Elementary Functions - 5.2 Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.2.E1 Γ ⁡ ( z ) = ∫ 0 ∞ e - t ⁢ t z - 1 ⁢ d t Euler-Gamma 𝑧 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝑧 1 𝑡 {\displaystyle{\displaystyle\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1% }\mathrm{d}t}}
\EulerGamma@{z} = \int_{0}^{\infty}e^{-t}t^{z-1}\diff{t}		 ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} 
GAMMA(z) = int(exp(- t)*(t)^(z - 1), t = 0..infinity)

Gamma[z] == Integrate[Exp[- t]*(t)^(z - 1), {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 5]
5.2.E2 ψ ⁡ ( z ) = Γ ′ ⁡ ( z ) / Γ ⁡ ( z ) digamma 𝑧 diffop Euler-Gamma 1 𝑧 Euler-Gamma 𝑧 {\displaystyle{\displaystyle\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma% \left(z\right)}}
\digamma@{z} = \EulerGamma'@{z}/\EulerGamma@{z}		 ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} 
Psi(z) = diff( GAMMA(z), z$(1) )/GAMMA(z)

PolyGamma[z] == D[Gamma[z], {z, 1}]/Gamma[z]
Successful Successful - Successful [Tested: 1]
5.2#Ex1 ( a ) 0 = 1 Pochhammer 𝑎 0 1 {\displaystyle{\displaystyle{\left(a\right)_{0}}=1}}
\Pochhammersym{a}{0} = 1		 

pochhammer(a, 0) = 1

Pochhammer[a, 0] == 1
Successful Successful - Successful [Tested: 6]
5.2.E5 ( a ) n = Γ ⁡ ( a + n ) / Γ ⁡ ( a ) Pochhammer 𝑎 𝑛 Euler-Gamma 𝑎 𝑛 Euler-Gamma 𝑎 {\displaystyle{\displaystyle{\left(a\right)_{n}}=\Gamma\left(a+n\right)/\Gamma% \left(a\right)}}
\Pochhammersym{a}{n} = \EulerGamma@{a+n}/\EulerGamma@{a}		 ℜ ⁡ ( a + n ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑎 𝑛 0 𝑎 0 {\displaystyle{\displaystyle\Re(a+n)>0,\Re a>0}} 
pochhammer(a, n) = GAMMA(a + n)/GAMMA(a)

Pochhammer[a, n] == Gamma[a + n]/Gamma[a]
Successful Successful - Successful [Tested: 3]
5.2.E6 ( - a ) n = ( - 1 ) n ⁢ ( a - n + 1 ) n Pochhammer 𝑎 𝑛 superscript 1 𝑛 Pochhammer 𝑎 𝑛 1 𝑛 {\displaystyle{\displaystyle{\left(-a\right)_{n}}=(-1)^{n}{\left(a-n+1\right)_% {n}}}}
\Pochhammersym{-a}{n} = (-1)^{n}\Pochhammersym{a-n+1}{n}		 

pochhammer(- a, n) = (- 1)^(n)* pochhammer(a - n + 1, n)

Pochhammer[- a, n] == (- 1)^(n)* Pochhammer[a - n + 1, n]
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
5.2#Ex3 ( a ) 2 ⁢ n = 2 2 ⁢ n ⁢ ( a 2 ) n ⁢ ( a + 1 2 ) n Pochhammer 𝑎 2 𝑛 superscript 2 2 𝑛 Pochhammer 𝑎 2 𝑛 Pochhammer 𝑎 1 2 𝑛 {\displaystyle{\displaystyle{\left(a\right)_{2n}}=2^{2n}{\left(\frac{a}{2}% \right)_{n}}{\left(\frac{a+1}{2}\right)_{n}}}}
\Pochhammersym{a}{2n} = 2^{2n}\Pochhammersym{\frac{a}{2}}{n}\Pochhammersym{\frac{a+1}{2}}{n}		 

pochhammer(a, 2*n) = (2)^(2*n)* pochhammer((a)/(2), n)*pochhammer((a + 1)/(2), n)

Pochhammer[a, 2*n] == (2)^(2*n)* Pochhammer[Divide[a,2], n]*Pochhammer[Divide[a + 1,2], n]
Successful Successful - Successful [Tested: 18]
5.2#Ex4 ( a ) 2 ⁢ n + 1 = 2 2 ⁢ n + 1 ⁢ ( a 2 ) n + 1 ⁢ ( a + 1 2 ) n Pochhammer 𝑎 2 𝑛 1 superscript 2 2 𝑛 1 Pochhammer 𝑎 2 𝑛 1 Pochhammer 𝑎 1 2 𝑛 {\displaystyle{\displaystyle{\left(a\right)_{2n+1}}=2^{2n+1}{\left(\frac{a}{2}% \right)_{n+1}}{\left(\frac{a+1}{2}\right)_{n}}}}
\Pochhammersym{a}{2n+1} = 2^{2n+1}\Pochhammersym{\frac{a}{2}}{n+1}\Pochhammersym{\frac{a+1}{2}}{n}		 

pochhammer(a, 2*n + 1) = (2)^(2*n + 1)* pochhammer((a)/(2), n + 1)*pochhammer((a + 1)/(2), n)

Pochhammer[a, 2*n + 1] == (2)^(2*n + 1)* Pochhammer[Divide[a,2], n + 1]*Pochhammer[Divide[a + 1,2], n]
Successful Successful - Successful [Tested: 18]
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