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 A n = ∫ 0 ∞ t ⁢ e - z ⁢ t 1 + t 2 ⁢ ( t 2 1 + t 2 ) n ⁢ d t subscript 𝐴 𝑛 superscript subscript 0 𝑡 superscript 𝑒 𝑧 𝑡 1 superscript 𝑡 2 superscript superscript 𝑡 2 1 superscript 𝑡 2 𝑛 𝑡 {\displaystyle{\displaystyle A_{n}=\int_{0}^{\infty}\frac{te^{-zt}}{1+t^{2}}% \left(\frac{t^{2}}{1+t^{2}}\right)^{n}\mathrm{d}t}}
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((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 B n = ∫ 0 ∞ e - z ⁢ t 1 + t 2 ⁢ ( t 2 1 + t 2 ) n ⁢ d t subscript 𝐵 𝑛 superscript subscript 0 superscript 𝑒 𝑧 𝑡 1 superscript 𝑡 2 superscript superscript 𝑡 2 1 superscript 𝑡 2 𝑛 𝑡 {\displaystyle{\displaystyle B_{n}=\int_{0}^{\infty}\frac{e^{-zt}}{1+t^{2}}% \left(\frac{t^{2}}{1+t^{2}}\right)^{n}\mathrm{d}t}}
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((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 C n = ∫ 0 ∞ e - z ⁢ t ⁢ ( t 2 1 + t 2 ) n ⁢ d t subscript 𝐶 𝑛 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript superscript 𝑡 2 1 superscript 𝑡 2 𝑛 𝑡 {\displaystyle{\displaystyle C_{n}=\int_{0}^{\infty}e^{-zt}\left(\frac{t^{2}}{% 1+t^{2}}\right)^{n}\mathrm{d}t}}
C_{n} = \int_{0}^{\infty}e^{-zt}\left(\frac{t^{2}}{1+t^{2}}\right)^{n}\diff{t}		 

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 A n - 1 = A n + z 2 ⁢ n ⁢ C n subscript 𝐴 𝑛 1 subscript 𝐴 𝑛 𝑧 2 𝑛 subscript 𝐶 𝑛 {\displaystyle{\displaystyle A_{n-1}=A_{n}+\frac{z}{2n}C_{n}}}
A_{n-1} = A_{n}+\frac{z}{2n}C_{n}		 

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 B n - 1 = 2 ⁢ n ⁢ B n + z ⁢ A n - 1 2 ⁢ n - 1 subscript 𝐵 𝑛 1 2 𝑛 subscript 𝐵 𝑛 𝑧 subscript 𝐴 𝑛 1 2 𝑛 1 {\displaystyle{\displaystyle B_{n-1}=\frac{2nB_{n}+zA_{n-1}}{2n-1}}}
B_{n-1} = \frac{2nB_{n}+zA_{n-1}}{2n-1}		 

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 C n - 1 = C n + B n - 1 subscript 𝐶 𝑛 1 subscript 𝐶 𝑛 subscript 𝐵 𝑛 1 {\displaystyle{\displaystyle C_{n-1}=C_{n}+B_{n-1}}}
C_{n-1} = C_{n}+B_{n-1}		 

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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