Incomplete Gamma and Related Functions - 8.5 Confluent Hypergeometric Representations

From testwiki
Revision as of 16:48, 25 May 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.5.E1 γ ⁡ ( a , z ) = a - 1 ⁢ z a ⁢ e - z ⁢ M ⁡ ( 1 , 1 + a , z ) incomplete-gamma 𝑎 𝑧 superscript 𝑎 1 superscript 𝑧 𝑎 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-M 1 1 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(a,z\right)=a^{-1}z^{a}e^{-z}M\left(1,1% +a,z\right)}}
\incgamma@{a}{z} = a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
GAMMA(a)-GAMMA(a, z) = (a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z)

Gamma[a, 0, z] == (a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z]
Successful Successful - Successful [Tested: 7]
8.5.E1 a - 1 ⁢ z a ⁢ e - z ⁢ M ⁡ ( 1 , 1 + a , z ) = a - 1 ⁢ z a ⁢ M ⁡ ( a , 1 + a , - z ) superscript 𝑎 1 superscript 𝑧 𝑎 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-M 1 1 𝑎 𝑧 superscript 𝑎 1 superscript 𝑧 𝑎 Kummer-confluent-hypergeometric-M 𝑎 1 𝑎 𝑧 {\displaystyle{\displaystyle a^{-1}z^{a}e^{-z}M\left(1,1+a,z\right)=a^{-1}z^{a% }M\left(a,1+a,-z\right)}}
a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z} = a^{-1}z^{a}\KummerconfhyperM@{a}{1+a}{-z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
(a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z) = (a)^(- 1)* (z)^(a)* KummerM(a, 1 + a, - z)

(a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z] == (a)^(- 1)* (z)^(a)* Hypergeometric1F1[a, 1 + a, - z]
Successful Successful - Successful [Tested: 7]
8.5.E2 γ * ⁡ ( a , z ) = e - z ⁢ 𝐌 ⁡ ( 1 , 1 + a , z ) incomplete-gamma-star 𝑎 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-bold-M 1 1 𝑎 𝑧 {\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=e^{-z}{\mathbf{M}}\left% (1,1+a,z\right)}}
\scincgamma@{a}{z} = e^{-z}\OlverconfhyperM@{1}{1+a}{z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a)

Error
Successful Missing Macro Error - -
8.5.E2 e - z ⁢ 𝐌 ⁡ ( 1 , 1 + a , z ) = 𝐌 ⁡ ( a , 1 + a , - z ) superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-bold-M 1 1 𝑎 𝑧 Kummer-confluent-hypergeometric-bold-M 𝑎 1 𝑎 𝑧 {\displaystyle{\displaystyle e^{-z}{\mathbf{M}}\left(1,1+a,z\right)={\mathbf{M% }}\left(a,1+a,-z\right)}}
e^{-z}\OlverconfhyperM@{1}{1+a}{z} = \OlverconfhyperM@{a}{1+a}{-z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a) = KummerM(a, 1 + a, - z)/GAMMA(1 + a)

Exp[- z]*Hypergeometric1F1Regularized[1, 1 + a, z] == Hypergeometric1F1Regularized[a, 1 + a, - z]
Successful Successful -
Failed [7 / 42]
Result: Indeterminate
Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


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

... skip entries to safe data
8.5.E3 Γ ⁡ ( a , z ) = e - z ⁢ U ⁡ ( 1 - a , 1 - a , z ) incomplete-Gamma 𝑎 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 1 𝑎 1 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(a,z\right)=e^{-z}U\left(1-a,1-a,z% \right)}}
\incGamma@{a}{z} = e^{-z}\KummerconfhyperU@{1-a}{1-a}{z}		 

GAMMA(a, z) = exp(- z)*KummerU(1 - a, 1 - a, z)

Gamma[a, z] == Exp[- z]*HypergeometricU[1 - a, 1 - a, z]
Successful Successful - Successful [Tested: 42]
8.5.E3 e - z ⁢ U ⁡ ( 1 - a , 1 - a , z ) = z a ⁢ e - z ⁢ U ⁡ ( 1 , 1 + a , z ) superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 1 𝑎 1 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 1 1 𝑎 𝑧 {\displaystyle{\displaystyle e^{-z}U\left(1-a,1-a,z\right)=z^{a}e^{-z}U\left(1% ,1+a,z\right)}}
e^{-z}\KummerconfhyperU@{1-a}{1-a}{z} = z^{a}e^{-z}\KummerconfhyperU@{1}{1+a}{z}		 

exp(- z)*KummerU(1 - a, 1 - a, z) = (z)^(a)* exp(- z)*KummerU(1, 1 + a, z)

Exp[- z]*HypergeometricU[1 - a, 1 - a, z] == (z)^(a)* Exp[- z]*HypergeometricU[1, 1 + a, z]
Successful Successful - Successful [Tested: 42]
8.5.E4 γ ⁡ ( a , z ) = a - 1 ⁢ z 1 2 ⁢ a - 1 2 ⁢ e - 1 2 ⁢ z ⁢ M 1 2 ⁢ a - 1 2 , 1 2 ⁢ a ⁡ ( z ) incomplete-gamma 𝑎 𝑧 superscript 𝑎 1 superscript 𝑧 1 2 𝑎 1 2 superscript 𝑒 1 2 𝑧 Whittaker-confluent-hypergeometric-M 1 2 𝑎 1 2 1 2 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(a,z\right)=a^{-1}z^{\frac{1}{2}a-\frac% {1}{2}}e^{-\frac{1}{2}z}M_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right% )}}
\incgamma@{a}{z} = a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}\WhittakerconfhyperM{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
GAMMA(a)-GAMMA(a, z) = (a)^(- 1)* (z)^((1)/(2)*a -(1)/(2))* exp(-(1)/(2)*z)*WhittakerM((1)/(2)*a -(1)/(2), (1)/(2)*a, z)

Gamma[a, 0, z] == (a)^(- 1)* (z)^(Divide[1,2]*a -Divide[1,2])* Exp[-Divide[1,2]*z]*WhittakerM[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z]
Successful Successful - Successful [Tested: 21]
8.5.E5 Γ ⁡ ( a , z ) = e - 1 2 ⁢ z ⁢ z 1 2 ⁢ a - 1 2 ⁢ W 1 2 ⁢ a - 1 2 , 1 2 ⁢ a ⁡ ( z ) incomplete-Gamma 𝑎 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝑎 1 2 Whittaker-confluent-hypergeometric-W 1 2 𝑎 1 2 1 2 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1% }{2}a-\frac{1}{2}}W_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right)}}
\incGamma@{a}{z} = e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}\WhittakerconfhyperW{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}		 

GAMMA(a, z) = exp(-(1)/(2)*z)*(z)^((1)/(2)*a -(1)/(2))* WhittakerW((1)/(2)*a -(1)/(2), (1)/(2)*a, z)

Gamma[a, z] == Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]*a -Divide[1,2])* WhittakerW[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z]
Successful Successful - Successful [Tested: 42]
Retrieved from "https://lct.wmflabs.org/w/index.php?title=8.5&oldid=515"