Exponential, Logarithmic, Sine, and Cosine Integrals - 6.10 Other Series Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
6.10#Ex1 c 0 = 1 subscript 𝑐 0 1 {\displaystyle{\displaystyle c_{0}=1}}
c_{0} = 1		 

c[0] = 1
 
Subscript[c, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex2 c 1 = - 1 subscript 𝑐 1 1 {\displaystyle{\displaystyle c_{1}=-1}}
c_{1} = -1		 

c[1] = - 1
 
Subscript[c, 1] == - 1
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex3 c 2 = 1 2 subscript 𝑐 2 1 2 {\displaystyle{\displaystyle c_{2}=\tfrac{1}{2}}}
c_{2} = \tfrac{1}{2}		 

c[2] = (1)/(2)
 
Subscript[c, 2] == Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex4 c 3 = - 1 3 subscript 𝑐 3 1 3 {\displaystyle{\displaystyle c_{3}=-\tfrac{1}{3}}}
c_{3} = -\tfrac{1}{3}		 

c[3] = -(1)/(3)
 
Subscript[c, 3] == -Divide[1,3]
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex5 c 4 = 1 6 subscript 𝑐 4 1 6 {\displaystyle{\displaystyle c_{4}=\tfrac{1}{6}}}
c_{4} = \tfrac{1}{6}		 

c[4] = (1)/(6)
 
Subscript[c, 4] == Divide[1,6]
 Skipped - no semantic math Skipped - no semantic math - -
6.10.E3 c k = - βˆ‘ j = 0 k - 1 c j k - j subscript 𝑐 π‘˜ superscript subscript 𝑗 0 π‘˜ 1 subscript 𝑐 𝑗 π‘˜ 𝑗 {\displaystyle{\displaystyle c_{k}=-\sum_{j=0}^{k-1}\frac{c_{j}}{k-j}}}
c_{k} = -\sum_{j=0}^{k-1}\frac{c_{j}}{k-j}		 

c[k] = - sum((c[j])/(k - j), j = 0..k - 1)
 
Subscript[c, k] == - Sum[Divide[Subscript[c, j],k - j], {j, 0, k - 1}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
6.10.E4 Si ⁑ ( z ) = z ⁒ βˆ‘ n = 0 ∞ ( 𝗃 n ⁑ ( 1 2 ⁒ z ) ) 2 sine-integral 𝑧 𝑧 superscript subscript 𝑛 0 superscript spherical-Bessel-J 𝑛 1 2 𝑧 2 {\displaystyle{\displaystyle\mathrm{Si}\left(z\right)=z\sum_{n=0}^{\infty}% \left(\mathsf{j}_{n}\left(\tfrac{1}{2}z\right)\right)^{2}}}
\sinint@{z} = z\sum_{n=0}^{\infty}\left(\sphBesselJ{n}@{\tfrac{1}{2}z}\right)^{2}		 

Error

SinIntegral[z] == z*Sum[(SphericalBesselJ[n, Divide[1,2]*z])^(2), {n, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 7]
6.10.E6 Ei ⁑ ( x ) = Ξ³ + ln ⁑ | x | + βˆ‘ n = 0 ∞ ( - 1 ) n ⁒ ( x - a n ) ⁒ ( 𝗂 n ( 1 ) ⁑ ( 1 2 ⁒ x ) ) 2 exponential-integral-Ei π‘₯ π‘₯ superscript subscript 𝑛 0 superscript 1 𝑛 π‘₯ subscript π‘Ž 𝑛 superscript spherical-Bessel-I-1 𝑛 1 2 π‘₯ 2 {\displaystyle{\displaystyle\mathrm{Ei}\left(x\right)=\gamma+\ln\left|x\right|% +\sum_{n=0}^{\infty}(-1)^{n}(x-a_{n})\left({\mathsf{i}^{(1)}_{n}}\left(\tfrac{% 1}{2}x\right)\right)^{2}}}
\expintEi@{x} = \EulerConstant+\ln@@{\abs{x}}+\sum_{n=0}^{\infty}(-1)^{n}(x-a_{n})\left(\modsphBesseli{1}{n}@{\tfrac{1}{2}x}\right)^{2}		 x β‰  0 π‘₯ 0 {\displaystyle{\displaystyle x\neq 0}} 
Error

ExpIntegralEi[x] == EulerGamma + Log[Abs[x]]+ Sum[(- 1)^(n)*(x -((2*n + 1)*(1 -(- 1)^(n)+ PolyGamma[n + 1]- PolyGamma[1])))*(Sqrt[Divide[Pi, Divide[1,2]*x]/2] BesselI[(-1)^(1-1)*(n + 1/2), n])^(2), {n, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [3 / 3]
Result: Plus[2.318604676120101, Times[-1.0, NSum[Times[2.0943951023931953, Power[-1, n], Power[BesselI[Plus[Rational[1, 2], n], n], 2], Plus[1.5, Times[-1, Plus[1, Times[2, n]], Plus[1, Power[-1, Plus[1, n]], EulerGamma, PolyGamma[0, Plus[1, n]]]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5]}


Result: Plus[0.570151420521586, Times[-1.0, NSum[Times[6.283185307179586, Power[-1, n], Power[BesselI[Plus[Rational[1, 2], n], n], 2], Plus[0.5, Times[-1, Plus[1, Times[2, n]], Plus[1, Power[-1, Plus[1, n]], EulerGamma, PolyGamma[0, Plus[1, n]]]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5]}

... skip entries to safe data
6.10.E8 Ein ⁑ ( z ) = z ⁒ e - z / 2 ⁒ ( 𝗂 0 ( 1 ) ⁑ ( 1 2 ⁒ z ) + βˆ‘ n = 1 ∞ 2 ⁒ n + 1 n ⁒ ( n + 1 ) ⁒ 𝗂 n ( 1 ) ⁑ ( 1 2 ⁒ z ) ) complementary-exponential-integral 𝑧 𝑧 superscript 𝑒 𝑧 2 spherical-Bessel-I-1 0 1 2 𝑧 superscript subscript 𝑛 1 2 𝑛 1 𝑛 𝑛 1 spherical-Bessel-I-1 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\mathrm{Ein}\left(z\right)=ze^{-z/2}\left({\mathsf% {i}^{(1)}_{0}}\left(\tfrac{1}{2}z\right)+\sum_{n=1}^{\infty}\dfrac{2n+1}{n(n+1% )}{\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}z\right)\right)}}
\expintEin@{z} = ze^{-z/2}\left(\modsphBesseli{1}{0}@{\tfrac{1}{2}z}+\sum_{n=1}^{\infty}\dfrac{2n+1}{n(n+1)}\modsphBesseli{1}{n}@{\tfrac{1}{2}z}\right)		 

Error

ExpIntegralE[1, z] + Ln[z] + EulerGamma == z*Exp[- z/2]*(Sqrt[Divide[Pi, Divide[1,2]*z]/2] BesselI[(-1)^(1-1)*(0 + 1/2), 0]+ Sum[Divide[2*n + 1,n*(n + 1)]*Sqrt[Divide[Pi, Divide[1,2]*z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n], {n, 1, Infinity}, GenerateConditions->None])
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.7449988338501623, -0.19274910655694033], Ln[Complex[0.8660254037844387, 0.49999999999999994]], Times[Complex[-0.6244291912543926, -0.17523758490546462], NSum[Times[Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Power[n, -1], Power[Plus[1, n], -1], Plus[1, Times[2, n]], Power[Pi, Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.31531322950964413, -1.02230061506208], Ln[Complex[-0.4999999999999998, 0.8660254037844387]], Times[Complex[0.11615580955286336, -1.278760766761026], NSum[Times[Power[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]], Rational[1, 2]], Power[n, -1], Power[Plus[1, n], -1], Plus[1, Times[2, n]], Power[Pi, Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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