13.2

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
13.2.E1	${\displaystyle{\displaystyle z\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(b-z)\frac{\mathrm{d}w}{\mathrm{d}z}-aw=0}}$ z\deriv[2]{w}{z}+(b-z)\deriv{w}{z}-aw = 0		z*diff(w, [z$(2)])+(b - z)*diff(w, z)- a*w = 0	z*D[w, {z, 2}]+(b - z)*D[w, z]- a*w == 0	Failure	Failure	Failed [300 / 300] Result: 1.299038106+.7500000000*I Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: 1.299038106+.7500000000*I Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.299038105676658, 0.7499999999999999] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.299038105676658, 0.7499999999999999] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E2	${\displaystyle{\displaystyle M\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{{\left(b\right)_{s}}s!}z^{s}}}$ \KummerconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\Pochhammersym{b}{s}s!}z^{s}		KummerM(a, b, z) = sum((pochhammer(a, s))/(pochhammer(b, s)*factorial(s))*(z)^(s), s = 0..infinity)	Hypergeometric1F1[a, b, z] == Sum[Divide[Pochhammer[a, s],Pochhammer[b, s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 252]
13.2.E3	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{\Gamma\left(b+s\right)s!}z^{s}}}$ \OlverconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\EulerGamma@{b+s}s!}z^{s}	${\displaystyle{\displaystyle\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = sum((pochhammer(a, s))/(GAMMA(b + s)*factorial(s))*(z)^(s), s = 0..infinity)	Hypergeometric1F1Regularized[a, b, z] == Sum[Divide[Pochhammer[a, s],Gamma[b + s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Failed [35 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E4	${\displaystyle{\displaystyle M\left(a,b,z\right)=\Gamma\left(b\right){\mathbf{M}}\left(a,b,z\right)}}$ \KummerconfhyperM@{a}{b}{z} = \EulerGamma@{b}\OlverconfhyperM@{a}{b}{z}	${\displaystyle{\displaystyle\Re b>0,\Re(b+s)>0}}$	KummerM(a, b, z) = GAMMA(b)*KummerM(a, b, z)/GAMMA(b)	Hypergeometric1F1[a, b, z] == Gamma[b]*Hypergeometric1F1Regularized[a, b, z]	Successful	Successful	-	Successful [Tested: 126]
13.2.E5	${\displaystyle{\displaystyle\lim_{b\to-n}\frac{M\left(a,b,z\right)}{\Gamma\left(b\right)}={\mathbf{M}}\left(a,-n,z\right)}}$ \lim_{b\to-n}\frac{\KummerconfhyperM@{a}{b}{z}}{\EulerGamma@{b}} = \OlverconfhyperM@{a}{-n}{z}	${\displaystyle{\displaystyle\Re b>0,\Re((-n)+s)>0}}$	limit((KummerM(a, b, z))/(GAMMA(b)), b = - n) = KummerM(a, - n, z)/GAMMA(- n)	Limit[Divide[Hypergeometric1F1[a, b, z],Gamma[b]], b -> - n, GenerateConditions->None] == Hypergeometric1F1Regularized[a, - n, z]	Failure	Successful	Successful [Tested: 0]	Failed [112 / 126] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.2.E5	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,-n,z\right)=\frac{{\left(a\right)_{n+1}}}{(n+1)!}z^{n+1}M\left(a+n+1,n+2,z\right)}}$ \OlverconfhyperM@{a}{-n}{z} = \frac{\Pochhammersym{a}{n+1}}{(n+1)!}z^{n+1}\KummerconfhyperM@{a+n+1}{n+2}{z}	${\displaystyle{\displaystyle\Re b>0,\Re((-n)+s)>0}}$	KummerM(a, - n, z)/GAMMA(- n) = (pochhammer(a, n + 1))/(factorial(n + 1))*(z)^(n + 1)* KummerM(a + n + 1, n + 2, z)	Hypergeometric1F1Regularized[a, - n, z] == Divide[Pochhammer[a, n + 1],(n + 1)!]*(z)^(n + 1)* Hypergeometric1F1[a + n + 1, n + 2, z]	Failure	Failure	Error	Successful [Tested: 126]
13.2.E7	${\displaystyle{\displaystyle U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)}}$ \KummerconfhyperU@{-m}{b}{z} = (-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z}		KummerU(- m, b, z) = (- 1)^(m)* pochhammer(b, m)*KummerM(- m, b, z)	HypergeometricU[- m, b, z] == (- 1)^(m)* Pochhammer[b, m]*Hypergeometric1F1[- m, b, z]	Failure	Failure	Error	Failed [7 / 126] Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E7	${\displaystyle{\displaystyle(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)=(-1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(b+s\right)_{m-s}}(-z)^{s}}}$ (-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{b+s}{m-s}(-z)^{s}		(- 1)^(m)* pochhammer(b, m)*KummerM(- m, b, z) = (- 1)^(m)* sum(binomial(m,s)*pochhammer(b + s, m - s)*(- z)^(s), s = 0..m)	(- 1)^(m)* Pochhammer[b, m]*Hypergeometric1F1[- m, b, z] == (- 1)^(m)* Sum[Binomial[m,s]*Pochhammer[b + s, m - s]*(- z)^(s), {s, 0, m}, GenerateConditions->None]	Successful	Successful	Skip - symbolical successful subtest	Failed [21 / 126] Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.2.E8	${\displaystyle{\displaystyle U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}M\left(-n,1-a-n,z\right)}}$ \KummerconfhyperU@{a}{a+n+1}{z} = \frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z}		KummerU(a, a + n + 1, z) = ((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))*KummerM(- n, 1 - a - n, z)	HypergeometricU[a, a + n + 1, z] == Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]*Hypergeometric1F1[- n, 1 - a - n, z]	Failure	Failure	Error	Failed [7 / 126] Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E8	${\displaystyle{\displaystyle\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}M\left(-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{s}{\left(a\right)_{s}}z^{-s}}}$ \frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z} = z^{-a}\sum_{s=0}^{n}\binom{n}{s}\Pochhammersym{a}{s}z^{-s}		((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))*KummerM(- n, 1 - a - n, z) = (z)^(- a)* sum(binomial(n,s)*pochhammer(a, s)*(z)^(- s), s = 0..n)	Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]*Hypergeometric1F1[- n, 1 - a - n, z] == (z)^(- a)* Sum[Binomial[n,s]*Pochhammer[a, s]*(z)^(- s), {s, 0, n}, GenerateConditions->None]	Failure	Failure	Error	Failed [7 / 126] Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E9	${\displaystyle{\displaystyle U\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\Gamma\left(a-n\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+\psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right)+\frac{1}{\Gamma\left(a\right)}\sum_{k=1}^{n}\frac{(k-1)!{\left(1-a+k\right)_{n-k}}}{(n-k)!}z^{-k}}}$ \KummerconfhyperU@{a}{n+1}{z} = \frac{(-1)^{n+1}}{n!\EulerGamma@{a-n}}\sum_{k=0}^{\infty}\frac{\Pochhammersym{a}{k}}{\Pochhammersym{n+1}{k}k!}z^{k}\left(\ln@@{z}+\digamma@{a+k}-\digamma@{1+k}-\digamma@{n+k+1}\right)+\frac{1}{\EulerGamma@{a}}\sum_{k=1}^{n}\frac{(k-1)!\Pochhammersym{1-a+k}{n-k}}{(n-k)!}z^{-k}	${\displaystyle{\displaystyle\Re(a-n)>0,\Re a>0}}$	KummerU(a, n + 1, z) = ((- 1)^(n + 1))/(factorial(n)*GAMMA(a - n))*sum((pochhammer(a, k))/(pochhammer(n + 1, k)*factorial(k))*(z)^(k)*(ln(z)+ Psi(a + k)- Psi(1 + k)- Psi(n + k + 1)), k = 0..infinity)+(1)/(GAMMA(a))*sum((factorial(k - 1)*pochhammer(1 - a + k, n - k))/(factorial(n - k))*(z)^(- k), k = 1..n)	HypergeometricU[a, n + 1, z] == Divide[(- 1)^(n + 1),(n)!*Gamma[a - n]]*Sum[Divide[Pochhammer[a, k],Pochhammer[n + 1, k]*(k)!]*(z)^(k)*(Log[z]+ PolyGamma[a + k]- PolyGamma[1 + k]- PolyGamma[n + k + 1]), {k, 0, Infinity}, GenerateConditions->None]+Divide[1,Gamma[a]]*Sum[Divide[(k - 1)!*Pochhammer[1 - a + k, n - k],(n - k)!]*(z)^(- k), {k, 1, n}, GenerateConditions->None]	Aborted	Aborted	Failed [7 / 14] Result: Float(undefined)+Float(undefined)*I Test Values: {a = 2, z = 1/2*3^(1/2)+1/2*I, n = 1} Result: Float(undefined)+Float(undefined)*I Test Values: {a = 2, z = -1/2+1/2*I*3^(1/2), n = 1} ... skip entries to safe data	Skipped - Because timed out
13.2.E10	${\displaystyle{\displaystyle U\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_{m}}M\left(-m,n+1,z\right)}}$ \KummerconfhyperU@{-m}{n+1}{z} = (-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z}		KummerU(- m, n + 1, z) = (- 1)^(m)* pochhammer(n + 1, m)*KummerM(- m, n + 1, z)	HypergeometricU[- m, n + 1, z] == (- 1)^(m)* Pochhammer[n + 1, m]*Hypergeometric1F1[- m, n + 1, z]	Failure	Failure	Successful [Tested: 63]	Successful [Tested: 63]
13.2.E10	${\displaystyle{\displaystyle(-1)^{m}{\left(n+1\right)_{m}}M\left(-m,n+1,z\right)=(-1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(n+s+1\right)_{m-s}}(-z)^{s}}}$ (-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{n+s+1}{m-s}(-z)^{s}		(- 1)^(m)* pochhammer(n + 1, m)*KummerM(- m, n + 1, z) = (- 1)^(m)* sum(binomial(m,s)*pochhammer(n + s + 1, m - s)*(- z)^(s), s = 0..m)	(- 1)^(m)* Pochhammer[n + 1, m]*Hypergeometric1F1[- m, n + 1, z] == (- 1)^(m)* Sum[Binomial[m,s]*Pochhammer[n + s + 1, m - s]*(- z)^(s), {s, 0, m}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 63]	Successful [Tested: 63]
13.2.E11	${\displaystyle{\displaystyle U\left(a,-n,z\right)=z^{n+1}U\left(a+n+1,n+2,z\right)}}$ \KummerconfhyperU@{a}{-n}{z} = z^{n+1}\KummerconfhyperU@{a+n+1}{n+2}{z}		KummerU(a, - n, z) = (z)^(n + 1)* KummerU(a + n + 1, n + 2, z)	HypergeometricU[a, - n, z] == (z)^(n + 1)* HypergeometricU[a + n + 1, n + 2, z]	Failure	Successful	Successful [Tested: 126]	Successful [Tested: 126]
13.2.E12	${\displaystyle{\displaystyle U\left(a,b,ze^{2\pi\mathrm{i}m}\right)=\frac{2\pi\mathrm{i}e^{-\pi\mathrm{i}bm}\sin\left(\pi bm\right)}{\Gamma\left(1+a-b\right)\sin\left(\pi b\right)}{\mathbf{M}}\left(a,b,z\right)+e^{-2\pi\mathrm{i}bm}U\left(a,b,z\right)}}$ \KummerconfhyperU@{a}{b}{ze^{2\pi\iunit m}} = \frac{2\pi\iunit e^{-\pi\iunit bm}\sin@{\pi bm}}{\EulerGamma@{1+a-b}\sin@{\pi b}}\OlverconfhyperM@{a}{b}{z}+e^{-2\pi\iunit bm}\KummerconfhyperU@{a}{b}{z}	${\displaystyle{\displaystyle\Re(1+a-b)>0,\Re(b+s)>0}}$	KummerU(a, b, z*exp(2*Pi*I*m)) = (2*Pi*I*exp(- Pi*I*b*m)*sin(Pi*b*m))/(GAMMA(1 + a - b)*sin(Pi*b))*KummerM(a, b, z)/GAMMA(b)+ exp(- 2*Pi*I*b*m)*KummerU(a, b, z)	HypergeometricU[a, b, z*Exp[2*Pi*I*m]] == Divide[2*Pi*I*Exp[- Pi*I*b*m]*Sin[Pi*b*m],Gamma[1 + a - b]*Sin[Pi*b]]*Hypergeometric1F1Regularized[a, b, z]+ Exp[- 2*Pi*I*b*m]*HypergeometricU[a, b, z]	Failure	Failure	Failed [230 / 300] Result: -.101548209-1.031304846*I Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, m = 1} Result: -.101548218-1.031304823*I Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, m = 3} ... skip entries to safe data	Failed [230 / 300] Result: Complex[-0.10154820915393259, -1.0313048488210503] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.1015482091539317, -1.03130484882105] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.2.E33	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right)\right\}=\sin\left(\pi b\right)z^{-b}e^{z}/\pi}}$ \Wronskian@{\OlverconfhyperM@{a}{b}{z},z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z}} = \sin@{\pi b}z^{-b}e^{z}/\pi	${\displaystyle{\displaystyle\Re(b+s)>0,\Re((2-b)+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))*diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b)) = sin(Pi*b)*(z)^(- b)* exp(z)/Pi	Wronskian[{Hypergeometric1F1Regularized[a, b, z], (z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z]}, z] == Sin[Pi*b]*(z)^(- b)* Exp[z]/Pi	Failure	Failure	Error	Successful [Tested: 252]
13.2.E34	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),U\left(a,b,z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a\right)}}}$ \Wronskian@{\OlverconfhyperM@{a}{b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a}}	${\displaystyle{\displaystyle\Re a>0,\Re(b+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))*diff(KummerU(a, b, z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(KummerU(a, b, z)) = -((z)^(- b)* exp(z))/(GAMMA(a))	Wronskian[{Hypergeometric1F1Regularized[a, b, z], HypergeometricU[a, b, z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[a]]	Failure	Failure	Error	Successful [Tested: 126]
13.2.E35	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=\ifrac{e^{-b\pi\mathrm{i}}z^{-b}e^{z}}{\Gamma\left(b-a\right)}}}$ \Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = \ifrac{e^{- b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}	${\displaystyle{\displaystyle\Re(b-a)>0,\Re(b+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)) = (exp(- b*Pi*I)*(z)^(- b)* exp(z))/(GAMMA(b - a))	Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z] == Divide[Exp[- b*Pi*I]*(z)^(- b)* Exp[z],Gamma[b - a]]	Failure	Failure	Failed [23 / 105] Result: -.6693440963-2.281274239*I Test Values: {a = -3/2, b = 3/2, z = 1/2*3^(1/2)+1/2*I} Result: -.4620307839+.3929465556*I Test Values: {a = -3/2, b = 3/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [20 / 105] Result: Complex[-0.6693440961046373, -2.2812742393329124] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.46203078407110554, 0.39294655583435506] 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
13.2.E35	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=\ifrac{e^{+b\pi\mathrm{i}}z^{-b}e^{z}}{\Gamma\left(b-a\right)}}}$ \Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = \ifrac{e^{+ b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}	${\displaystyle{\displaystyle\Re(b-a)>0,\Re(b+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)) = (exp(+ b*Pi*I)*(z)^(- b)* exp(z))/(GAMMA(b - a))	Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z] == Divide[Exp[+ b*Pi*I]*(z)^(- b)* Exp[z],Gamma[b - a]]	Failure	Failure	Failed [53 / 105] Result: -1.068139482+1.255929884*I Test Values: {a = -3/2, b = 3/2, z = 1/2-1/2*I*3^(1/2)} Result: .1184211651-.4036057902*I Test Values: {a = -3/2, b = 3/2, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [50 / 105] Result: Complex[-1.0681394822800954, 1.2559298845291709] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} Result: Complex[0.11842116492450601, -0.40360579036441874] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.2.E36	${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),U\left(a,b,z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a-b+1\right)}}}$ \Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a-b+1}}	${\displaystyle{\displaystyle\Re(a-b+1)>0,\Re((2-b)+s)>0}}$	((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(KummerU(a, b, z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(KummerU(a, b, z)) = -((z)^(- b)* exp(z))/(GAMMA(a - b + 1))	Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], HypergeometricU[a, b, z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[a - b + 1]]	Failure	Failure	Error	Successful [Tested: 161]
13.2.E37	${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),e^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(1-a\right)}}}$ \Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}	${\displaystyle{\displaystyle\Re(1-a)>0,\Re((2-b)+s)>0}}$	((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)) = -((z)^(- b)* exp(z))/(GAMMA(1 - a))	Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[1 - a]]	Failure	Aborted	Error	Successful [Tested: 168]
13.2.E37	${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(1-a\right)}}}$ \Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}	${\displaystyle{\displaystyle\Re(1-a)>0,\Re((2-b)+s)>0}}$	((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)) = -((z)^(- b)* exp(z))/(GAMMA(1 - a))	Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[1 - a]]	Failure	Aborted	Error	Successful [Tested: 168]
13.2.E38	${\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=e^{+(a-b)\pi\mathrm{i}}z^{-b}e^{z}}}$ \Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = e^{+(a-b)\pi\iunit}z^{-b}e^{z}		(KummerU(a, b, z))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff(KummerU(a, b, z), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)) = exp(+(a - b)*Pi*I)*(z)^(- b)* exp(z)	Wronskian[{HypergeometricU[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z] == Exp[+(a - b)*Pi*I]*(z)^(- b)* Exp[z]	Failure	Aborted	Failed [38 / 252] Result: 4.753561418-.1121990572*I Test Values: {a = -3/2, b = -2, z = 1/2*3^(1/2)+1/2*I} Result: -1.142634185-.4073142366*I Test Values: {a = -3/2, b = -2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [32 / 252] Result: Complex[4.753561408836843, -0.1121990577209182] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.1426341834354088, -0.40731423683768475] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E38	${\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=e^{-(a-b)\pi\mathrm{i}}z^{-b}e^{z}}}$ \Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = e^{-(a-b)\pi\iunit}z^{-b}e^{z}		(KummerU(a, b, z))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff(KummerU(a, b, z), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)) = exp(-(a - b)*Pi*I)*(z)^(- b)* exp(z)	Wronskian[{HypergeometricU[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z] == Exp[-(a - b)*Pi*I]*(z)^(- b)* Exp[z]	Failure	Aborted	Failed [80 / 252] Result: .5941419621-3.243473855*I Test Values: {a = -3/2, b = -2, z = 1/2-1/2*I*3^(1/2)} Result: -.4376938533+.7184072077*I Test Values: {a = -3/2, b = -2, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [80 / 252] Result: Complex[0.5941419683502733, -3.243473853028733] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} Result: Complex[-0.4376938536795689, 0.7184072074542298] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.2.E39	${\displaystyle{\displaystyle M\left(a,b,z\right)=e^{z}M\left(b-a,b,-z\right)}}$ \KummerconfhyperM@{a}{b}{z} = e^{z}\KummerconfhyperM@{b-a}{b}{-z}		KummerM(a, b, z) = exp(z)*KummerM(b - a, b, - z)	Hypergeometric1F1[a, b, z] == Exp[z]*Hypergeometric1F1[b - a, b, - z]	Failure	Successful	Error	Failed [42 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E40	${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{1-b}U\left(a-b+1,2-b,z\right)}}$ \KummerconfhyperU@{a}{b}{z} = z^{1-b}\KummerconfhyperU@{a-b+1}{2-b}{z}		KummerU(a, b, z) = (z)^(1 - b)* KummerU(a - b + 1, 2 - b, z)	HypergeometricU[a, b, z] == (z)^(1 - b)* HypergeometricU[a - b + 1, 2 - b, z]	Successful	Successful	-	Successful [Tested: 252]
13.2.E41	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=\frac{e^{-a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^{+(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)}}$ \frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{- a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{+(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}	${\displaystyle{\displaystyle\Re b>0,\Re(b-a)>0,\Re a>0}}$	(1)/(GAMMA(b))*KummerM(a, b, z) = (exp(- a*Pi*I))/(GAMMA(b - a))*KummerU(a, b, z)+(exp(+(b - a)*Pi*I))/(GAMMA(a))*exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)	Divide[1,Gamma[b]]*Hypergeometric1F1[a, b, z] == Divide[Exp[- a*Pi*I],Gamma[b - a]]*HypergeometricU[a, b, z]+Divide[Exp[+(b - a)*Pi*I],Gamma[a]]*Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]	Failure	Failure	Failed [6 / 21] Result: 3.583210384+1.512741910*I Test Values: {a = 3/2, b = 2, z = 1/2*3^(1/2)+1/2*I} Result: 1.096602540+.7868998856*I Test Values: {a = 3/2, b = 2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [6 / 21] Result: Complex[3.583210382577498, 1.512741908514331] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.096602539454242, 0.7868998849931845] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E41	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=\frac{e^{+a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^{-(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)}}$ \frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{+ a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{-(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}	${\displaystyle{\displaystyle\Re b>0,\Re(b-a)>0,\Re a>0}}$	(1)/(GAMMA(b))*KummerM(a, b, z) = (exp(+ a*Pi*I))/(GAMMA(b - a))*KummerU(a, b, z)+(exp(-(b - a)*Pi*I))/(GAMMA(a))*exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)	Divide[1,Gamma[b]]*Hypergeometric1F1[a, b, z] == Divide[Exp[+ a*Pi*I],Gamma[b - a]]*HypergeometricU[a, b, z]+Divide[Exp[-(b - a)*Pi*I],Gamma[a]]*Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]	Failure	Failure	Failed [15 / 21] Result: 2.239690726-1.798422043*I Test Values: {a = 3/2, b = 2, z = 1/2-1/2*I*3^(1/2)} Result: .9984283068-.3592011980*I Test Values: {a = 3/2, b = 2, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [15 / 21] Result: Complex[2.239690726834086, -1.7984220417127512] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} Result: Complex[0.9984283065924617, -0.35920119796837185] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.2.E42	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}M\left(a,b,z\right)+\frac{\Gamma\left(b-1\right)}{\Gamma\left(a\right)}z^{1-b}M\left(a-b+1,2-b,z\right)}}$ \KummerconfhyperU@{a}{b}{z} = \frac{\EulerGamma@{1-b}}{\EulerGamma@{a-b+1}}\KummerconfhyperM@{a}{b}{z}+\frac{\EulerGamma@{b-1}}{\EulerGamma@{a}}z^{1-b}\KummerconfhyperM@{a-b+1}{2-b}{z}	${\displaystyle{\displaystyle\Re(1-b)>0,\Re(a-b+1)>0,\Re(b-1)>0,\Re a>0}}$	KummerU(a, b, z) = (GAMMA(1 - b))/(GAMMA(a - b + 1))*KummerM(a, b, z)+(GAMMA(b - 1))/(GAMMA(a))*(z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)	HypergeometricU[a, b, z] == Divide[Gamma[1 - b],Gamma[a - b + 1]]*Hypergeometric1F1[a, b, z]+Divide[Gamma[b - 1],Gamma[a]]*(z)^(1 - b)* Hypergeometric1F1[a - b + 1, 2 - b, z]	Successful	Successful	-	-