/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, zt](t)^(c - 1)(1 - t)^(b - c - 1), {t, 0, 1}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 126]
13.4.E3	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{% 2}-\frac{1}{2}b}}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}% b-\frac{1}{2}}J_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\Re a>0,\Re((b-1)+k+1)>0,\Re(b+s)>0}}$	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, 2sqrt(zt)), 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, 2Sqrt[zt]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Error	Skipped - Because timed out
13.4.E4	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)% }\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}`	${\displaystyle{\displaystyle\Re a>0,\|\operatorname{ph}{z}\|<\frac{1}{2}\pi}}$	KummerU(a, b, z) = (1)/(GAMMA(a))int(exp(- zt)(t)^(a - 1)(1 + t)^(b - a - 1), t = 0..infinity)	HypergeometricU[a, b, z] == Divide[1,Gamma[a]]Integrate[Exp[- zt](t)^(a - 1)(1 + t)^(b - a - 1), {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 90]
13.4.E5	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a% \right)\Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^% {-t}t^{a-1}}{t+z}\mathrm{d}t}}$ `\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}`	${\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	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{% \Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t% \right)e^{-t}t^{b+n-1}}{t+z}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\pi,-\Re b<n,n<1+% \Re\left(a-b\right),\Re(1+a-b)>0,\Re(b+s)>0}}$	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	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1% }{2}b}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}\\int_{0}^{\infty}e^{-t}% t^{a-\frac{1}{2}b-\frac{1}{2}}K_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ `\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}`	${\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, 2sqrt(zt)), 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, 2Sqrt[zt]], {t, 0, Infinity}, GenerateConditions->None]	Successful	Aborted	-	Skipped - Because timed out
13.4.E8	${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{c-a}\\int_{0}^{\infty}e^{% -zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left(a,a-b+1;c;-t\right)\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\frac{1}{2}\pi}}$	KummerU(a, b, z) = (z)^(c - a)* int(exp(- zt)(t)^(c - 1)* hypergeom([a , a - b + 1], [c], - t), t = 0..infinity)	HypergeometricU[a, b, z] == (z)^(c - a)* Integrate[Exp[- zt](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 = 1/23^(1/2)+1/2I} Result: Float(undefined)+Float(undefined)I Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2-1/2I3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
13.4.E9	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1% +a-b\right)}{2\pi\mathrm{i}\Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(% t-1)^{b-a-1}}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\Re a>0,\Re(1+a-b)>0,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = (GAMMA(1 + a - b))/(2PiIGAMMA(a))int(exp(zt)(t)^(a - 1)*(t - 1)^(b - a - 1), t = 0..(1 +))	Hypergeometric1F1Regularized[a, b, z] == Divide[Gamma[1 + a - b],2PiIGamma[a]]Integrate[Exp[zt](t)^(a - 1)*(t - 1)^(b - a - 1), {t, 0, (1 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E10	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-a\pi\mathrm{i}}% \frac{\Gamma\left(1-a\right)}{2\pi\mathrm{i}\Gamma\left(b-a\right)}\int_{1}^{(% 0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\Re\left(b-a\right)>0,\Re(1-a)>0,\Re(b-a)>0,\Re(b+% s)>0}}$	KummerM(a, b, z)/GAMMA(b) = exp(- aPiI)(GAMMA(1 - a))/(2PiIGAMMA(b - a))int(exp(zt)(t)^(a - 1)(1 - t)^(b - a - 1), t = 1..(0 +))	Hypergeometric1F1Regularized[a, b, z] == Exp[- aPiI]Divide[Gamma[1 - a],2PiIGamma[b - a]]Integrate[Exp[zt](t)^(a - 1)(1 - t)^(b - a - 1), {t, 1, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E11	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-b\pi\mathrm{i}}% \Gamma\left(1-a\right)\Gamma\left(1+a-b\right)\\frac{1}{4\pi^{2}}\int_{\alpha% }^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\Re(1-a)>0,\Re(1+a-b)>0,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = exp(- bPiI)GAMMA(1 - a)GAMMA(1 + a - b)(1)/(4(Pi)^(2))int(exp(zt)(t)^(a - 1)(1 - t)^(b - a - 1), t = alpha..(0 + , 1 + , 0 - , 1 -))	Hypergeometric1F1Regularized[a, b, z] == Exp[- bPiI]Gamma[1 - a]Gamma[1 + a - b]Divide[1,4(Pi)^(2)]Integrate[Exp[zt](t)^(a - 1)(1 - t)^(b - a - 1), {t, \[Alpha], (0 + , 1 + , 0 - , 1 -)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E12	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b% \right)}{2\pi\mathrm{i}}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;c;\ifrac{1}{t}\right)\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi,\Re b% >0,\Re(c+s)>0}}$	KummerM(a, c, z)/GAMMA(c) = (GAMMA(b))/(2PiI)(z)^(1 - b) int(exp(zt)(t)^(- b)* hypergeom([a , b], [c], (1)/(t)), t = - infinity..(0 + , 1 +))	Hypergeometric1F1Regularized[a, c, z] == Divide[Gamma[b],2PiI](z)^(1 - b) Integrate[Exp[zt](t)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[1,t]], {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E13	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi% \mathrm{i}}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-% a}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\frac{1}{2}\pi,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = ((z)^(1 - b))/(2PiI)int(exp(zt)(t)^(- b)(1 -(1)/(t))^(- a), t = - infinity..(0 + , 1 +))	Hypergeometric1F1Regularized[a, b, z] == Divide[(z)^(1 - b),2PiI]Integrate[Exp[zt](t)^(- b)(1 -Divide[1,t])^(- a), {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E14	${\displaystyle{\displaystyle U\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{% \Gamma\left(1-a\right)}{2\pi\mathrm{i}}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t% )^{b-a-1}}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi,\Re% (1-a)>0}}$	KummerU(a, b, z) = exp(- aPiI)(GAMMA(1 - a))/(2PiI)int(exp(- zt)(t)^(a - 1)*(1 + t)^(b - a - 1), t = infinity..(0 +))	HypergeometricU[a, b, z] == Exp[- aPiI]Divide[Gamma[1 - a],2PiI]Integrate[Exp[- zt](t)^(a - 1)*(1 + t)^(b - a - 1), {t, Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E15	${\displaystyle{\displaystyle\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)% \Gamma\left(c-b+1\right)}=\frac{z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e% ^{zt}t^{-c}{{}_{2}{\mathbf{F}}_{1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)% \mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi,\Re c% >0,\Re(c-b+1)>0}}$	(KummerU(a, b, z))/(GAMMA(c)GAMMA(c - b + 1)) = ((z)^(1 - c))/(2PiI)int(exp(zt)(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),2PiI]Integrate[Exp[zt](t)^(- c)* HypergeometricPFQRegularized[{a , c}, {a + c - b + 1}, 1 -Divide[1,t]], {t, - Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E16	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi% \mathrm{i}\Gamma\left(a\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}% \frac{\Gamma\left(a+t\right)\Gamma\left(-t\right)}{\Gamma\left(b+t\right)}z^{t% }\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi,\Re a>0,\Re% (a+t)>0,\Re(-t)>0,\Re(b+t)>0,\Re(b+s)>0}}$	KummerM(a, b, - z)/GAMMA(b) = (1)/(2PiIGAMMA(a))int((GAMMA(a + t)GAMMA(- t))/(GAMMA(b + t))(z)^(t), t = - Iinfinity..Iinfinity)	Hypergeometric1F1Regularized[a, b, - z] == Divide[1,2PiIGamma[a]]Integrate[Divide[Gamma[a + t]Gamma[- t],Gamma[b + t]](z)^(t), {t, - IInfinity, IInfinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.4.E17	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{-a}}{2\pi\mathrm{i}}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(1+a-b+t\right)\Gamma\left(-t\right)}{\Gamma\left(a\right)\Gamma\left(1+a% -b\right)}z^{-t}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\tfrac{3}{2}\pi,\Re(a+t)>0,% \Re(1+a-b+t)>0,\Re(-t)>0,\Re a>0,\Re(1+a-b)>0}}$	KummerU(a, b, z) = ((z)^(- a))/(2PiI)int((GAMMA(a + t)GAMMA(1 + a - b + t)GAMMA(- t))/(GAMMA(a)GAMMA(1 + a - b))(z)^(- t), t = - Iinfinity..I*infinity)	HypergeometricU[a, b, z] == Divide[(z)^(- a),2PiI]Integrate[Divide[Gamma[a + t]Gamma[1 + a - b + t]Gamma[- t],Gamma[a]Gamma[1 + a - b]](z)^(- t), {t, - IInfinity, I*Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.4.E18	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-b}e^{z}}{2\pi% \mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(b-1+t% \right)\Gamma\left(t\right)}{\Gamma\left(a+t\right)}z^{-t}\mathrm{d}t}}$ `\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}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi,\Re(b-1+t)>% 0,\Re t>0,\Re(a+t)>0}}$	KummerU(a, b, z) = ((z)^(1 - b)* exp(z))/(2PiI)int((GAMMA(b - 1 + t)GAMMA(t))/(GAMMA(a + t))(z)^(- t), t = - Iinfinity..I*infinity)	HypergeometricU[a, b, z] == Divide[(z)^(1 - b)* Exp[z],2PiI]Integrate[Divide[Gamma[b - 1 + t]Gamma[t],Gamma[a + t]](z)^(- t), {t, - IInfinity, I*Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out

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-| [https://dlmf.nist.gov/13.4.E1 13.4.E1] || [[Item:Q4364|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{b} > \realpart@@{a}, \realpart@@{a} > 0, \realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
+| [https://dlmf.nist.gov/13.4.E1 13.4.E1] || <math qid="Q4364">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{b} > \realpart@@{a}, \realpart@@{a} > 0, \realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
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-| [https://dlmf.nist.gov/13.4.E2 13.4.E2] || [[Item:Q4365|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{b} > \realpart@@{c}, \realpart@@{c} > 0, \realpart@@{(b-c)} > 0, \realpart@@{(b+s)} > 0, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 126]
+| [https://dlmf.nist.gov/13.4.E2 13.4.E2] || <math qid="Q4365">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{b} > \realpart@@{c}, \realpart@@{c} > 0, \realpart@@{(b-c)} > 0, \realpart@@{(b+s)} > 0, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 126]
 |-
-| [https://dlmf.nist.gov/13.4.E3 13.4.E3] || [[Item:Q4366|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{((b-1)+k+1)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
+| [https://dlmf.nist.gov/13.4.E3 13.4.E3] || <math qid="Q4366">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{((b-1)+k+1)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/13.4.E4 13.4.E4] || [[Item:Q4367|<math>\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0, |\phase{z}| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (1)/(GAMMA(a))*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 90]
+| [https://dlmf.nist.gov/13.4.E4 13.4.E4] || <math qid="Q4367">\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0, |\phase{z}| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (1)/(GAMMA(a))*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 90]
 |-
-| [https://dlmf.nist.gov/13.4.E5 13.4.E5] || [[Item:Q4368|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \pi, \realpart@@{a} > \max\left(\realpart@@{b-1}, \realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+| [https://dlmf.nist.gov/13.4.E5 13.4.E5] || <math qid="Q4368">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \pi, \realpart@@{a} > \max\left(\realpart@@{b-1}, \realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/13.4.E6 13.4.E6] || [[Item:Q4369|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \pi, -\realpart@@{b} < n, n < 1+\realpart@{a-b}, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+| [https://dlmf.nist.gov/13.4.E6 13.4.E6] || <math qid="Q4369">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \pi, -\realpart@@{b} < n, n < 1+\realpart@{a-b}, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/13.4.E7 13.4.E7] || [[Item:Q4370|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{a} > \max\left(\realpart@@{b-1}, \realpart@@{a} > 0, \realpart@@{(a-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
+| [https://dlmf.nist.gov/13.4.E7 13.4.E7] || <math qid="Q4370">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{a} > \max\left(\realpart@@{b-1}, \realpart@@{a} > 0, \realpart@@{(a-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/13.4.E8 13.4.E8] || [[Item:Q4371|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (z)^(c - a)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([a , a - b + 1], [c], - t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [294 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+| [https://dlmf.nist.gov/13.4.E8 13.4.E8] || <math qid="Q4371">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (z)^(c - a)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([a , a - b + 1], [c], - t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [294 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/13.4.E9 13.4.E9] || [[Item:Q4372|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/13.4.E9 13.4.E9] || <math qid="Q4372">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/13.4.E10 13.4.E10] || [[Item:Q4373|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@{b-a} > 0, \realpart@@{(1-a)} > 0, \realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/13.4.E10 13.4.E10] || <math qid="Q4373">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@{b-a} > 0, \realpart@@{(1-a)} > 0, \realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/13.4.E11 13.4.E11] || [[Item:Q4374|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{(1-a)} > 0, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 -))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/13.4.E11 13.4.E11] || <math qid="Q4374">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{(1-a)} > 0, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 -))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/13.4.E12 13.4.E12] || [[Item:Q4375|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{b} > 0, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/13.4.E12 13.4.E12] || <math qid="Q4375">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{b} > 0, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/13.4.E13 13.4.E13] || [[Item:Q4376|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \frac{1}{2}\pi, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/13.4.E13 13.4.E13] || <math qid="Q4376">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \frac{1}{2}\pi, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/13.4.E14 13.4.E14] || [[Item:Q4377|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{(1-a)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/13.4.E14 13.4.E14] || <math qid="Q4377">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{(1-a)} > 0</math> || <syntaxhighlight lang=mathematica>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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/13.4.E15 13.4.E15] || [[Item:Q4378|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{c} > 0, \realpart@@{(c-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>(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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/13.4.E15 13.4.E15] || <math qid="Q4378">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{c} > 0, \realpart@@{(c-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>(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 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/13.4.E16 13.4.E16] || [[Item:Q4379|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{a} > 0, \realpart@@{(a+t)} > 0, \realpart@@{(-t)} > 0, \realpart@@{(b+t)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+| [https://dlmf.nist.gov/13.4.E16 13.4.E16] || <math qid="Q4379">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{a} > 0, \realpart@@{(a+t)} > 0, \realpart@@{(-t)} > 0, \realpart@@{(b+t)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/13.4.E17 13.4.E17] || [[Item:Q4380|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{3}{2}\pi, \realpart@@{(a+t)} > 0, \realpart@@{(1+a-b+t)} > 0, \realpart@@{(-t)} > 0, \realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+| [https://dlmf.nist.gov/13.4.E17 13.4.E17] || <math qid="Q4380">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{3}{2}\pi, \realpart@@{(a+t)} > 0, \realpart@@{(1+a-b+t)} > 0, \realpart@@{(-t)} > 0, \realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/13.4.E18 13.4.E18] || [[Item:Q4381|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(b-1+t)} > 0, \realpart@@{t} > 0, \realpart@@{(a+t)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+| [https://dlmf.nist.gov/13.4.E18 13.4.E18] || <math qid="Q4381">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(b-1+t)} > 0, \realpart@@{t} > 0, \realpart@@{(a+t)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
 |}
 </div>

13.4: Difference between revisions

Latest revision as of 11:32, 28 June 2021

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