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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{0} = 1}
c_{0} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[0] = 1
 
Subscript[c, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{1} = -1}
c_{1} = -1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[1] = - 1
 
Subscript[c, 1] == - 1
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{2} = \tfrac{1}{2}}
c_{2} = \tfrac{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[2] = (1)/(2)
 
Subscript[c, 2] == Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{3} = -\tfrac{1}{3}}
c_{3} = -\tfrac{1}{3}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[3] = -(1)/(3)
 
Subscript[c, 3] == -Divide[1,3]
 Skipped - no semantic math Skipped - no semantic math - -
6.10#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{4} = \tfrac{1}{6}}
c_{4} = \tfrac{1}{6}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[4] = (1)/(6)
 
Subscript[c, 4] == Divide[1,6]
 Skipped - no semantic math Skipped - no semantic math - -
6.10.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sinint@{z} = z\sum_{n=0}^{\infty}\left(\sphBesselJ{n}@{\tfrac{1}{2}z}\right)^{2}}
\sinint@{z} = z\sum_{n=0}^{\infty}\left(\sphBesselJ{n}@{\tfrac{1}{2}z}\right)^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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)}
\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)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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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