Exponential, Logarithmic, Sine, and Cosine Integrals - 6.18 Methods of Computation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
6.18#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{n} = \int_{0}^{\infty}\frac{te^{-zt}}{1+t^{2}}\left(\frac{t^{2}}{1+t^{2}}\right)^{n}\diff{t}}
A_{n} = \int_{0}^{\infty}\frac{te^{-zt}}{1+t^{2}}\left(\frac{t^{2}}{1+t^{2}}\right)^{n}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[n] = int((t*exp(- z*t))/(1 + (t)^(2))*(((t)^(2))/(1 + (t)^(2)))^(n), t = 0..infinity)

Subscript[A, n] == Integrate[Divide[t*Exp[- z*t],1 + (t)^(2)]*(Divide[(t)^(2),1 + (t)^(2)])^(n), {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [210 / 210]
Result: .7485296696+.6226310704*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, A[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .8043767351+.5871300239*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, A[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[0.7485296693535908, 0.622631070403298]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8043767348683764, 0.5871300238783713]
Test Values: {Rule[n, 2], Rule[z, 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
6.18#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle B_{n} = \int_{0}^{\infty}\frac{e^{-zt}}{1+t^{2}}\left(\frac{t^{2}}{1+t^{2}}\right)^{n}\diff{t}}
B_{n} = \int_{0}^{\infty}\frac{e^{-zt}}{1+t^{2}}\left(\frac{t^{2}}{1+t^{2}}\right)^{n}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
B[n] = int((exp(- z*t))/(1 + (t)^(2))*(((t)^(2))/(1 + (t)^(2)))^(n), t = 0..infinity)

Subscript[B, n] == Integrate[Divide[Exp[- z*t],1 + (t)^(2)]*(Divide[(t)^(2),1 + (t)^(2)])^(n), {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [210 / 210]
Result: .7390515864+.5822189558*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, B[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .8115624973+.5498007781*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, B[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[0.7390515861602941, 0.5822189558055343]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8115624970800986, 0.549800778092373]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
6.18#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle C_{n} = \int_{0}^{\infty}e^{-zt}\left(\frac{t^{2}}{1+t^{2}}\right)^{n}\diff{t}}
C_{n} = \int_{0}^{\infty}e^{-zt}\left(\frac{t^{2}}{1+t^{2}}\right)^{n}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
C[n] = int(exp(- z*t)*(((t)^(2))/(1 + (t)^(2)))^(n), t = 0..infinity)

Subscript[C, n] == Integrate[Exp[- z*t]*(Divide[(t)^(2),1 + (t)^(2)])^(n), {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [210 / 210]
Result: .6165937696+.8168923194*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, C[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .7435675872+.7346733636*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, C[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[0.6165937693596737, 0.8168923194411848]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.7435675869838186, 0.7346733636356504]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
6.18#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{n-1} = A_{n}+\frac{z}{2n}C_{n}}
A_{n-1} = A_{n}+\frac{z}{2n}C_{n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[n - 1] = A[n]+(z)/(2*n)*C[n]
 
Subscript[A, n - 1] == Subscript[A, n]+Divide[z,2*n]*Subscript[C, n]
 Skipped - no semantic math Skipped - no semantic math - -
6.18#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle B_{n-1} = \frac{2nB_{n}+zA_{n-1}}{2n-1}}
B_{n-1} = \frac{2nB_{n}+zA_{n-1}}{2n-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
B[n - 1] = (2*n*B[n]+ z*A[n - 1])/(2*n - 1)
 
Subscript[B, n - 1] == Divide[2*n*Subscript[B, n]+ z*Subscript[A, n - 1],2*n - 1]
 Skipped - no semantic math Skipped - no semantic math - -
6.18#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle C_{n-1} = C_{n}+B_{n-1}}
C_{n-1} = C_{n}+B_{n-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
C[n - 1] = C[n]+ B[n - 1]
 
Subscript[C, n - 1] == Subscript[C, n]+ Subscript[B, n - 1]
 Skipped - no semantic math Skipped - no semantic math - -
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