Exponential, Logarithmic, Sine, and Cosine Integrals - 6.7 Integral Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
6.7.E1 ∫ 0 ∞ e - a ⁢ t t + b ⁢ d t = ∫ 0 ∞ e i ⁢ a ⁢ t t + i ⁢ b ⁢ d t superscript subscript 0 superscript 𝑒 𝑎 𝑡 𝑡 𝑏 𝑡 superscript subscript 0 superscript 𝑒 𝑖 𝑎 𝑡 𝑡 𝑖 𝑏 𝑡 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{e^{-at}}{t+b}\mathrm{d}t=% \int_{0}^{\infty}\frac{e^{iat}}{t+ib}\mathrm{d}t}}
\int_{0}^{\infty}\frac{e^{-at}}{t+b}\diff{t} = \int_{0}^{\infty}\frac{e^{iat}}{t+ib}\diff{t}		 a > 0 , b > 0 formulae-sequence 𝑎 0 𝑏 0 {\displaystyle{\displaystyle a>0,b>0}} 
int((exp(- a*t))/(t + b), t = 0..infinity) = int((exp(I*a*t))/(t + I*b), t = 0..infinity)

Integrate[Divide[Exp[- a*t],t + b], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[Exp[I*a*t],t + I*b], {t, 0, Infinity}, GenerateConditions->None]
Successful Aborted Skip - symbolical successful subtest Successful [Tested: 9]
6.7.E1 ∫ 0 ∞ e i ⁢ a ⁢ t t + i ⁢ b ⁢ d t = e a ⁢ b ⁢ E 1 ⁡ ( a ⁢ b ) superscript subscript 0 superscript 𝑒 𝑖 𝑎 𝑡 𝑡 𝑖 𝑏 𝑡 superscript 𝑒 𝑎 𝑏 exponential-integral 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{e^{iat}}{t+ib}\mathrm{d}t=e% ^{ab}E_{1}\left(ab\right)}}
\int_{0}^{\infty}\frac{e^{iat}}{t+ib}\diff{t} = e^{ab}\expintE@{ab}		 a > 0 , b > 0 formulae-sequence 𝑎 0 𝑏 0 {\displaystyle{\displaystyle a>0,b>0}} 
int((exp(I*a*t))/(t + I*b), t = 0..infinity) = exp(a*b)*Ei(a*b)

Integrate[Divide[Exp[I*a*t],t + I*b], {t, 0, Infinity}, GenerateConditions->None] == Exp[a*b]*ExpIntegralE[1, a*b]
Failure Failure
Failed [9 / 9]
Result: -56.03273673
Test Values: {a = 1.5, b = 1.5}


Result: -1.835422085
Test Values: {a = 1.5, b = .5}

... skip entries to safe data
 Successful [Tested: 9]
6.7.E2 e x ⁢ ∫ 0 α e - x ⁢ t 1 - t ⁢ d t = Ei ⁡ ( x ) - Ei ⁡ ( ( 1 - α ) ⁢ x ) superscript 𝑒 𝑥 superscript subscript 0 𝛼 superscript 𝑒 𝑥 𝑡 1 𝑡 𝑡 exponential-integral-Ei 𝑥 exponential-integral-Ei 1 𝛼 𝑥 {\displaystyle{\displaystyle e^{x}\int_{0}^{\alpha}\frac{e^{-xt}}{1-t}\mathrm{% d}t=\mathrm{Ei}\left(x\right)-\mathrm{Ei}\left((1-\alpha)x\right)}}
e^{x}\int_{0}^{\alpha}\frac{e^{-xt}}{1-t}\diff{t} = \expintEi@{x}-\expintEi@{(1-\alpha)x}		 0 ≤ α , α < 1 , x > 0 formulae-sequence 0 𝛼 formulae-sequence 𝛼 1 𝑥 0 {\displaystyle{\displaystyle 0\leq\alpha,\alpha<1,x>0}} 
Error

Exp[x]*Integrate[Divide[Exp[- x*t],1 - t], {t, 0, \[Alpha]}, GenerateConditions->None] == ExpIntegralEi[x]- ExpIntegralEi[(1 - \[Alpha])*x]
Missing Macro Error Failure - Successful [Tested: 3]
6.7.E3 ∫ x ∞ e i ⁢ t a 2 + t 2 ⁢ d t = i 2 ⁢ a ⁢ ( e a ⁢ E 1 ⁡ ( a - i ⁢ x ) - e - a ⁢ E 1 ⁡ ( - a - i ⁢ x ) ) superscript subscript 𝑥 superscript 𝑒 𝑖 𝑡 superscript 𝑎 2 superscript 𝑡 2 𝑡 𝑖 2 𝑎 superscript 𝑒 𝑎 exponential-integral 𝑎 𝑖 𝑥 superscript 𝑒 𝑎 exponential-integral 𝑎 𝑖 𝑥 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{e^{it}}{a^{2}+t^{2}}\mathrm% {d}t=\frac{i}{2a}\left(e^{a}E_{1}\left(a-ix\right)-e^{-a}E_{1}\left(-a-ix% \right)\right)}}
\int_{x}^{\infty}\frac{e^{it}}{a^{2}+t^{2}}\diff{t} = \frac{i}{2a}\left(e^{a}\expintE@{a-ix}-e^{-a}\expintE@{-a-ix}\right)		 a > 0 , x > 0 formulae-sequence 𝑎 0 𝑥 0 {\displaystyle{\displaystyle a>0,x>0}} 
int((exp(I*t))/((a)^(2)+ (t)^(2)), t = x..infinity) = (I)/(2*a)*(exp(a)*Ei(a - I*x)- exp(- a)*Ei(- a - I*x))

Integrate[Divide[Exp[I*t],(a)^(2)+ (t)^(2)], {t, x, Infinity}, GenerateConditions->None] == Divide[I,2*a]*(Exp[a]*ExpIntegralE[1, a - I*x]- Exp[- a]*ExpIntegralE[1, - a - I*x])
Failure Aborted
Failed [9 / 9]
Result: -5.458727175-3.178550596*I
Test Values: {a = 1.5, x = 1.5}


Result: -1.924923680-4.406791455*I
Test Values: {a = 1.5, x = .5}

... skip entries to safe data
 Skipped - Because timed out
6.7.E4 ∫ x ∞ t ⁢ e i ⁢ t a 2 + t 2 ⁢ d t = 1 2 ⁢ ( e a ⁢ E 1 ⁡ ( a - i ⁢ x ) + e - a ⁢ E 1 ⁡ ( - a - i ⁢ x ) ) superscript subscript 𝑥 𝑡 superscript 𝑒 𝑖 𝑡 superscript 𝑎 2 superscript 𝑡 2 𝑡 1 2 superscript 𝑒 𝑎 exponential-integral 𝑎 𝑖 𝑥 superscript 𝑒 𝑎 exponential-integral 𝑎 𝑖 𝑥 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{te^{it}}{a^{2}+t^{2}}% \mathrm{d}t=\tfrac{1}{2}\left(e^{a}E_{1}\left(a-ix\right)+e^{-a}E_{1}\left(-a-% ix\right)\right)}}
\int_{x}^{\infty}\frac{te^{it}}{a^{2}+t^{2}}\diff{t} = \tfrac{1}{2}\left(e^{a}\expintE@{a-ix}+e^{-a}\expintE@{-a-ix}\right)		 a > 0 , x > 0 formulae-sequence 𝑎 0 𝑥 0 {\displaystyle{\displaystyle a>0,x>0}} 
int((t*exp(I*t))/((a)^(2)+ (t)^(2)), t = x..infinity) = (1)/(2)*(exp(a)*Ei(a - I*x)+ exp(- a)*Ei(- a - I*x))

Integrate[Divide[t*Exp[I*t],(a)^(2)+ (t)^(2)], {t, x, Infinity}, GenerateConditions->None] == Divide[1,2]*(Exp[a]*ExpIntegralE[1, a - I*x]+ Exp[- a]*ExpIntegralE[1, - a - I*x])
Failure Aborted
Failed [9 / 9]
Result: -5.267453009+8.746914637*I
Test Values: {a = 1.5, x = 1.5}


Result: -7.302877906+3.948990541*I
Test Values: {a = 1.5, x = .5}

... skip entries to safe data
 Successful [Tested: 9]
6.7.E5 ∫ x ∞ e - t a 2 + t 2 ⁢ d t = - 1 2 ⁢ a ⁢ i ⁢ ( e i ⁢ a ⁢ E 1 ⁡ ( x + i ⁢ a ) - e - i ⁢ a ⁢ E 1 ⁡ ( x - i ⁢ a ) ) superscript subscript 𝑥 superscript 𝑒 𝑡 superscript 𝑎 2 superscript 𝑡 2 𝑡 1 2 𝑎 𝑖 superscript 𝑒 𝑖 𝑎 exponential-integral 𝑥 𝑖 𝑎 superscript 𝑒 𝑖 𝑎 exponential-integral 𝑥 𝑖 𝑎 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{e^{-t}}{a^{2}+t^{2}}\mathrm% {d}t=-\frac{1}{2ai}\left(e^{ia}E_{1}\left(x+ia\right)-e^{-ia}E_{1}\left(x-ia% \right)\right)}}
\int_{x}^{\infty}\frac{e^{-t}}{a^{2}+t^{2}}\diff{t} = -\frac{1}{2ai}\left(e^{ia}\expintE@{x+ia}-e^{-ia}\expintE@{x-ia}\right)		 a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}} 
int((exp(- t))/((a)^(2)+ (t)^(2)), t = x..infinity) = -(1)/(2*a*I)*(exp(I*a)*Ei(x + I*a)- exp(- I*a)*Ei(x - I*a))

Integrate[Divide[Exp[- t],(a)^(2)+ (t)^(2)], {t, x, Infinity}, GenerateConditions->None] == -Divide[1,2*a*I]*(Exp[I*a]*ExpIntegralE[1, x + I*a]- Exp[- I*a]*ExpIntegralE[1, x - I*a])
Failure Aborted
Failed [9 / 9]
Result: 1.667239755-0.*I
Test Values: {a = 1.5, x = 1.5, x = 3/2}


Result: .7611670238-0.*I
Test Values: {a = 1.5, x = .5, x = 3/2}

... skip entries to safe data
 Skipped - Because timed out
6.7.E6 ∫ x ∞ t ⁢ e - t a 2 + t 2 ⁢ d t = 1 2 ⁢ ( e i ⁢ a ⁢ E 1 ⁡ ( x + i ⁢ a ) + e - i ⁢ a ⁢ E 1 ⁡ ( x - i ⁢ a ) ) superscript subscript 𝑥 𝑡 superscript 𝑒 𝑡 superscript 𝑎 2 superscript 𝑡 2 𝑡 1 2 superscript 𝑒 𝑖 𝑎 exponential-integral 𝑥 𝑖 𝑎 superscript 𝑒 𝑖 𝑎 exponential-integral 𝑥 𝑖 𝑎 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{te^{-t}}{a^{2}+t^{2}}% \mathrm{d}t=\tfrac{1}{2}\left(e^{ia}E_{1}\left(x+ia\right)+e^{-ia}E_{1}\left(x% -ia\right)\right)}}
\int_{x}^{\infty}\frac{te^{-t}}{a^{2}+t^{2}}\diff{t} = \tfrac{1}{2}\left(e^{ia}\expintE@{x+ia}+e^{-ia}\expintE@{x-ia}\right)		 a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}} 
int((t*exp(- t))/((a)^(2)+ (t)^(2)), t = x..infinity) = (1)/(2)*(exp(I*a)*Ei(x + I*a)+ exp(- I*a)*Ei(x - I*a))

Integrate[Divide[t*Exp[- t],(a)^(2)+ (t)^(2)], {t, x, Infinity}, GenerateConditions->None] == Divide[1,2]*(Exp[I*a]*ExpIntegralE[1, x + I*a]+ Exp[- I*a]*ExpIntegralE[1, x - I*a])
Failure Aborted
Failed [9 / 9]
Result: 3.610851888+0.*I
Test Values: {a = 1.5, x = 1.5, x = 3/2}


Result: 2.934911868+0.*I
Test Values: {a = 1.5, x = .5, x = 3/2}

... skip entries to safe data
 Skipped - Because timed out
6.7.E7 ∫ 0 1 e - a ⁢ t ⁢ sin ⁡ ( b ⁢ t ) t ⁢ d t = ℑ ⁡ Ein ⁡ ( a + i ⁢ b ) superscript subscript 0 1 superscript 𝑒 𝑎 𝑡 𝑏 𝑡 𝑡 𝑡 complementary-exponential-integral 𝑎 𝑖 𝑏 {\displaystyle{\displaystyle\int_{0}^{1}\frac{e^{-at}\sin\left(bt\right)}{t}% \mathrm{d}t=\Im\mathrm{Ein}\left(a+ib\right)}}
\int_{0}^{1}\frac{e^{-at}\sin@{bt}}{t}\diff{t} = \imagpart@@{\expintEin@{a+ib}}		 

Error

Integrate[Divide[Exp[- a*t]*Sin[b*t],t], {t, 0, 1}, GenerateConditions->None] == Im[ExpIntegralE[1, a + I*b] + Ln[a + I*b] + EulerGamma]
Missing Macro Error Failure -
Failed [1 / 1]
Result: Plus[Complex[0.7167380515760515, 5.551115123125783*^-17], Times[-1.0, Plus[-0.06866011182139653, Im[Ln[Complex[1.5, 1.5]]]]]]
Test Values: {Rule[a, Rational[3, 2]], Rule[b, Rational[3, 2]]}

6.7.E8 ∫ 0 1 e - a ⁢ t ⁢ ( 1 - cos ⁡ ( b ⁢ t ) ) t ⁢ d t = ℜ ⁡ Ein ⁡ ( a + i ⁢ b ) - Ein ⁡ ( a ) superscript subscript 0 1 superscript 𝑒 𝑎 𝑡 1 𝑏 𝑡 𝑡 𝑡 complementary-exponential-integral 𝑎 𝑖 𝑏 complementary-exponential-integral 𝑎 {\displaystyle{\displaystyle\int_{0}^{1}\frac{e^{-at}(1-\cos\left(bt\right))}{% t}\mathrm{d}t=\Re\mathrm{Ein}\left(a+ib\right)-\mathrm{Ein}\left(a\right)}}
\int_{0}^{1}\frac{e^{-at}(1-\cos@{bt})}{t}\diff{t} = \realpart@@{\expintEin@{a+ib}}-\expintEin@{a}		 

Error

Integrate[Divide[Exp[- a*t]*(1 - Cos[b*t]),t], {t, 0, 1}, GenerateConditions->None] == Re[ExpIntegralE[1, a + I*b] + Ln[a + I*b] + EulerGamma]- ExpIntegralE[1, a] + Ln[a] + EulerGamma
Missing Macro Error Failure -
Failed [1 / 1]
Result: Plus[Complex[-0.8490131893081223, 0.0], Times[-1.0, Ln[1.5]], Times[-1.0, Plus[-0.04115544978502889, Re[Ln[Complex[1.5, 1.5]]]]]]
Test Values: {Rule[a, Rational[3, 2]], Rule[b, Rational[3, 2]]}

6.7.E9 si ⁡ ( z ) = - ∫ 0 π / 2 e - z ⁢ cos ⁡ t ⁢ cos ⁡ ( z ⁢ sin ⁡ t ) ⁢ d t shifted-sine-integral 𝑧 superscript subscript 0 𝜋 2 superscript 𝑒 𝑧 𝑡 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{si}\left(z\right)=-\int_{0}^{\pi/2}e^{-z% \cos t}\cos\left(z\sin t\right)\mathrm{d}t}}
\shiftsinint@{z} = -\int_{0}^{\pi/2}e^{-z\cos@@{t}}\cos@{z\sin@@{t}}\diff{t}		 

Ssi(z) = - int(exp(- z*cos(t))*cos(z*sin(t)), t = 0..Pi/2)

SinIntegral[z] - Pi/2 == - Integrate[Exp[- z*Cos[t]]*Cos[z*Sin[t]], {t, 0, Pi/2}, GenerateConditions->None]
Failure Aborted
Failed [2 / 7]
Result: -3.141592654+.1e-9*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: -3.141592653+0.*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Skipped - Because timed out
6.7.E13 ∫ 0 ∞ sin ⁡ t t + z ⁢ d t = ∫ 0 ∞ e - z ⁢ t t 2 + 1 ⁢ d t superscript subscript 0 𝑡 𝑡 𝑧 𝑡 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\sin t}{t+z}\mathrm{d}t=% \int_{0}^{\infty}\frac{e^{-zt}}{t^{2}+1}\mathrm{d}t}}
\int_{0}^{\infty}\frac{\sin@@{t}}{t+z}\diff{t} = \int_{0}^{\infty}\frac{e^{-zt}}{t^{2}+1}\diff{t}		 

int((sin(t))/(t + z), t = 0..infinity) = int((exp(- z*t))/((t)^(2)+ 1), t = 0..infinity)

Integrate[Divide[Sin[t],t + z], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[Exp[- z*t],(t)^(2)+ 1], {t, 0, Infinity}, GenerateConditions->None]
Failure Successful Error Successful [Tested: 7]
6.7.E14 ∫ 0 ∞ cos ⁡ t t + z ⁢ d t = ∫ 0 ∞ t ⁢ e - z ⁢ t t 2 + 1 ⁢ d t superscript subscript 0 𝑡 𝑡 𝑧 𝑡 superscript subscript 0 𝑡 superscript 𝑒 𝑧 𝑡 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\cos t}{t+z}\mathrm{d}t=% \int_{0}^{\infty}\frac{te^{-zt}}{t^{2}+1}\mathrm{d}t}}
\int_{0}^{\infty}\frac{\cos@@{t}}{t+z}\diff{t} = \int_{0}^{\infty}\frac{te^{-zt}}{t^{2}+1}\diff{t}		 

int((cos(t))/(t + z), t = 0..infinity) = int((t*exp(- z*t))/((t)^(2)+ 1), t = 0..infinity)

Integrate[Divide[Cos[t],t + z], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[t*Exp[- z*t],(t)^(2)+ 1], {t, 0, Infinity}, GenerateConditions->None]
Failure Successful Error Successful [Tested: 7]
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