Gamma Function - 5.7 Series Expansions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
5.7.E1	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(z\right)}=\sum_{k=1}^{\infty}c_{k}z^{k}}}$ \frac{1}{\EulerGamma@{z}} = \sum_{k=1}^{\infty}c_{k}z^{k}	${\displaystyle{\displaystyle\Re z>0}}$	(1)/(GAMMA(z)) = sum(c[k]*(z)^(k), k = 1..infinity)	Divide[1,Gamma[z]] == Sum[Subscript[c, k]*(z)^(k), {k, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [50 / 50] Result: 2.444337041-.9752791869*I Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = 1/2*3^(1/2)+1/2*I} Result: 2.444337041+1.756771621*I Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = -1/2+1/2*I*3^(1/2)} Result: -.287713767-.9752791869*I Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = 1/2-1/2*I*3^(1/2)} Result: -.287713767+1.756771621*I Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [50 / 50] Result: Plus[Complex[1.0783116366515544, 0.3907462172966202], Times[-1.0, NSum[Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[1, k]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[1.0783116366515544, 0.3907462172966202], Times[-1.0, NSum[Times[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
5.7.E3	${\displaystyle{\displaystyle\ln\Gamma\left(1+z\right)=-\ln\left(1+z\right)+z(1-\gamma)+\sum_{k=2}^{\infty}(-1)^{k}(\zeta\left(k\right)-1)\frac{z^{k}}{k}}}$ \ln@@{\EulerGamma@{1+z}} = -\ln@{1+z}+z(1-\EulerConstant)+\sum_{k=2}^{\infty}(-1)^{k}(\Riemannzeta@{k}-1)\frac{z^{k}}{k}	${\displaystyle{\displaystyle|z|<2,\Re(1+z)>0}}$	ln(GAMMA(1 + z)) = - ln(1 + z)+ z*(1 - gamma)+ sum((- 1)^(k)*(Zeta(k)- 1)*((z)^(k))/(k), k = 2..infinity)	Log[Gamma[1 + z]] == - Log[1 + z]+ z*(1 - EulerGamma)+ Sum[(- 1)^(k)*(Zeta[k]- 1)*Divide[(z)^(k),k], {k, 2, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 6]	Successful [Tested: 6]
5.7.E4	${\displaystyle{\displaystyle\psi\left(1+z\right)=-\gamma+\sum_{k=2}^{\infty}(-1)^{k}\zeta\left(k\right)z^{k-1}}}$ \digamma@{1+z} = -\EulerConstant+\sum_{k=2}^{\infty}(-1)^{k}\Riemannzeta@{k}z^{k-1}	${\displaystyle{\displaystyle|z|<1}}$	Psi(1 + z) = - gamma + sum((- 1)^(k)* Zeta(k)*(z)^(k - 1), k = 2..infinity)	PolyGamma[1 + z] == - EulerGamma + Sum[(- 1)^(k)* Zeta[k]*(z)^(k - 1), {k, 2, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 1]	Successful [Tested: 1]
5.7.E5	${\displaystyle{\displaystyle\psi\left(1+z\right)=\frac{1}{2z}-\frac{\pi}{2}\cot\left(\pi z\right)+\frac{1}{z^{2}-1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta\left(2k+1\right)-1)z^{2k}}}$ \digamma@{1+z} = \frac{1}{2z}-\frac{\pi}{2}\cot@{\pi z}+\frac{1}{z^{2}-1}+1-\EulerConstant-\sum_{k=1}^{\infty}(\Riemannzeta@{2k+1}-1)z^{2k}	${\displaystyle{\displaystyle|z|<2,z\neq 0}}$	Psi(1 + z) = (1)/(2*z)-(Pi)/(2)*cot(Pi*z)+(1)/((z)^(2)- 1)+ 1 - gamma - sum((Zeta(2*k + 1)- 1)*(z)^(2*k), k = 1..infinity)	PolyGamma[1 + z] == Divide[1,2*z]-Divide[Pi,2]*Cot[Pi*z]+Divide[1,(z)^(2)- 1]+ 1 - EulerGamma - Sum[(Zeta[2*k + 1]- 1)*(z)^(2*k), {k, 1, Infinity}, GenerateConditions->None]	Error	Successful	-	Successful [Tested: 6]
5.7.E6	${\displaystyle{\displaystyle\psi\left(z\right)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}}}$ \digamma@{z} = -\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}		Psi(z) = - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity)	PolyGamma[z] == - EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
5.7.E6	${\displaystyle{\displaystyle-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}=-\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right)}}$ -\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)} = -\EulerConstant+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right)		- gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity) = - gamma + sum((1)/(k + 1)-(1)/(k + z), k = 0..infinity)	- EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}, GenerateConditions->None] == - EulerGamma + Sum[Divide[1,k + 1]-Divide[1,k + z], {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
5.7.E7	${\displaystyle{\displaystyle\psi\left(\frac{z+1}{2}\right)-\psi\left(\frac{z}{2}\right)=2\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+z}}}$ \digamma@{\frac{z+1}{2}}-\digamma@{\frac{z}{2}} = 2\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+z}		Psi((z + 1)/(2))- Psi((z)/(2)) = 2*sum(((- 1)^(k))/(k + z), k = 0..infinity)	PolyGamma[Divide[z + 1,2]]- PolyGamma[Divide[z,2]] == 2*Sum[Divide[(- 1)^(k),k + z], {k, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
5.7.E8	${\displaystyle{\displaystyle\Im\psi\left(1+\mathrm{i}y\right)=\sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}}}$ \imagpart@@{\digamma@{1+\iunit y}} = \sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}		Im(Psi(1 + I*y)) = sum((y)/((k)^(2)+ (y)^(2)), k = 1..infinity)	Im[PolyGamma[1 + I*y]] == Sum[Divide[y,(k)^(2)+ (y)^(2)], {k, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 6]	Successful [Tested: 6]