Parabolic Cylinder Functions - 13.2 Definitions and Basic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.2.E1 z ⁢ d 2 w d z 2 + ( b - z ) ⁢ d w d z - a ⁢ w = 0 𝑧 derivative 𝑤 𝑧 2 𝑏 𝑧 derivative 𝑤 𝑧 𝑎 𝑤 0 {\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 M ⁡ ( a , b , z ) = ∑ s = 0 ∞ ( a ) s ( b ) s ⁢ s ! ⁢ z s Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑏 𝑠 𝑠 superscript 𝑧 𝑠 {\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 𝐌 ⁡ ( a , b , z ) = ∑ s = 0 ∞ ( a ) s Γ ⁡ ( b + s ) ⁢ s ! ⁢ z s Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 𝑠 superscript 𝑧 𝑠 {\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}		 ℜ ⁡ ( b + s ) > 0 𝑏 𝑠 0 {\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 M ⁡ ( a , b , z ) = Γ ⁡ ( b ) ⁢ 𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 Euler-Gamma 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\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}		 ℜ ⁡ b > 0 , ℜ ⁡ ( b + s ) > 0 formulae-sequence 𝑏 0 𝑏 𝑠 0 {\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 lim b → - n ⁡ M ⁡ ( a , b , z ) Γ ⁡ ( b ) = 𝐌 ⁡ ( a , - n , z ) subscript → 𝑏 𝑛 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 Euler-Gamma 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑛 𝑧 {\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}		 ℜ ⁡ b > 0 , ℜ ⁡ ( ( - n ) + s ) > 0 formulae-sequence 𝑏 0 𝑛 𝑠 0 {\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 𝐌 ⁡ ( a , - n , z ) = ( a ) n + 1 ( n + 1 ) ! ⁢ z n + 1 ⁢ M ⁡ ( a + n + 1 , n + 2 , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑛 𝑧 Pochhammer 𝑎 𝑛 1 𝑛 1 superscript 𝑧 𝑛 1 Kummer-confluent-hypergeometric-M 𝑎 𝑛 1 𝑛 2 𝑧 {\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}		 ℜ ⁡ b > 0 , ℜ ⁡ ( ( - n ) + s ) > 0 formulae-sequence 𝑏 0 𝑛 𝑠 0 {\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 U ⁡ ( - m , b , z ) = ( - 1 ) m ⁢ ( b ) m ⁢ M ⁡ ( - m , b , z ) Kummer-confluent-hypergeometric-U 𝑚 𝑏 𝑧 superscript 1 𝑚 Pochhammer 𝑏 𝑚 Kummer-confluent-hypergeometric-M 𝑚 𝑏 𝑧 {\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 ( - 1 ) m ⁢ ( b ) m ⁢ M ⁡ ( - m , b , z ) = ( - 1 ) m ⁢ ∑ s = 0 m ( m s ) ⁢ ( b + s ) m - s ⁢ ( - z ) s superscript 1 𝑚 Pochhammer 𝑏 𝑚 Kummer-confluent-hypergeometric-M 𝑚 𝑏 𝑧 superscript 1 𝑚 superscript subscript 𝑠 0 𝑚 binomial 𝑚 𝑠 Pochhammer 𝑏 𝑠 𝑚 𝑠 superscript 𝑧 𝑠 {\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 U ⁡ ( a , a + n + 1 , z ) = ( - 1 ) n ⁢ ( 1 - a - n ) n z a + n ⁢ M ⁡ ( - n , 1 - a - n , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑎 𝑛 1 𝑧 superscript 1 𝑛 Pochhammer 1 𝑎 𝑛 𝑛 superscript 𝑧 𝑎 𝑛 Kummer-confluent-hypergeometric-M 𝑛 1 𝑎 𝑛 𝑧 {\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 ( - 1 ) n ⁢ ( 1 - a - n ) n z a + n ⁢ M ⁡ ( - n , 1 - a - n , z ) = z - a ⁢ ∑ s = 0 n ( n s ) ⁢ ( a ) s ⁢ z - s superscript 1 𝑛 Pochhammer 1 𝑎 𝑛 𝑛 superscript 𝑧 𝑎 𝑛 Kummer-confluent-hypergeometric-M 𝑛 1 𝑎 𝑛 𝑧 superscript 𝑧 𝑎 superscript subscript 𝑠 0 𝑛 binomial 𝑛 𝑠 Pochhammer 𝑎 𝑠 superscript 𝑧 𝑠 {\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 U ⁡ ( a , n + 1 , z ) = ( - 1 ) n + 1 n ! ⁢ Γ ⁡ ( a - n ) ⁢ ∑ k = 0 ∞ ( a ) k ( n + 1 ) k ⁢ k ! ⁢ z k ⁢ ( ln ⁡ z + ψ ⁡ ( a + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k + 1 ) ) + 1 Γ ⁡ ( a ) ⁢ ∑ k = 1 n ( k - 1 ) ! ⁢ ( 1 - a + k ) n - k ( n - k ) ! ⁢ z - k Kummer-confluent-hypergeometric-U 𝑎 𝑛 1 𝑧 superscript 1 𝑛 1 𝑛 Euler-Gamma 𝑎 𝑛 superscript subscript 𝑘 0 Pochhammer 𝑎 𝑘 Pochhammer 𝑛 1 𝑘 𝑘 superscript 𝑧 𝑘 𝑧 digamma 𝑎 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 1 1 Euler-Gamma 𝑎 superscript subscript 𝑘 1 𝑛 𝑘 1 Pochhammer 1 𝑎 𝑘 𝑛 𝑘 𝑛 𝑘 superscript 𝑧 𝑘 {\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}		 ℜ ⁡ ( a - n ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑎 𝑛 0 𝑎 0 {\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 U ⁡ ( - m , n + 1 , z ) = ( - 1 ) m ⁢ ( n + 1 ) m ⁢ M ⁡ ( - m , n + 1 , z ) Kummer-confluent-hypergeometric-U 𝑚 𝑛 1 𝑧 superscript 1 𝑚 Pochhammer 𝑛 1 𝑚 Kummer-confluent-hypergeometric-M 𝑚 𝑛 1 𝑧 {\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 ( - 1 ) m ⁢ ( n + 1 ) m ⁢ M ⁡ ( - m , n + 1 , z ) = ( - 1 ) m ⁢ ∑ s = 0 m ( m s ) ⁢ ( n + s + 1 ) m - s ⁢ ( - z ) s superscript 1 𝑚 Pochhammer 𝑛 1 𝑚 Kummer-confluent-hypergeometric-M 𝑚 𝑛 1 𝑧 superscript 1 𝑚 superscript subscript 𝑠 0 𝑚 binomial 𝑚 𝑠 Pochhammer 𝑛 𝑠 1 𝑚 𝑠 superscript 𝑧 𝑠 {\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 U ⁡ ( a , - n , z ) = z n + 1 ⁢ U ⁡ ( a + n + 1 , n + 2 , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑛 𝑧 superscript 𝑧 𝑛 1 Kummer-confluent-hypergeometric-U 𝑎 𝑛 1 𝑛 2 𝑧 {\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 U ⁡ ( a , b , z ⁢ e 2 ⁢ π ⁢ i ⁢ m ) = 2 ⁢ π ⁢ i ⁢ e - π ⁢ i ⁢ b ⁢ m ⁢ sin ⁡ ( π ⁢ b ⁢ m ) Γ ⁡ ( 1 + a - b ) ⁢ sin ⁡ ( π ⁢ b ) ⁢ 𝐌 ⁡ ( a , b , z ) + e - 2 ⁢ π ⁢ i ⁢ b ⁢ m ⁢ U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑒 2 𝜋 imaginary-unit 𝑚 2 𝜋 imaginary-unit superscript 𝑒 𝜋 imaginary-unit 𝑏 𝑚 𝜋 𝑏 𝑚 Euler-Gamma 1 𝑎 𝑏 𝜋 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑒 2 𝜋 imaginary-unit 𝑏 𝑚 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\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}		 ℜ ⁡ ( 1 + a - b ) > 0 , ℜ ⁡ ( b + s ) > 0 formulae-sequence 1 𝑎 𝑏 0 𝑏 𝑠 0 {\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 𝒲 ⁡ { 𝐌 ⁡ ( a , b , z ) , z 1 - b ⁢ 𝐌 ⁡ ( a - b + 1 , 2 - b , z ) } = sin ⁡ ( π ⁢ b ) ⁢ z - b ⁢ e z / π Wronskian Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 1 2 𝑏 𝑧 𝜋 𝑏 superscript 𝑧 𝑏 superscript 𝑒 𝑧 𝜋 {\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		 ℜ ⁡ ( b + s ) > 0 , ℜ ⁡ ( ( 2 - b ) + s ) > 0 formulae-sequence 𝑏 𝑠 0 2 𝑏 𝑠 0 {\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 𝒲 ⁡ { 𝐌 ⁡ ( a , b , z ) , U ⁡ ( a , b , z ) } = - z - b ⁢ e z / Γ ⁡ ( a ) Wronskian Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑧 𝑏 superscript 𝑒 𝑧 Euler-Gamma 𝑎 {\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}}		 ℜ ⁡ a > 0 , ℜ ⁡ ( b + s ) > 0 formulae-sequence 𝑎 0 𝑏 𝑠 0 {\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 𝒲 ⁡ { 𝐌 ⁡ ( a , b , z ) , e z ⁢ U ⁡ ( b - a , b , e + π ⁢ i ⁢ z ) } = e - b ⁢ π ⁢ i ⁢ z - b ⁢ e z / Γ ⁡ ( b - a ) Wronskian Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑒 𝑏 𝜋 imaginary-unit superscript 𝑧 𝑏 superscript 𝑒 𝑧 Euler-Gamma 𝑏 𝑎 {\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}}		 ℜ ⁡ ( b - a ) > 0 , ℜ ⁡ ( b + s ) > 0 formulae-sequence 𝑏 𝑎 0 𝑏 𝑠 0 {\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 𝒲 ⁡ { 𝐌 ⁡ ( a , b , z ) , e z ⁢ U ⁡ ( b - a , b , e - π ⁢ i ⁢ z ) } = e + b ⁢ π ⁢ i ⁢ z - b ⁢ e z / Γ ⁡ ( b - a ) Wronskian Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑒 𝑏 𝜋 imaginary-unit superscript 𝑧 𝑏 superscript 𝑒 𝑧 Euler-Gamma 𝑏 𝑎 {\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}}		 ℜ ⁡ ( b - a ) > 0 , ℜ ⁡ ( b + s ) > 0 formulae-sequence 𝑏 𝑎 0 𝑏 𝑠 0 {\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 𝒲 ⁡ { z 1 - b ⁢ 𝐌 ⁡ ( a - b + 1 , 2 - b , z ) , U ⁡ ( a , b , z ) } = - z - b ⁢ e z / Γ ⁡ ( a - b + 1 ) Wronskian superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 1 2 𝑏 𝑧 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑧 𝑏 superscript 𝑒 𝑧 Euler-Gamma 𝑎 𝑏 1 {\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}}		 ℜ ⁡ ( a - b + 1 ) > 0 , ℜ ⁡ ( ( 2 - b ) + s ) > 0 formulae-sequence 𝑎 𝑏 1 0 2 𝑏 𝑠 0 {\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 𝒲 ⁡ { z 1 - b ⁢ 𝐌 ⁡ ( a - b + 1 , 2 - b , z ) , e z ⁢ U ⁡ ( b - a , b , e + π ⁢ i ⁢ z ) } = - z - b ⁢ e z / Γ ⁡ ( 1 - a ) Wronskian superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 1 2 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑧 𝑏 superscript 𝑒 𝑧 Euler-Gamma 1 𝑎 {\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}}		 ℜ ⁡ ( 1 - a ) > 0 , ℜ ⁡ ( ( 2 - b ) + s ) > 0 formulae-sequence 1 𝑎 0 2 𝑏 𝑠 0 {\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 𝒲 ⁡ { z 1 - b ⁢ 𝐌 ⁡ ( a - b + 1 , 2 - b , z ) , e z ⁢ U ⁡ ( b - a , b , e - π ⁢ i ⁢ z ) } = - z - b ⁢ e z / Γ ⁡ ( 1 - a ) Wronskian superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 1 2 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑧 𝑏 superscript 𝑒 𝑧 Euler-Gamma 1 𝑎 {\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}}		 ℜ ⁡ ( 1 - a ) > 0 , ℜ ⁡ ( ( 2 - b ) + s ) > 0 formulae-sequence 1 𝑎 0 2 𝑏 𝑠 0 {\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 𝒲 ⁡ { U ⁡ ( a , b , z ) , e z ⁢ U ⁡ ( b - a , b , e + π ⁢ i ⁢ z ) } = e + ( a - b ) ⁢ π ⁢ i ⁢ z - b ⁢ e z Wronskian Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑒 𝑎 𝑏 𝜋 imaginary-unit superscript 𝑧 𝑏 superscript 𝑒 𝑧 {\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 𝒲 ⁡ { U ⁡ ( a , b , z ) , e z ⁢ U ⁡ ( b - a , b , e - π ⁢ i ⁢ z ) } = e - ( a - b ) ⁢ π ⁢ i ⁢ z - b ⁢ e z Wronskian Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 superscript 𝑒 𝑎 𝑏 𝜋 imaginary-unit superscript 𝑧 𝑏 superscript 𝑒 𝑧 {\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 M ⁡ ( a , b , z ) = e z ⁢ M ⁡ ( b - a , b , - z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-M 𝑏 𝑎 𝑏 𝑧 {\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 U ⁡ ( a , b , z ) = z 1 - b ⁢ U ⁡ ( a - b + 1 , 2 - b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-U 𝑎 𝑏 1 2 𝑏 𝑧 {\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 1 Γ ⁡ ( b ) ⁢ M ⁡ ( a , b , z ) = e - a ⁢ π ⁢ i Γ ⁡ ( b - a ) ⁢ U ⁡ ( a , b , z ) + e + ( b - a ) ⁢ π ⁢ i Γ ⁡ ( a ) ⁢ e z ⁢ U ⁡ ( b - a , b , e + π ⁢ i ⁢ z ) 1 Euler-Gamma 𝑏 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 superscript 𝑒 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑏 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑒 𝑏 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑎 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 {\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}		 ℜ ⁡ b > 0 , ℜ ⁡ ( b - a ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑏 0 formulae-sequence 𝑏 𝑎 0 𝑎 0 {\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 1 Γ ⁡ ( b ) ⁢ M ⁡ ( a , b , z ) = e + a ⁢ π ⁢ i Γ ⁡ ( b - a ) ⁢ U ⁡ ( a , b , z ) + e - ( b - a ) ⁢ π ⁢ i Γ ⁡ ( a ) ⁢ e z ⁢ U ⁡ ( b - a , b , e - π ⁢ i ⁢ z ) 1 Euler-Gamma 𝑏 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 superscript 𝑒 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑏 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑒 𝑏 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑎 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 superscript 𝑒 𝜋 imaginary-unit 𝑧 {\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}		 ℜ ⁡ b > 0 , ℜ ⁡ ( b - a ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑏 0 formulae-sequence 𝑏 𝑎 0 𝑎 0 {\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 U ⁡ ( a , b , z ) = Γ ⁡ ( 1 - b ) Γ ⁡ ( a - b + 1 ) ⁢ M ⁡ ( a , b , z ) + Γ ⁡ ( b - 1 ) Γ ⁡ ( a ) ⁢ z 1 - b ⁢ M ⁡ ( a - b + 1 , 2 - b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 Euler-Gamma 1 𝑏 Euler-Gamma 𝑎 𝑏 1 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 Euler-Gamma 𝑏 1 Euler-Gamma 𝑎 superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-M 𝑎 𝑏 1 2 𝑏 𝑧 {\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}		 ℜ ⁡ ( 1 - b ) > 0 , ℜ ⁡ ( a - b + 1 ) > 0 , ℜ ⁡ ( b - 1 ) > 0 , ℜ ⁡ a > 0 formulae-sequence 1 𝑏 0 formulae-sequence 𝑎 𝑏 1 0 formulae-sequence 𝑏 1 0 𝑎 0 {\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 - -
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