Asymptotic Approximations - 2.5 Mellin Transform Methods

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.5.E8 I ⁢ ( x ) = ∫ 0 ∞ J ν 2 ⁡ ( x ⁢ t ) 1 + t ⁢ d t 𝐼 𝑥 superscript subscript 0 Bessel-J 𝜈 2 𝑥 𝑡 1 𝑡 𝑡 {\displaystyle{\displaystyle I(x)=\int_{0}^{\infty}\frac{{J_{\nu}^{2}}\left(xt% \right)}{1+t}\mathrm{d}t}}
I(x) = \int_{0}^{\infty}\frac{\BesselJ{\nu}^{2}@{xt}}{1+t}\diff{t}		 ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\nu>-\tfrac{1}{2}}} 
I(x) = int(((BesselJ(nu, x*t))^(2))/(1 + t), t = 0..infinity)

I[x] == Integrate[Divide[(BesselJ[\[Nu], x*t])^(2),1 + t], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [90 / 90]
Result: Float(infinity)+Float(infinity)*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, nu = 1.5, x = 1.5}


Result: Float(infinity)+Float(infinity)*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, nu = 1.5, x = .5}


Result: Float(infinity)+Float(infinity)*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, nu = 1.5, x = 2}


Result: Float(infinity)+.7500000000*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, nu = .5, x = 1.5}

... skip entries to safe data
 Skipped - Because timed out
2.5.E12 a n = 2 n - 1 ⁢ Γ ⁡ ( ν + 1 2 ⁢ n ) Γ 2 ⁡ ( 1 - 1 2 ⁢ n ) ⁢ Γ ⁡ ( 1 + ν - 1 2 ⁢ n ) ⁢ Γ ⁡ ( n ) subscript 𝑎 𝑛 superscript 2 𝑛 1 Euler-Gamma 𝜈 1 2 𝑛 Euler-Gamma 2 1 1 2 𝑛 Euler-Gamma 1 𝜈 1 2 𝑛 Euler-Gamma 𝑛 {\displaystyle{\displaystyle a_{n}=\frac{2^{n-1}\Gamma\left(\nu+\tfrac{1}{2}n% \right)}{{\Gamma^{2}}\left(1-\tfrac{1}{2}n\right)\Gamma\left(1+\nu-\tfrac{1}{2% }n\right)\Gamma\left(n\right)}}}
a_{n} = \frac{2^{n-1}\EulerGamma@{\nu+\tfrac{1}{2}n}}{\EulerGamma^{2}@{1-\tfrac{1}{2}n}\EulerGamma@{1+\nu-\tfrac{1}{2}n}\EulerGamma@{n}}		 ℜ ⁡ ( ν + 1 2 ⁢ n ) > 0 , ℜ ⁡ ( 1 - 1 2 ⁢ n ) > 0 , ℜ ⁡ ( 1 + ν - 1 2 ⁢ n ) > 0 , ℜ ⁡ n > 0 formulae-sequence 𝜈 1 2 𝑛 0 formulae-sequence 1 1 2 𝑛 0 formulae-sequence 1 𝜈 1 2 𝑛 0 𝑛 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{1}{2}n)>0,\Re(1-\tfrac{1}{2}n)>0,% \Re(1+\nu-\tfrac{1}{2}n)>0,\Re n>0}} 
a[n] = ((2)^(n - 1)* GAMMA(nu +(1)/(2)*n))/((GAMMA(1 -(1)/(2)*n))^(2)* GAMMA(1 + nu -(1)/(2)*n)*GAMMA(n))

Subscript[a, n] == Divide[(2)^(n - 1)* Gamma[\[Nu]+Divide[1,2]*n],(Gamma[1 -Divide[1,2]*n])^(2)* Gamma[1 + \[Nu]-Divide[1,2]*n]*Gamma[n]]
Failure Failure
Failed [300 / 300]
Result: .5477155179+.5000000000*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .8660254040+.5000000000*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: .8262366682+.3621677762*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -.8183098861+.8660254040*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5477155176006481, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.5.E13 b n = - a n ⁢ ( ln ⁡ 2 + 1 2 ⁢ ψ ⁡ ( ν + 1 2 ⁢ n ) + ψ ⁡ ( 1 - 1 2 ⁢ n ) + 1 2 ⁢ ψ ⁡ ( 1 + ν - 1 2 ⁢ n ) - ψ ⁡ ( n ) ) subscript 𝑏 𝑛 subscript 𝑎 𝑛 2 1 2 digamma 𝜈 1 2 𝑛 digamma 1 1 2 𝑛 1 2 digamma 1 𝜈 1 2 𝑛 digamma 𝑛 {\displaystyle{\displaystyle b_{n}=-a_{n}\left(\ln 2+\tfrac{1}{2}\psi\left(\nu% +\tfrac{1}{2}n\right)+\psi\left(1-\tfrac{1}{2}n\right)+\tfrac{1}{2}\psi\left(1% +\nu-\tfrac{1}{2}n\right)-\psi\left(n\right)\right)}}
b_{n} = -a_{n}\left(\ln@@{2}+\tfrac{1}{2}\digamma@{\nu+\tfrac{1}{2}n}+\digamma@{1-\tfrac{1}{2}n}+\tfrac{1}{2}\digamma@{1+\nu-\tfrac{1}{2}n}-\digamma@{n}\right)		 

b[n] = - a[n]*(ln(2)+(1)/(2)*Psi(nu +(1)/(2)*n)+ Psi(1 -(1)/(2)*n)+(1)/(2)*Psi(1 + nu -(1)/(2)*n)- Psi(n))

Subscript[b, n] == - Subscript[a, n]*(Log[2]+Divide[1,2]*PolyGamma[\[Nu]+Divide[1,2]*n]+ PolyGamma[1 -Divide[1,2]*n]+Divide[1,2]*PolyGamma[1 + \[Nu]-Divide[1,2]*n]- PolyGamma[n])
Failure Failure
Failed [300 / 300]
Result: .386290893e-1+.5914576348*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: Float(infinity)+Float(infinity)*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: .719377583e-1+1.226073019*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -1.327396315+.9574830388*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, b[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.0386290885385151, 0.59145763437721]
Test Values: {Rule[n, 1], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.3273963152459234, 0.9574830381616488]
Test Values: {Rule[n, 1], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.5.E24 h 2 ⁢ ( t ) = h ⁢ ( t ) - h 1 ⁢ ( t ) subscript ℎ 2 𝑡 ℎ 𝑡 subscript ℎ 1 𝑡 {\displaystyle{\displaystyle h_{2}(t)=h(t)-h_{1}(t)}}
h_{2}(t) = h(t)-h_{1}(t)		 

h[2](t) = h(t)- h[1](t)
 
Subscript[h, 2][t] == h[t]- Subscript[h, 1][t]
 Skipped - no semantic math Skipped - no semantic math - -
2.5.E30 I j ⁢ k ⁢ ( x ) = ∫ 0 ∞ f j ⁢ ( t ) ⁢ h k ⁢ ( x ⁢ t ) ⁢ d t subscript 𝐼 𝑗 𝑘 𝑥 superscript subscript 0 subscript 𝑓 𝑗 𝑡 subscript ℎ 𝑘 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle I_{jk}(x)=\int_{0}^{\infty}f_{j}(t)h_{k}(xt)% \mathrm{d}t}}
I_{jk}(x) = \int_{0}^{\infty}f_{j}(t)h_{k}(xt)\diff{t}		 

I[j, k](x) = int(f[j](t)* h[k](x*t), t = 0..infinity)

Subscript[I, j, k][x] == Integrate[Subscript[f, j][t]* Subscript[h, k][x*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {x = 1.5, I[j*k] = 1/2*3^(1/2)+1/2*I, f[j] = 1/2*3^(1/2)+1/2*I, h[k] = 1/2*3^(1/2)+1/2*I, j = 1, k = 1}


Result: Float(infinity)+Float(infinity)*I
Test Values: {x = 1.5, I[j*k] = 1/2*3^(1/2)+1/2*I, f[j] = 1/2*3^(1/2)+1/2*I, h[k] = 1/2*3^(1/2)+1/2*I, j = 1, k = 2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {x = 1.5, I[j*k] = 1/2*3^(1/2)+1/2*I, f[j] = 1/2*3^(1/2)+1/2*I, h[k] = 1/2*3^(1/2)+1/2*I, j = 1, k = 3}


Result: Float(infinity)+Float(infinity)*I
Test Values: {x = 1.5, I[j*k] = 1/2*3^(1/2)+1/2*I, f[j] = 1/2*3^(1/2)+1/2*I, h[k] = 1/2*3^(1/2)+1/2*I, j = 2, k = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.59977280929447116972275470162594*^+83839, -2.77088778626521950864048398971341*^+83839]
Test Values: {Rule[j, 1], Rule[k, 1], Rule[x, 1.5], Rule[Subscript[Complex[0, 1], Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.59977280929447116972275470162594*^+83839, -2.77088778626521950864048398971341*^+83839]
Test Values: {Rule[j, 1], Rule[k, 2], Rule[x, 1.5], Rule[Subscript[Complex[0, 1], Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.5.E31 I 21 ⁢ ( x ) = 0 subscript 𝐼 21 𝑥 0 {\displaystyle{\displaystyle I_{21}(x)=0}}
I_{21}(x) = 0		 x ≥ 1 𝑥 1 {\displaystyle{\displaystyle x\geq 1}} 
I[21](x) = 0
 
Subscript[I, 21][x] == 0
 Skipped - no semantic math Skipped - no semantic math - -
2.5.E33 I j ⁢ k ⁢ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ p j ⁢ k - i ⁢ ∞ p j ⁢ k + i ⁢ ∞ x - z ⁢ G j ⁢ k ⁢ ( z ) ⁢ d z subscript 𝐼 𝑗 𝑘 𝑥 1 2 𝜋 𝑖 superscript subscript subscript 𝑝 𝑗 𝑘 𝑖 subscript 𝑝 𝑗 𝑘 𝑖 superscript 𝑥 𝑧 subscript 𝐺 𝑗 𝑘 𝑧 𝑧 {\displaystyle{\displaystyle I_{jk}(x)=\frac{1}{2\pi i}\int_{p_{jk}-i\infty}^{% p_{jk}+i\infty}x^{-z}G_{jk}(z)\mathrm{d}z}}
I_{jk}(x) = \frac{1}{2\pi i}\int_{p_{jk}-i\infty}^{p_{jk}+i\infty}x^{-z}G_{jk}(z)\diff{z}		 

I[j, k](x) = (1)/(2*Pi*I)*int((x)^(-(x + y*I))* G[j, k]*((x + y*I)), (x + y*I) = p[j, k]- I*infinity..p[j, k]+ I*infinity)

Subscript[I, j, k][x] == Divide[1,2*Pi*I]*Integrate[(x)^(-(x + y*I))* Subscript[G, j, k]*((x + y*I)), {(x + y*I), Subscript[p, j, k]- I*Infinity, Subscript[p, j, k]+ I*Infinity}, GenerateConditions->None]
Error Failure -
Failed [300 / 300]
Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[Complex[0.0, 0.15915494309189535], NIntegrate[Complex[1.0861132213040667, 0.3920349523216481]
Test Values: {Complex[1.5, -1.5], DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}]]], {Rule[j, 1], Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[Complex[0, 1], Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[G, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[Complex[0.0, 0.15915494309189535], NIntegrate[Complex[1.0861132213040667, 0.3920349523216481]
Test Values: {Complex[1.5, -1.5], DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}]]], {Rule[j, 1], Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[Complex[0, 1], Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[G, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.5.E35 I j ⁢ k ⁢ ( x ) = ∑ p j ⁢ k < ℜ ⁡ z < q j ⁢ k res ⁡ [ - x - z ⁢ G j ⁢ k ⁢ ( z ) ] + E j ⁢ k ⁢ ( x ) subscript 𝐼 𝑗 𝑘 𝑥 subscript subscript 𝑝 𝑗 𝑘 𝑧 subscript 𝑞 𝑗 𝑘 residue delimited-[] superscript 𝑥 𝑧 subscript 𝐺 𝑗 𝑘 𝑧 subscript 𝐸 𝑗 𝑘 𝑥 {\displaystyle{\displaystyle I_{jk}(x)=\sum_{p_{jk}<\Re z<q_{jk}}\Residue\left% [-x^{-z}G_{jk}(z)\right]+E_{jk}(x)}}
I_{jk}(x) = \sum_{p_{jk}<\realpart@@{z}<q_{jk}}\Residue\left[-x^{-z}G_{jk}(z)\right]+E_{jk}(x)		 

I[j, k](x) = sum(*(- (x)^(-(x + y*I))* G[j, k]*((x + y*I))), Re(x + y*I) = p[j, k] + 1..q[j, k] - 1)+ E[j, k](x)

Subscript[I, j, k][x] == Sum[*(- (x)^(-(x + y*I))* Subscript[G, j, k]*((x + y*I))), {Re[x + y*I], Subscript[p, j, k] + 1, Subscript[q, j, k] - 1}, GenerateConditions->None]+ Subscript[E, j, k][x]
Error Failure - Error
2.5.E36 E j ⁢ k ⁢ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ q j ⁢ k - i ⁢ ∞ q j ⁢ k + i ⁢ ∞ x - z ⁢ G j ⁢ k ⁢ ( z ) ⁢ d z subscript 𝐸 𝑗 𝑘 𝑥 1 2 𝜋 𝑖 superscript subscript subscript 𝑞 𝑗 𝑘 𝑖 subscript 𝑞 𝑗 𝑘 𝑖 superscript 𝑥 𝑧 subscript 𝐺 𝑗 𝑘 𝑧 𝑧 {\displaystyle{\displaystyle E_{jk}(x)=\frac{1}{2\pi i}\int_{q_{jk}-i\infty}^{% q_{jk}+i\infty}x^{-z}G_{jk}(z)\mathrm{d}z}}
E_{jk}(x) = \frac{1}{2\pi i}\int_{q_{jk}-i\infty}^{q_{jk}+i\infty}x^{-z}G_{jk}(z)\diff{z}		 

E[j, k](x) = (1)/(2*Pi*I)*int((x)^(-(x + y*I))* G[j, k]*((x + y*I)), (x + y*I) = q[j, k]- I*infinity..q[j, k]+ I*infinity)

Subscript[E, j, k][x] == Divide[1,2*Pi*I]*Integrate[(x)^(-(x + y*I))* Subscript[G, j, k]*((x + y*I)), {(x + y*I), Subscript[q, j, k]- I*Infinity, Subscript[q, j, k]+ I*Infinity}, GenerateConditions->None]
Error Failure -
Failed [300 / 300]
Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[Complex[0.0, 0.15915494309189535], NIntegrate[Complex[1.0861132213040667, 0.3920349523216481]
Test Values: {Complex[1.5, -1.5], DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}]]], {Rule[j, 1], Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[E, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[G, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[q, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[Complex[0.0, 0.15915494309189535], NIntegrate[Complex[1.0861132213040667, 0.3920349523216481]
Test Values: {Complex[1.5, -1.5], DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}]]], {Rule[j, 1], Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[E, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[G, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[q, Times[j, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.5.E39 I j ⁢ ( x ) = ∫ 0 ∞ e - t ⁢ h j ⁢ ( x ⁢ t ) ⁢ d t subscript 𝐼 𝑗 𝑥 superscript subscript 0 superscript 𝑒 𝑡 subscript ℎ 𝑗 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle I_{j}(x)=\int_{0}^{\infty}e^{-t}h_{j}(xt)\mathrm{% d}t}}
I_{j}(x) = \int_{0}^{\infty}e^{-t}h_{j}(xt)\diff{t}		 

I[j](x) = int(exp(- t)*h[j](x*t), t = 0..infinity)

Subscript[I, j][x] == Integrate[Exp[- t]*Subscript[h, j][x*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [270 / 300]
Result: 2.049038106-.5490381060*I
Test Values: {x = 1.5, I[j] = 1/2*3^(1/2)+1/2*I, h[j] = -1/2+1/2*I*3^(1/2), j = 1, j = 1}


Result: 2.049038106-.5490381060*I
Test Values: {x = 1.5, I[j] = 1/2*3^(1/2)+1/2*I, h[j] = -1/2+1/2*I*3^(1/2), j = 2, j = 1}


Result: 2.049038106-.5490381060*I
Test Values: {x = 1.5, I[j] = 1/2*3^(1/2)+1/2*I, h[j] = -1/2+1/2*I*3^(1/2), j = 3, j = 1}


Result: .5490381060+2.049038106*I
Test Values: {x = 1.5, I[j] = 1/2*3^(1/2)+1/2*I, h[j] = 1/2-1/2*I*3^(1/2), j = 1, j = 1}

... skip entries to safe data
 Skip - No test values generated
2.5.E46 res z = k ⁡ [ - ζ z - 1 ⁢ Γ ⁡ ( 1 - z ) ⁢ π ⁢ csc ⁡ ( π ⁢ z ) ] = ( - ln ⁡ ζ + ψ ⁡ ( k ) ) ⁢ ζ k - 1 ( k - 1 ) ! subscript residue 𝑧 𝑘 delimited-[] superscript 𝜁 𝑧 1 Euler-Gamma 1 𝑧 𝜋 𝜋 𝑧 𝜁 digamma 𝑘 superscript 𝜁 𝑘 1 𝑘 1 {\displaystyle{\displaystyle\Residue_{z=k}\left[-\zeta^{z-1}\Gamma\left(1-z% \right)\pi\csc\left(\pi z\right)\right]=\left(-\ln\zeta+\psi\left(k\right)% \right)\dfrac{\zeta^{k-1}}{(k-1)!}}}
\Residue_{z=k}\left[-\zeta^{z-1}\EulerGamma@{1-z}\pi\csc@{\pi z}\right] = \left(-\ln@@{\zeta}+\digamma@{k}\right)\dfrac{\zeta^{k-1}}{(k-1)!}		 ℜ ⁡ ( 1 - z ) > 0 1 𝑧 0 {\displaystyle{\displaystyle\Re(1-z)>0}} 
[z = k]*(- (zeta)^(z - 1)* GAMMA(1 - z)*Pi*csc(Pi*z)) = (- ln(zeta)+ Psi(k))*((zeta)^(k - 1))/(factorial(k - 1))

Subscript[, z == k]*(- \[Zeta]^(z - 1)* Gamma[1 - z]*Pi*Csc[Pi*z]) == (- Log[\[Zeta]]+ PolyGamma[k])*Divide[\[Zeta]^(k - 1),(k - 1)!]
Failure Failure Error
Failed [50 / 50]
Result: Plus[Complex[0.5772156649015329, 0.5235987755982988], Subscript[Null, False][Complex[1.288067451091007, -0.9403972809133088]]]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.5772156649015329, 2.0943951023931953], Subscript[Null, False][Complex[0.48475507921827343, -0.541984224121457]]]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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