Results of Exponential, Logarithmic, Sine, and Cosine Integrals

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
6.2.E1 E 1 ⁑ ( z ) = ∫ z ∞ e - t t ⁒ d t exponential-integral 𝑧 superscript subscript 𝑧 superscript 𝑒 𝑑 𝑑 𝑑 {\displaystyle{\displaystyle E_{1}\left(z\right)=\int_{z}^{\infty}\frac{e^{-t}% }{t}\mathrm{d}t}} Ei(z)= int((exp(- t))/(t), t = z..infinity) -ExpIntegralEi[-(z)]= Integrate[Divide[Exp[- t],t], {t, z, Infinity}] Failure Failure Skip Successful
6.2.E2 E 1 ⁑ ( z ) = e - z ⁒ ∫ 0 ∞ e - t t + z ⁒ d t exponential-integral 𝑧 superscript 𝑒 𝑧 superscript subscript 0 superscript 𝑒 𝑑 𝑑 𝑧 𝑑 {\displaystyle{\displaystyle E_{1}\left(z\right)=e^{-z}\int_{0}^{\infty}\frac{% e^{-t}}{t+z}\mathrm{d}t}} Ei(z)= exp(- z)*int((exp(- t))/(t + z), t = 0..infinity) -ExpIntegralEi[-(z)]= Exp[- z]*Integrate[Divide[Exp[- t],t + z], {t, 0, Infinity}] Failure Failure Skip
Fail
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.2.E3 Ein ⁑ ( z ) = ∫ 0 z 1 - e - t t ⁒ d t complementary-exponential-integral 𝑧 superscript subscript 0 𝑧 1 superscript 𝑒 𝑑 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Ein}\left(z\right)=\int_{0}^{z}\frac{1-e^{% -t}}{t}\mathrm{d}t}} Error -ExpIntegralEi[-(z)] + Ln[z] + EulerGamma = Integrate[Divide[1 - Exp[- t],t], {t, 0, z}] Error Failure - Successful
6.2.E4 E 1 ⁑ ( z ) = Ein ⁑ ( z ) - ln ⁑ z - Ξ³ exponential-integral 𝑧 complementary-exponential-integral 𝑧 𝑧 {\displaystyle{\displaystyle E_{1}\left(z\right)=\mathrm{Ein}\left(z\right)-% \ln z-\gamma}} Error -ExpIntegralEi[-(z)]= -ExpIntegralEi[-(z)] + Ln[z] + EulerGamma - Log[z]- EulerGamma Error Failure - Successful
6.2.E6 Ei ⁑ ( - x ) = - ∫ x ∞ e - t t ⁒ d t exponential-integral-Ei π‘₯ superscript subscript π‘₯ superscript 𝑒 𝑑 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Ei}\left(-x\right)=-\int_{x}^{\infty}\frac% {e^{-t}}{t}\mathrm{d}t}} Error -ExpIntegralEi[-(- x)]= - Integrate[Divide[Exp[- t],t], {t, x, Infinity}] Error Failure -
Fail
-1.6757338819604166 <- {Rule[x, 1]}
-4.905333845293828 <- {Rule[x, 2]}
-9.920784189531217 <- {Rule[x, 3]}
6.2.E6 - ∫ x ∞ e - t t ⁒ d t = - E 1 ⁑ ( x ) superscript subscript π‘₯ superscript 𝑒 𝑑 𝑑 𝑑 exponential-integral π‘₯ {\displaystyle{\displaystyle-\int_{x}^{\infty}\frac{e^{-t}}{t}\mathrm{d}t=-E_{% 1}\left(x\right)}} - int((exp(- t))/(t), t = x..infinity)= - Ei(x) - Integrate[Divide[Exp[- t],t], {t, x, Infinity}]= - -ExpIntegralEi[-(x)] Failure Failure Skip Successful
6.2.E7 Ei ⁑ ( + x ) = - Ein ⁑ ( - x ) + ln ⁑ x + Ξ³ exponential-integral-Ei π‘₯ complementary-exponential-integral π‘₯ π‘₯ {\displaystyle{\displaystyle\mathrm{Ei}\left(+x\right)=-\mathrm{Ein}\left(-x% \right)+\ln x+\gamma}} Error -ExpIntegralEi[-(+ x)]= - -ExpIntegralEi[-(- x)] + Ln[- x] + EulerGamma + Log[x]+ EulerGamma Error Failure - Successful
6.2.E7 Ei ⁑ ( - x ) = - Ein ⁑ ( + x ) + ln ⁑ x + Ξ³ exponential-integral-Ei π‘₯ complementary-exponential-integral π‘₯ π‘₯ {\displaystyle{\displaystyle\mathrm{Ei}\left(-x\right)=-\mathrm{Ein}\left(+x% \right)+\ln x+\gamma}} Error -ExpIntegralEi[-(- x)]= - -ExpIntegralEi[-(+ x)] + Ln[+ x] + EulerGamma + Log[x]+ EulerGamma Error Failure - Successful
6.2.E9 Si ⁑ ( z ) = ∫ 0 z sin ⁑ t t ⁒ d t sine-integral 𝑧 superscript subscript 0 𝑧 𝑑 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Si}\left(z\right)=\int_{0}^{z}\frac{\sin t% }{t}\mathrm{d}t}} Si(z)= int((sin(t))/(t), t = 0..z) SinIntegral[z]= Integrate[Divide[Sin[t],t], {t, 0, z}] Successful Successful - -
6.2.E10 si ⁑ ( z ) = - ∫ z ∞ sin ⁑ t t ⁒ d t shifted-sine-integral 𝑧 superscript subscript 𝑧 𝑑 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{si}\left(z\right)=-\int_{z}^{\infty}\frac{% \sin t}{t}\mathrm{d}t}} Ssi(z)= - int((sin(t))/(t), t = z..infinity) SinIntegral[z] - Pi/2 = - Integrate[Divide[Sin[t],t], {t, z, Infinity}] Successful Successful - -
6.2.E10 - ∫ z ∞ sin ⁑ t t ⁒ d t = Si ⁑ ( z ) - 1 2 ⁒ Ο€ superscript subscript 𝑧 𝑑 𝑑 𝑑 sine-integral 𝑧 1 2 πœ‹ {\displaystyle{\displaystyle-\int_{z}^{\infty}\frac{\sin t}{t}\mathrm{d}t=% \mathrm{Si}\left(z\right)-\tfrac{1}{2}\pi}} - int((sin(t))/(t), t = z..infinity)= Si(z)-(1)/(2)*Pi - Integrate[Divide[Sin[t],t], {t, z, Infinity}]= SinIntegral[z]-Divide[1,2]*Pi Successful Successful - -
6.2.E11 Ci ⁑ ( z ) = - ∫ z ∞ cos ⁑ t t ⁒ d t cosine-integral 𝑧 superscript subscript 𝑧 𝑑 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Ci}(z)=-\int_{z}^{\infty}\frac{\cos t}{t}% \mathrm{d}t}} Ci(z)= - int((cos(t))/(t), t = z..infinity) CosIntegral[z]= - Integrate[Divide[Cos[t],t], {t, z, Infinity}] Successful Failure - Successful
6.2#Ex1 lim x β†’ ∞ ⁑ Si ⁑ ( x ) = 1 2 ⁒ Ο€ subscript β†’ π‘₯ sine-integral π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle\lim_{x\to\infty}\mathrm{Si}\left(x\right)=\tfrac{% 1}{2}\pi}} limit(Si(x), x = infinity)=(1)/(2)*Pi Limit[SinIntegral[x], x -> Infinity]=Divide[1,2]*Pi Successful Successful - -
6.2#Ex2 lim x β†’ ∞ ⁑ Ci ⁑ ( x ) = 0 subscript β†’ π‘₯ cosine-integral π‘₯ 0 {\displaystyle{\displaystyle\lim_{x\to\infty}\mathrm{Ci}\left(x\right)=0}} limit(Ci(x), x = infinity)= 0 Limit[CosIntegral[x], x -> Infinity]= 0 Successful Successful - -
6.2.E15 Shi ⁑ ( z ) = ∫ 0 z sinh ⁑ t t ⁒ d t hyperbolic-sine-integral 𝑧 superscript subscript 0 𝑧 𝑑 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Shi}\left(z\right)=\int_{0}^{z}\frac{\sinh t% }{t}\mathrm{d}t}} Shi(z)= int((sinh(t))/(t), t = 0..z) SinhIntegral[z]= Integrate[Divide[Sinh[t],t], {t, 0, z}] Successful Successful - -
6.2.E16 Chi ⁑ ( z ) = Ξ³ + ln ⁑ z + ∫ 0 z cosh ⁑ t - 1 t ⁒ d t hyperbolic-cosine-integral 𝑧 𝑧 superscript subscript 0 𝑧 𝑑 1 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{Chi}\left(z\right)=\gamma+\ln z+\int_{0}^{% z}\frac{\cosh t-1}{t}\mathrm{d}t}} Chi(z)= gamma + ln(z)+ int((cosh(t)- 1)/(t), t = 0..z) CoshIntegral[z]= EulerGamma + Log[z]+ Integrate[Divide[Cosh[t]- 1,t], {t, 0, z}] Successful Successful - -
6.4.E1 E 1 ⁑ ( z ) = Ein ⁑ ( z ) - Ln ⁑ z - Ξ³ exponential-integral 𝑧 complementary-exponential-integral 𝑧 multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle E_{1}\left(z\right)=\mathrm{Ein}\left(z\right)-% \operatorname{Ln}z-\gamma}} Error -ExpIntegralEi[-(z)]= -ExpIntegralEi[-(z)] + Ln[z] + EulerGamma - Log[z]- EulerGamma Error Failure - Successful
6.4.E2 E 1 ⁑ ( z ⁒ e 2 ⁒ m ⁒ Ο€ ⁒ i ) = E 1 ⁑ ( z ) - 2 ⁒ m ⁒ Ο€ ⁒ i exponential-integral 𝑧 superscript 𝑒 2 π‘š πœ‹ 𝑖 exponential-integral 𝑧 2 π‘š πœ‹ 𝑖 {\displaystyle{\displaystyle E_{1}\left(ze^{2m\pi i}\right)=E_{1}\left(z\right% )-2m\pi i}} Ei(z*exp(2*m*Pi*I))= Ei(z)- 2*m*Pi*I -ExpIntegralEi[-(z*Exp[2*m*Pi*I])]= -ExpIntegralEi[-(z)]- 2*m*Pi*I Failure Failure
Fail
.6e-8+18.84955592*I <- {z = 2^(1/2)+I*2^(1/2), m = 3}
-.6e-8+18.84955592*I <- {z = 2^(1/2)-I*2^(1/2), m = 3}
-.34e-9+18.84955592*I <- {z = -2^(1/2)-I*2^(1/2), m = 3}
.34e-9+18.84955592*I <- {z = -2^(1/2)+I*2^(1/2), m = 3}
Fail
Complex[0.0, 18.84955592153876] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 18.84955592153876] <- {Rule[m, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 18.84955592153876] <- {Rule[m, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 18.84955592153876] <- {Rule[m, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.4.E3 E 1 ⁑ ( z ⁒ e + Ο€ ⁒ i ) = Ein ⁑ ( - z ) - ln ⁑ z - Ξ³ - Ο€ ⁒ i exponential-integral 𝑧 superscript 𝑒 πœ‹ 𝑖 complementary-exponential-integral 𝑧 𝑧 πœ‹ 𝑖 {\displaystyle{\displaystyle E_{1}\left(ze^{+\pi i}\right)=\mathrm{Ein}\left(-% z\right)-\ln z-\gamma-\pi i}} Error -ExpIntegralEi[-(z*Exp[+ Pi*I])]= -ExpIntegralEi[-(- z)] + Ln[- z] + EulerGamma - Log[z]- EulerGamma - Pi*I Error Failure - Successful
6.4.E3 E 1 ⁑ ( z ⁒ e - Ο€ ⁒ i ) = Ein ⁑ ( - z ) - ln ⁑ z - Ξ³ + Ο€ ⁒ i exponential-integral 𝑧 superscript 𝑒 πœ‹ 𝑖 complementary-exponential-integral 𝑧 𝑧 πœ‹ 𝑖 {\displaystyle{\displaystyle E_{1}\left(ze^{-\pi i}\right)=\mathrm{Ein}\left(-% z\right)-\ln z-\gamma+\pi i}} Error -ExpIntegralEi[-(z*Exp[- Pi*I])]= -ExpIntegralEi[-(- z)] + Ln[- z] + EulerGamma - Log[z]- EulerGamma + Pi*I Error Failure - Successful
6.4.E4 Ci ⁑ ( z ⁒ e + Ο€ ⁒ i ) = + Ο€ ⁒ i + Ci ⁑ ( z ) cosine-integral 𝑧 superscript 𝑒 πœ‹ 𝑖 πœ‹ 𝑖 cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Ci}\left(ze^{+\pi i}\right)=+\pi i+\mathrm% {Ci}\left(z\right)}} Ci(z*exp(+ Pi*I))= + Pi*I + Ci(z) CosIntegral[z*Exp[+ Pi*I]]= + Pi*I + CosIntegral[z] Failure Failure
Fail
0.-6.283185307*I <- {z = 2^(1/2)+I*2^(1/2)}
0.-6.283185307*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.0, -6.283185307179586] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -6.283185307179586] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.4.E4 Ci ⁑ ( z ⁒ e - Ο€ ⁒ i ) = - Ο€ ⁒ i + Ci ⁑ ( z ) cosine-integral 𝑧 superscript 𝑒 πœ‹ 𝑖 πœ‹ 𝑖 cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Ci}\left(ze^{-\pi i}\right)=-\pi i+\mathrm% {Ci}\left(z\right)}} Ci(z*exp(- Pi*I))= - Pi*I + Ci(z) CosIntegral[z*Exp[- Pi*I]]= - Pi*I + CosIntegral[z] Failure Failure
Fail
0.+6.283185307*I <- {z = 2^(1/2)-I*2^(1/2)}
0.+6.283185307*I <- {z = -2^(1/2)-I*2^(1/2)}
Fail
Complex[0.0, 6.283185307179586] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
6.4.E5 Chi ⁑ ( z ⁒ e + Ο€ ⁒ i ) = + Ο€ ⁒ i + Chi ⁑ ( z ) hyperbolic-cosine-integral 𝑧 superscript 𝑒 πœ‹ 𝑖 πœ‹ 𝑖 hyperbolic-cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Chi}\left(ze^{+\pi i}\right)=+\pi i+% \mathrm{Chi}\left(z\right)}} Chi(z*exp(+ Pi*I))= + Pi*I + Chi(z) CoshIntegral[z*Exp[+ Pi*I]]= + Pi*I + CoshIntegral[z] Failure Failure
Fail
0.-6.283185307*I <- {z = 2^(1/2)+I*2^(1/2)}
0.-6.283185307*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.0, -6.283185307179586] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -6.283185307179586] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.4.E5 Chi ⁑ ( z ⁒ e - Ο€ ⁒ i ) = - Ο€ ⁒ i + Chi ⁑ ( z ) hyperbolic-cosine-integral 𝑧 superscript 𝑒 πœ‹ 𝑖 πœ‹ 𝑖 hyperbolic-cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Chi}\left(ze^{-\pi i}\right)=-\pi i+% \mathrm{Chi}\left(z\right)}} Chi(z*exp(- Pi*I))= - Pi*I + Chi(z) CoshIntegral[z*Exp[- Pi*I]]= - Pi*I + CoshIntegral[z] Failure Failure
Fail
0.+6.283185307*I <- {z = 2^(1/2)-I*2^(1/2)}
0.+6.283185307*I <- {z = -2^(1/2)-I*2^(1/2)}
Fail
Complex[0.0, 6.283185307179586] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
6.5.E1 E 1 ⁑ ( - x + i ⁒ 0 ) = - Ei ⁑ ( x ) - i ⁒ Ο€ exponential-integral π‘₯ 𝑖 0 exponential-integral-Ei π‘₯ 𝑖 πœ‹ {\displaystyle{\displaystyle E_{1}\left(-x+i0\right)=-\mathrm{Ei}\left(x\right% )-i\pi}} Error -ExpIntegralEi[-(- x + I*0)]= - -ExpIntegralEi[-(x)]- I*Pi Error Failure -
Fail
Complex[-1.6757338819604166, 3.141592653589793] <- {Rule[x, 1]}
Complex[-4.905333845293828, 3.141592653589793] <- {Rule[x, 2]}
Complex[-9.920784189531217, 3.141592653589793] <- {Rule[x, 3]}
6.5.E1 E 1 ⁑ ( - x - i ⁒ 0 ) = - Ei ⁑ ( x ) + i ⁒ Ο€ exponential-integral π‘₯ 𝑖 0 exponential-integral-Ei π‘₯ 𝑖 πœ‹ {\displaystyle{\displaystyle E_{1}\left(-x-i0\right)=-\mathrm{Ei}\left(x\right% )+i\pi}} Error -ExpIntegralEi[-(- x - I*0)]= - -ExpIntegralEi[-(x)]+ I*Pi Error Failure -
Fail
Complex[-1.6757338819604166, -3.141592653589793] <- {Rule[x, 1]}
Complex[-4.905333845293828, -3.141592653589793] <- {Rule[x, 2]}
Complex[-9.920784189531217, -3.141592653589793] <- {Rule[x, 3]}
6.5.E2 Ei ⁑ ( x ) = - 1 2 ⁒ ( E 1 ⁑ ( - x + i ⁒ 0 ) + E 1 ⁑ ( - x - i ⁒ 0 ) ) exponential-integral-Ei π‘₯ 1 2 exponential-integral π‘₯ 𝑖 0 exponential-integral π‘₯ 𝑖 0 {\displaystyle{\displaystyle\mathrm{Ei}\left(x\right)=-\tfrac{1}{2}(E_{1}\left% (-x+i0\right)+E_{1}\left(-x-i0\right))}} Error -ExpIntegralEi[-(x)]= -Divide[1,2]*(-ExpIntegralEi[-(- x + I*0)]+ -ExpIntegralEi[-(- x - I*0)]) Error Failure -
Fail
-1.6757338819604166 <- {Rule[x, 1]}
-4.905333845293828 <- {Rule[x, 2]}
-9.920784189531217 <- {Rule[x, 3]}
6.5.E3 1 2 ⁒ ( Ei ⁑ ( x ) + E 1 ⁑ ( x ) ) = Shi ⁑ ( x ) 1 2 exponential-integral-Ei π‘₯ exponential-integral π‘₯ hyperbolic-sine-integral π‘₯ {\displaystyle{\displaystyle\tfrac{1}{2}(\mathrm{Ei}\left(x\right)+E_{1}\left(% x\right))=\mathrm{Shi}\left(x\right)}} Error Divide[1,2]*(-ExpIntegralEi[-(x)]+ -ExpIntegralEi[-(x)])= SinhIntegral[x] Error Failure -
Fail
-0.8378669409802083 <- {Rule[x, 1]}
-2.4526669226469147 <- {Rule[x, 2]}
-4.960392094765611 <- {Rule[x, 3]}
6.5.E3 Shi ⁑ ( x ) = - i ⁒ Si ⁑ ( i ⁒ x ) hyperbolic-sine-integral π‘₯ 𝑖 sine-integral 𝑖 π‘₯ {\displaystyle{\displaystyle\mathrm{Shi}\left(x\right)=-i\mathrm{Si}\left(ix% \right)}} Shi(x)= - I*Si(I*x) SinhIntegral[x]= - I*SinIntegral[I*x] Successful Successful - -
6.5.E4 1 2 ⁒ ( Ei ⁑ ( x ) - E 1 ⁑ ( x ) ) = Chi ⁑ ( x ) 1 2 exponential-integral-Ei π‘₯ exponential-integral π‘₯ hyperbolic-cosine-integral π‘₯ {\displaystyle{\displaystyle\tfrac{1}{2}(\mathrm{Ei}\left(x\right)-E_{1}\left(% x\right))=\mathrm{Chi}\left(x\right)}} Error Divide[1,2]*(-ExpIntegralEi[-(x)]- -ExpIntegralEi[-(x)])= CoshIntegral[x] Error Failure -
Fail
-0.8378669409802083 <- {Rule[x, 1]}
-2.452666922646914 <- {Rule[x, 2]}
-4.960392094765608 <- {Rule[x, 3]}
6.5.E4 Chi ⁑ ( x ) = Ci ⁑ ( i ⁒ x ) - 1 2 ⁒ Ο€ ⁒ i hyperbolic-cosine-integral π‘₯ cosine-integral 𝑖 π‘₯ 1 2 πœ‹ 𝑖 {\displaystyle{\displaystyle\mathrm{Chi}\left(x\right)=\mathrm{Ci}\left(ix% \right)-\tfrac{1}{2}\pi i}} Chi(x)= Ci(I*x)-(1)/(2)*Pi*I CoshIntegral[x]= CosIntegral[I*x]-Divide[1,2]*Pi*I Failure Failure Successful Successful
6.5.E5 Si ⁑ ( z ) = 1 2 ⁒ i ⁒ ( E 1 ⁑ ( - i ⁒ z ) - E 1 ⁑ ( i ⁒ z ) ) + 1 2 ⁒ Ο€ sine-integral 𝑧 1 2 𝑖 exponential-integral 𝑖 𝑧 exponential-integral 𝑖 𝑧 1 2 πœ‹ {\displaystyle{\displaystyle\mathrm{Si}\left(z\right)=\tfrac{1}{2}i(E_{1}\left% (-iz\right)-E_{1}\left(iz\right))+\tfrac{1}{2}\pi}} Si(z)=(1)/(2)*I*(Ei(- I*z)- Ei(I*z))+(1)/(2)*Pi SinIntegral[z]=Divide[1,2]*I*(-ExpIntegralEi[-(- I*z)]- -ExpIntegralEi[-(I*z)])+Divide[1,2]*Pi Failure Failure
Fail
-3.141592654+0.*I <- {z = 2^(1/2)+I*2^(1/2)}
-3.141592654+0.*I <- {z = 2^(1/2)-I*2^(1/2)}
Fail
Complex[-3.141592653589793, 0.0] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.141592653589793, 0.0] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
6.5.E6 Ci ⁑ ( z ) = - 1 2 ⁒ ( E 1 ⁑ ( i ⁒ z ) + E 1 ⁑ ( - i ⁒ z ) ) cosine-integral 𝑧 1 2 exponential-integral 𝑖 𝑧 exponential-integral 𝑖 𝑧 {\displaystyle{\displaystyle\mathrm{Ci}\left(z\right)=-\tfrac{1}{2}(E_{1}\left% (iz\right)+E_{1}\left(-iz\right))}} Ci(z)= -(1)/(2)*(Ei(I*z)+ Ei(- I*z)) CosIntegral[z]= -Divide[1,2]*(-ExpIntegralEi[-(I*z)]+ -ExpIntegralEi[-(- I*z)]) Failure Failure
Fail
2.208978234-.399630454*I <- {z = 2^(1/2)+I*2^(1/2)}
2.208978234+.399630454*I <- {z = 2^(1/2)-I*2^(1/2)}
2.208978234-3.541223107*I <- {z = -2^(1/2)-I*2^(1/2)}
2.208978234+3.541223107*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.6.E1 Ei ⁑ ( x ) = Ξ³ + ln ⁑ x + βˆ‘ n = 1 ∞ x n n ! ⁒ n exponential-integral-Ei π‘₯ π‘₯ superscript subscript 𝑛 1 superscript π‘₯ 𝑛 𝑛 𝑛 {\displaystyle{\displaystyle\mathrm{Ei}\left(x\right)=\gamma+\ln x+\sum_{n=1}^% {\infty}\frac{x^{n}}{n!\thinspace n}}} Error -ExpIntegralEi[-(x)]= EulerGamma + Log[x]+ Sum[Divide[(x)^(n),(n)!*n], {n, 1, Infinity}] Error Failure -
Fail
Complex[0.10555368991298714, 0.0] <- {Rule[x, Rational[1, 2]]}
6.6.E2 E 1 ⁑ ( z ) = - Ξ³ - ln ⁑ z - βˆ‘ n = 1 ∞ ( - 1 ) n ⁒ z n n ! ⁒ n exponential-integral 𝑧 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑧 𝑛 𝑛 𝑛 {\displaystyle{\displaystyle E_{1}\left(z\right)=-\gamma-\ln z-\sum_{n=1}^{% \infty}\frac{(-1)^{n}z^{n}}{n!\thinspace n}}} Ei(z)= - gamma - ln(z)- sum(((- 1)^(n)* (z)^(n))/(factorial(n)*n), n = 1..infinity) -ExpIntegralEi[-(z)]= - EulerGamma - Log[z]- Sum[Divide[(- 1)^(n)* (z)^(n),(n)!*n], {n, 1, Infinity}] Failure Failure Skip
Fail
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.6.E3 E 1 ⁑ ( z ) = - ln ⁑ z + e - z ⁒ βˆ‘ n = 0 ∞ z n n ! ⁒ ψ ⁑ ( n + 1 ) exponential-integral 𝑧 𝑧 superscript 𝑒 𝑧 superscript subscript 𝑛 0 superscript 𝑧 𝑛 𝑛 digamma 𝑛 1 {\displaystyle{\displaystyle E_{1}\left(z\right)=-\ln z+e^{-z}\sum_{n=0}^{% \infty}\frac{z^{n}}{n!}\psi\left(n+1\right)}} Ei(z)= - ln(z)+ exp(- z)*sum(((z)^(n))/(factorial(n))*Psi(n + 1), n = 0..infinity) -ExpIntegralEi[-(z)]= - Log[z]+ Exp[- z]*Sum[Divide[(z)^(n),(n)!]*PolyGamma[n + 1], {n, 0, Infinity}] Error Failure -
Fail
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.6.E4 Ein ⁑ ( z ) = βˆ‘ n = 1 ∞ ( - 1 ) n - 1 ⁒ z n n ! ⁒ n complementary-exponential-integral 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 1 superscript 𝑧 𝑛 𝑛 𝑛 {\displaystyle{\displaystyle\mathrm{Ein}\left(z\right)=\sum_{n=1}^{\infty}% \frac{(-1)^{n-1}z^{n}}{n!\thinspace n}}} Error -ExpIntegralEi[-(z)] + Ln[z] + EulerGamma = Sum[Divide[(- 1)^(n - 1)* (z)^(n),(n)!*n], {n, 1, Infinity}] Error Failure - Successful
6.6.E5 Si ⁑ ( z ) = βˆ‘ n = 0 ∞ ( - 1 ) n ⁒ z 2 ⁒ n + 1 ( 2 ⁒ n + 1 ) ! ⁒ ( 2 ⁒ n + 1 ) sine-integral 𝑧 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑧 2 𝑛 1 2 𝑛 1 2 𝑛 1 {\displaystyle{\displaystyle\mathrm{Si}\left(z\right)=\sum_{n=0}^{\infty}\frac% {(-1)^{n}z^{2n+1}}{(2n+1)!(2n+1)}}} Si(z)= sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(2*n + 1)*(2*n + 1)), n = 0..infinity) SinIntegral[z]= Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(2*n + 1)!*(2*n + 1)], {n, 0, Infinity}] Successful Successful - -
6.6.E6 Ci ⁑ ( z ) = Ξ³ + ln ⁑ z + βˆ‘ n = 1 ∞ ( - 1 ) n ⁒ z 2 ⁒ n ( 2 ⁒ n ) ! ⁒ ( 2 ⁒ n ) cosine-integral 𝑧 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑧 2 𝑛 2 𝑛 2 𝑛 {\displaystyle{\displaystyle\mathrm{Ci}\left(z\right)=\gamma+\ln z+\sum_{n=1}^% {\infty}\frac{(-1)^{n}z^{2n}}{(2n)!(2n)}}} Ci(z)= gamma + ln(z)+ sum(((- 1)^(n)* (z)^(2*n))/(factorial(2*n)*(2*n)), n = 1..infinity) CosIntegral[z]= EulerGamma + Log[z]+ Sum[Divide[(- 1)^(n)* (z)^(2*n),(2*n)!*(2*n)], {n, 1, Infinity}] Successful Successful - -
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 π‘Ž 𝑑 𝑑 𝑏 𝑑 superscript subscript 0 superscript 𝑒 𝑖 π‘Ž 𝑑 𝑑 𝑖 𝑏 𝑑 {\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((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}]= Integrate[Divide[Exp[I*a*t],t + I*b], {t, 0, Infinity}] Failure Failure Skip Error
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 π‘Ž 𝑑 𝑑 𝑖 𝑏 𝑑 superscript 𝑒 π‘Ž 𝑏 exponential-integral π‘Ž 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{e^{iat}}{t+ib}\mathrm{d}t=e% ^{ab}E_{1}\left(ab\right)}} 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}]= Exp[a*b]*-ExpIntegralEi[-(a*b)] Failure Failure Skip Error
6.7.E2 e x ⁒ ∫ 0 Ξ± e - x ⁒ t 1 - t ⁒ d t = Ei ⁑ ( x ) - Ei ⁑ ( ( 1 - Ξ± ) ⁒ x ) superscript 𝑒 π‘₯ superscript subscript 0 𝛼 superscript 𝑒 π‘₯ 𝑑 1 𝑑 𝑑 exponential-integral-Ei π‘₯ exponential-integral-Ei 1 𝛼 π‘₯ {\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)}} Error Exp[x]*Integrate[Divide[Exp[- x*t],1 - t], {t, 0, \[Alpha]}]= -ExpIntegralEi[-(x)]- -ExpIntegralEi[-((1 - \[Alpha])* x)] Error Failure -
Fail
Complex[-0.42316473444652486, 0.4289071640857123] <- {Rule[x, Rational[1, 2]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.42316473444652486, -0.4289071640857123] <- {Rule[x, Rational[1, 2]], Rule[Ξ±, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.36486876599473, 1.1603687293382365] <- {Rule[x, Rational[1, 2]], Rule[Ξ±, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.36486876599473, -1.1603687293382365] <- {Rule[x, Rational[1, 2]], Rule[Ξ±, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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 π‘₯ superscript 𝑒 𝑖 𝑑 superscript π‘Ž 2 superscript 𝑑 2 𝑑 𝑖 2 π‘Ž superscript 𝑒 π‘Ž exponential-integral π‘Ž 𝑖 π‘₯ superscript 𝑒 π‘Ž exponential-integral π‘Ž 𝑖 π‘₯ {\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((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}]=Divide[I,2*a]*(Exp[a]*-ExpIntegralEi[-(a - I*x)]- Exp[- a]*-ExpIntegralEi[-(- a - I*x)]) Failure Failure Skip Error
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 π‘₯ 𝑑 superscript 𝑒 𝑖 𝑑 superscript π‘Ž 2 superscript 𝑑 2 𝑑 1 2 superscript 𝑒 π‘Ž exponential-integral π‘Ž 𝑖 π‘₯ superscript 𝑒 π‘Ž exponential-integral π‘Ž 𝑖 π‘₯ {\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((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}]=Divide[1,2]*(Exp[a]*-ExpIntegralEi[-(a - I*x)]+ Exp[- a]*-ExpIntegralEi[-(- a - I*x)]) Failure Failure Skip Error
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 π‘₯ superscript 𝑒 𝑑 superscript π‘Ž 2 superscript 𝑑 2 𝑑 1 2 π‘Ž 𝑖 superscript 𝑒 𝑖 π‘Ž exponential-integral π‘₯ 𝑖 π‘Ž superscript 𝑒 𝑖 π‘Ž exponential-integral π‘₯ 𝑖 π‘Ž {\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((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}]= -Divide[1,2*a*I]*(Exp[I*a]*-ExpIntegralEi[-(x + I*a)]- Exp[- I*a]*-ExpIntegralEi[-(x - I*a)]) Error Failure - Error
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 π‘₯ 𝑑 superscript 𝑒 𝑑 superscript π‘Ž 2 superscript 𝑑 2 𝑑 1 2 superscript 𝑒 𝑖 π‘Ž exponential-integral π‘₯ 𝑖 π‘Ž superscript 𝑒 𝑖 π‘Ž exponential-integral π‘₯ 𝑖 π‘Ž {\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((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}]=Divide[1,2]*(Exp[I*a]*-ExpIntegralEi[-(x + I*a)]+ Exp[- I*a]*-ExpIntegralEi[-(x - I*a)]) Error Failure - Error
6.7.E7 ∫ 0 1 e - a ⁒ t ⁒ sin ⁑ ( b ⁒ t ) t ⁒ d t = β„‘ ⁑ Ein ⁑ ( a + i ⁒ b ) superscript subscript 0 1 superscript 𝑒 π‘Ž 𝑑 𝑏 𝑑 𝑑 𝑑 complementary-exponential-integral π‘Ž 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)}} Error Integrate[Divide[Exp[- a*t]*Sin[b*t],t], {t, 0, 1}]= Im[-ExpIntegralEi[-(a + I*b)] + Ln[a + I*b] + EulerGamma] Error Failure - Successful
6.7.E8 ∫ 0 1 e - a ⁒ t ⁒ ( 1 - cos ⁑ ( b ⁒ t ) ) t ⁒ d t = β„œ ⁑ Ein ⁑ ( a + i ⁒ b ) - Ein ⁑ ( a ) superscript subscript 0 1 superscript 𝑒 π‘Ž 𝑑 1 𝑏 𝑑 𝑑 𝑑 complementary-exponential-integral π‘Ž 𝑖 𝑏 complementary-exponential-integral π‘Ž {\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)}} Error Integrate[Divide[Exp[- a*t]*(1 - Cos[b*t]),t], {t, 0, 1}]= Re[-ExpIntegralEi[-(a + I*b)] + Ln[a + I*b] + EulerGamma]- -ExpIntegralEi[-(a)] + Ln[a] + EulerGamma Error Failure - Successful
6.7.E9 si ⁑ ( z ) = - ∫ 0 Ο€ / 2 e - z ⁒ cos ⁑ t ⁒ cos ⁑ ( z ⁒ sin ⁑ t ) ⁒ d t shifted-sine-integral 𝑧 superscript subscript 0 πœ‹ 2 superscript 𝑒 𝑧 𝑑 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathrm{si}\left(z\right)=-\int_{0}^{\pi/2}e^{-z% \cos t}\cos\left(z\sin t\right)\mathrm{d}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}] Failure Failure Skip Error
6.7.E13 ∫ 0 ∞ sin ⁑ t t + z ⁒ d t = ∫ 0 ∞ e - z ⁒ t t 2 + 1 ⁒ d t superscript subscript 0 𝑑 𝑑 𝑧 𝑑 superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 2 1 𝑑 {\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((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}]= Integrate[Divide[Exp[- z*t],(t)^(2)+ 1], {t, 0, Infinity}] Failure Failure Skip Error
6.7.E14 ∫ 0 ∞ cos ⁑ t t + z ⁒ d t = ∫ 0 ∞ t ⁒ e - z ⁒ t t 2 + 1 ⁒ d t superscript subscript 0 𝑑 𝑑 𝑧 𝑑 superscript subscript 0 𝑑 superscript 𝑒 𝑧 𝑑 superscript 𝑑 2 1 𝑑 {\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((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}]= Integrate[Divide[t*Exp[- z*t],(t)^(2)+ 1], {t, 0, Infinity}] Failure Failure Skip Successful
6.8.E1 1 2 ⁒ ln ⁑ ( 1 + 2 x ) < e x ⁒ E 1 ⁑ ( x ) 1 2 1 2 π‘₯ superscript 𝑒 π‘₯ exponential-integral π‘₯ {\displaystyle{\displaystyle\frac{1}{2}\ln\left(1+\frac{2}{x}\right)<e^{x}E_{1% }\left(x\right)}} (1)/(2)*ln(1 +(2)/(x))< exp(x)*Ei(x) Divide[1,2]*Log[1 +Divide[2,x]]< Exp[x]*-ExpIntegralEi[-(x)] Failure Failure Successful Successful
6.8.E1 e x ⁒ E 1 ⁑ ( x ) < ln ⁑ ( 1 + 1 x ) superscript 𝑒 π‘₯ exponential-integral π‘₯ 1 1 π‘₯ {\displaystyle{\displaystyle e^{x}E_{1}\left(x\right)<\ln\left(1+\frac{1}{x}% \right)}} exp(x)*Ei(x)< ln(1 +(1)/(x)) Exp[x]*-ExpIntegralEi[-(x)]< Log[1 +Divide[1,x]] Failure Failure
Fail
5.151464321 < .6931471806 <- {x = 1}
36.60711558 < .4054651081 <- {x = 2}
199.5263609 < .2876820722 <- {x = 3}
 Successful
6.8.E2 x x + 1 < x ⁒ e x ⁒ E 1 ⁑ ( x ) π‘₯ π‘₯ 1 π‘₯ superscript 𝑒 π‘₯ exponential-integral π‘₯ {\displaystyle{\displaystyle\frac{x}{x+1}<xe^{x}E_{1}\left(x\right)}} (x)/(x + 1)< x*exp(x)*Ei(x) Divide[x,x + 1]< x*Exp[x]*-ExpIntegralEi[-(x)] Failure Failure Successful Successful
6.8.E2 x ⁒ e x ⁒ E 1 ⁑ ( x ) < x + 1 x + 2 π‘₯ superscript 𝑒 π‘₯ exponential-integral π‘₯ π‘₯ 1 π‘₯ 2 {\displaystyle{\displaystyle xe^{x}E_{1}\left(x\right)<\frac{x+1}{x+2}}} x*exp(x)*Ei(x)<(x + 1)/(x + 2) x*Exp[x]*-ExpIntegralEi[-(x)]<Divide[x + 1,x + 2] Failure Failure
Fail
5.151464321 < .6666666667 <- {x = 1}
73.21423116 < .7500000000 <- {x = 2}
598.5790827 < .8000000000 <- {x = 3}
 Successful
6.8.E3 x ⁒ ( x + 3 ) x 2 + 4 ⁒ x + 2 < x ⁒ e x ⁒ E 1 ⁑ ( x ) π‘₯ π‘₯ 3 superscript π‘₯ 2 4 π‘₯ 2 π‘₯ superscript 𝑒 π‘₯ exponential-integral π‘₯ {\displaystyle{\displaystyle\frac{x(x+3)}{x^{2}+4x+2}<xe^{x}E_{1}\left(x\right% )}} (x*(x + 3))/((x)^(2)+ 4*x + 2)< x*exp(x)*Ei(x) Divide[x*(x + 3),(x)^(2)+ 4*x + 2]< x*Exp[x]*-ExpIntegralEi[-(x)] Failure Failure Successful Successful
6.8.E3 x ⁒ e x ⁒ E 1 ⁑ ( x ) < x 2 + 5 ⁒ x + 2 x 2 + 6 ⁒ x + 6 π‘₯ superscript 𝑒 π‘₯ exponential-integral π‘₯ superscript π‘₯ 2 5 π‘₯ 2 superscript π‘₯ 2 6 π‘₯ 6 {\displaystyle{\displaystyle xe^{x}E_{1}\left(x\right)<\frac{x^{2}+5x+2}{x^{2}% +6x+6}}} x*exp(x)*Ei(x)<((x)^(2)+ 5*x + 2)/((x)^(2)+ 6*x + 6) x*Exp[x]*-ExpIntegralEi[-(x)]<Divide[(x)^(2)+ 5*x + 2,(x)^(2)+ 6*x + 6] Failure Failure
Fail
5.151464321 < .6153846154 <- {x = 1}
73.21423116 < .7272727273 <- {x = 2}
598.5790827 < .7878787879 <- {x = 3}
 Successful
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}}} Error SinIntegral[z]= z*Sum[(SphericalBesselJ[n, Divide[1,2]*z])^(2), {n, 0, Infinity}] Error Successful - -
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}}} Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]*Divide[1,2]*x*))^(2), {n, 0, Infinity}] Error Error - -
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)}} Error \|Sqrt[1/2 Pi /$2] BesselI[-0 - 1/2, 0]*Divide[1,2]*z*+ Sum[Divide[2*n + 1,n*(n + 1)]*Sqrt[1/2 Pi /$2] BesselI[(-1)^(1-1)*n + 1/2, n]\|\|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]*Divide[1,2]*z, {n, 1, Infinity}]) Error Error - -
6.11.E1 E 1 ⁑ ( z ) = Ξ“ ⁑ ( 0 , z ) exponential-integral 𝑧 incomplete-Gamma 0 𝑧 {\displaystyle{\displaystyle E_{1}\left(z\right)=\Gamma\left(0,z\right)}} Ei(z)= GAMMA(0, z) -ExpIntegralEi[-(z)]= Gamma[0, z] Failure Failure
Fail
2.208978234+3.541223107*I <- {z = 2^(1/2)+I*2^(1/2)}
2.208978234-3.541223107*I <- {z = 2^(1/2)-I*2^(1/2)}
2.208978234-2.741962200*I <- {z = -2^(1/2)-I*2^(1/2)}
2.208978234+2.741962200*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.11.E2 E 1 ⁑ ( z ) = e - z ⁒ U ⁑ ( 1 , 1 , z ) exponential-integral 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 1 1 𝑧 {\displaystyle{\displaystyle E_{1}\left(z\right)=e^{-z}U\left(1,1,z\right)}} Ei(z)= exp(- z)*KummerU(1, 1, z) -ExpIntegralEi[-(z)]= Exp[- z]*HypergeometricU[1, 1, z] Failure Failure
Fail
2.208978234+3.541223107*I <- {z = 2^(1/2)+I*2^(1/2)}
2.208978234-3.541223107*I <- {z = 2^(1/2)-I*2^(1/2)}
2.208978234-2.741962200*I <- {z = -2^(1/2)-I*2^(1/2)}
2.208978234+2.741962200*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.14.E1 ∫ 0 ∞ e - a ⁒ t ⁒ E 1 ⁑ ( t ) ⁒ d t = 1 a ⁒ ln ⁑ ( 1 + a ) superscript subscript 0 superscript 𝑒 π‘Ž 𝑑 exponential-integral 𝑑 𝑑 1 π‘Ž 1 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}E_{1}\left(t\right)\mathrm% {d}t=\frac{1}{a}\ln\left(1+a\right)}} int(exp(- a*t)*Ei(t), t = 0..infinity)=(1)/(a)*ln(1 + a) Integrate[Exp[- a*t]*-ExpIntegralEi[-(t)], {t, 0, Infinity}]=Divide[1,a]*Log[1 + a] Failure Failure Skip Successful
6.14.E2 ∫ 0 ∞ e - a ⁒ t ⁒ Ci ⁑ ( t ) ⁒ d t = - 1 2 ⁒ a ⁒ ln ⁑ ( 1 + a 2 ) superscript subscript 0 superscript 𝑒 π‘Ž 𝑑 cosine-integral 𝑑 𝑑 1 2 π‘Ž 1 superscript π‘Ž 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\mathrm{Ci}\left(t\right)% \mathrm{d}t=-\frac{1}{2a}\ln\left(1+a^{2}\right)}} int(exp(- a*t)*Ci(t), t = 0..infinity)= -(1)/(2*a)*ln(1 + (a)^(2)) Integrate[Exp[- a*t]*CosIntegral[t], {t, 0, Infinity}]= -Divide[1,2*a]*Log[1 + (a)^(2)] Failure Failure Skip Error
6.14.E3 ∫ 0 ∞ e - a ⁒ t ⁒ si ⁑ ( t ) ⁒ d t = - 1 a ⁒ arctan ⁑ a superscript subscript 0 superscript 𝑒 π‘Ž 𝑑 shifted-sine-integral 𝑑 𝑑 1 π‘Ž π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\mathrm{si}\left(t\right)% \mathrm{d}t=-\frac{1}{a}\operatorname{arctan}a}} int(exp(- a*t)*Ssi(t), t = 0..infinity)= -(1)/(a)*arctan(a) Integrate[Exp[- a*t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}]= -Divide[1,a]*ArcTan[a] Failure Failure Skip Error
6.14.E4 ∫ 0 ∞ E 1 2 ⁑ ( t ) ⁒ d t = 2 ⁒ ln ⁑ 2 superscript subscript 0 exponential-integral 2 𝑑 𝑑 2 2 {\displaystyle{\displaystyle\int_{0}^{\infty}{E_{1}^{2}}\left(t\right)\mathrm{% d}t=2\ln 2}} int((Ei(t))^(2), t = 0..infinity)= 2*ln(2) Integrate[(-ExpIntegralEi[-(t)])^(2), {t, 0, Infinity}]= 2*Log[2] Failure Successful Skip -
6.14.E5 ∫ 0 ∞ cos ⁑ t ⁒ Ci ⁑ ( t ) ⁒ d t = ∫ 0 ∞ sin ⁑ t ⁒ si ⁑ ( t ) ⁒ d t superscript subscript 0 𝑑 cosine-integral 𝑑 𝑑 superscript subscript 0 𝑑 shifted-sine-integral 𝑑 𝑑 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos t\mathrm{Ci}\left(t\right)% \mathrm{d}t=\int_{0}^{\infty}\sin t\mathrm{si}\left(t\right)\mathrm{d}t}} int(cos(t)*Ci(t), t = 0..infinity)= int(sin(t)*Ssi(t), t = 0..infinity) Integrate[Cos[t]*CosIntegral[t], {t, 0, Infinity}]= Integrate[Sin[t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}] Failure Failure Skip
Fail
Complex[-2.199611725770543, -1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.199611725770543, 1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.6288153989756469, 1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.6288153989756469, -1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.14.E5 ∫ 0 ∞ sin ⁑ t ⁒ si ⁑ ( t ) ⁒ d t = - 1 4 ⁒ Ο€ superscript subscript 0 𝑑 shifted-sine-integral 𝑑 𝑑 1 4 πœ‹ {\displaystyle{\displaystyle\int_{0}^{\infty}\sin t\mathrm{si}\left(t\right)% \mathrm{d}t=-\tfrac{1}{4}\pi}} int(sin(t)*Ssi(t), t = 0..infinity)= -(1)/(4)*Pi Integrate[Sin[t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}]= -Divide[1,4]*Pi Successful Failure -
Fail
Complex[2.199611725770543, 1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.199611725770543, -1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.6288153989756469, -1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.6288153989756469, 1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[Sin[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.14.E6 ∫ 0 ∞ Ci 2 ⁑ ( t ) ⁒ d t = ∫ 0 ∞ si 2 ⁑ ( t ) ⁒ d t superscript subscript 0 cosine-integral 2 𝑑 𝑑 superscript subscript 0 shifted-sine-integral 2 𝑑 𝑑 {\displaystyle{\displaystyle\int_{0}^{\infty}{\mathrm{Ci}^{2}}\left(t\right)% \mathrm{d}t=\int_{0}^{\infty}{\mathrm{si}^{2}}\left(t\right)\mathrm{d}t}} int((Ci(t))^(2), t = 0..infinity)= int((Ssi(t))^(2), t = 0..infinity) Integrate[(CosIntegral[t])^(2), {t, 0, Infinity}]= Integrate[(SinIntegral[t] - Pi/2)^(2), {t, 0, Infinity}] Failure Successful Skip -
6.14.E6 ∫ 0 ∞ si 2 ⁑ ( t ) ⁒ d t = 1 2 ⁒ Ο€ superscript subscript 0 shifted-sine-integral 2 𝑑 𝑑 1 2 πœ‹ {\displaystyle{\displaystyle\int_{0}^{\infty}{\mathrm{si}^{2}}\left(t\right)% \mathrm{d}t=\tfrac{1}{2}\pi}} int((Ssi(t))^(2), t = 0..infinity)=(1)/(2)*Pi Integrate[(SinIntegral[t] - Pi/2)^(2), {t, 0, Infinity}]=Divide[1,2]*Pi Failure Successful Skip -
6.14.E7 ∫ 0 ∞ Ci ⁑ ( t ) ⁒ si ⁑ ( t ) ⁒ d t = ln ⁑ 2 superscript subscript 0 cosine-integral 𝑑 shifted-sine-integral 𝑑 𝑑 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\mathrm{Ci}\left(t\right)\mathrm{% si}\left(t\right)\mathrm{d}t=\ln 2}} int(Ci(t)*Ssi(t), t = 0..infinity)= ln(2) Integrate[CosIntegral[t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}]= Log[2] Failure Failure Skip
Fail
Complex[0.7210663818131499, 1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[CosIntegral[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.7210663818131499, -1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[CosIntegral[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.1073607429330403, -1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[CosIntegral[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.1073607429330403, 1.4142135623730951] <- {Rule[Integrate[Plus[Times[Rational[-1, 2], Pi], Times[CosIntegral[t], SinIntegral[t]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.15.E1 βˆ‘ n = 1 ∞ Ci ⁑ ( Ο€ ⁒ n ) = 1 2 ⁒ ( ln ⁑ 2 - Ξ³ ) superscript subscript 𝑛 1 cosine-integral πœ‹ 𝑛 1 2 2 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\mathrm{Ci}\left(\pi n\right)=% \tfrac{1}{2}(\ln 2-\gamma)}} sum(Ci(Pi*n), n = 1..infinity)=(1)/(2)*(ln(2)- gamma) Sum[CosIntegral[Pi*n], {n, 1, Infinity}]=Divide[1,2]*(Log[2]- EulerGamma) Failure Failure Skip
Fail
Complex[1.356247804543889, 1.4142135623730951] <- {Rule[Sum[CosIntegral[Times[n, Pi]], {n, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.356247804543889, -1.4142135623730951] <- {Rule[Sum[CosIntegral[Times[n, Pi]], {n, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4721793202023012, -1.4142135623730951] <- {Rule[Sum[CosIntegral[Times[n, Pi]], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4721793202023012, 1.4142135623730951] <- {Rule[Sum[CosIntegral[Times[n, Pi]], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.15.E2 βˆ‘ n = 1 ∞ si ⁑ ( Ο€ ⁒ n ) n = 1 2 ⁒ Ο€ ⁒ ( ln ⁑ Ο€ - 1 ) superscript subscript 𝑛 1 shifted-sine-integral πœ‹ 𝑛 𝑛 1 2 πœ‹ πœ‹ 1 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\mathrm{si}\left(\pi n% \right)}{n}=\tfrac{1}{2}\pi(\ln\pi-1)}} sum((Ssi(Pi*n))/(n), n = 1..infinity)=(1)/(2)*Pi*(ln(Pi)- 1) Sum[Divide[SinIntegral[Pi*n] - Pi/2,n], {n, 1, Infinity}]=Divide[1,2]*Pi*(Log[Pi]- 1) Failure Failure Skip
Fail
Complex[1.1868723893034128, 1.4142135623730951] <- {Rule[Sum[Times[Power[n, -1], Plus[Times[Rational[-1, 2], Pi], SinIntegral[Times[n, Pi]]]], {n, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.1868723893034128, -1.4142135623730951] <- {Rule[Sum[Times[Power[n, -1], Plus[Times[Rational[-1, 2], Pi], SinIntegral[Times[n, Pi]]]], {n, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6415547354427775, -1.4142135623730951] <- {Rule[Sum[Times[Power[n, -1], Plus[Times[Rational[-1, 2], Pi], SinIntegral[Times[n, Pi]]]], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6415547354427775, 1.4142135623730951] <- {Rule[Sum[Times[Power[n, -1], Plus[Times[Rational[-1, 2], Pi], SinIntegral[Times[n, Pi]]]], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.15.E3 βˆ‘ n = 1 ∞ ( - 1 ) n ⁒ Ci ⁑ ( 2 ⁒ Ο€ ⁒ n ) = 1 - ln ⁑ 2 - 1 2 ⁒ Ξ³ superscript subscript 𝑛 1 superscript 1 𝑛 cosine-integral 2 πœ‹ 𝑛 1 2 1 2 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}(-1)^{n}\mathrm{Ci}\left(2\pi n% \right)=1-\ln 2-\tfrac{1}{2}\gamma}} sum((- 1)^(n)* Ci(2*Pi*n), n = 1..infinity)= 1 - ln(2)-(1)/(2)*gamma Sum[(- 1)^(n)* CosIntegral[2*Pi*n], {n, 1, Infinity}]= 1 - Log[2]-Divide[1,2]*EulerGamma Failure Failure Skip
Fail
Complex[1.395968575383807, 1.4142135623730951] <- {Rule[Sum[Times[Power[-1, n], CosIntegral[Times[2, n, Pi]]], {n, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.395968575383807, -1.4142135623730951] <- {Rule[Sum[Times[Power[-1, n], CosIntegral[Times[2, n, Pi]]], {n, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4324585493623831, -1.4142135623730951] <- {Rule[Sum[Times[Power[-1, n], CosIntegral[Times[2, n, Pi]]], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4324585493623831, 1.4142135623730951] <- {Rule[Sum[Times[Power[-1, n], CosIntegral[Times[2, n, Pi]]], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.15.E4 βˆ‘ n = 1 ∞ ( - 1 ) n ⁒ si ⁑ ( 2 ⁒ Ο€ ⁒ n ) n = Ο€ ⁒ ( 3 2 ⁒ ln ⁑ 2 - 1 ) superscript subscript 𝑛 1 superscript 1 𝑛 shifted-sine-integral 2 πœ‹ 𝑛 𝑛 πœ‹ 3 2 2 1 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}(-1)^{n}\frac{\mathrm{si}\left(% 2\pi n\right)}{n}=\pi(\tfrac{3}{2}\ln 2-1)}} sum((- 1)^(n)*(Ssi(2*Pi*n))/(n), n = 1..infinity)= Pi*((3)/(2)*ln(2)- 1) Sum[(- 1)^(n)*Divide[SinIntegral[2*Pi*n] - Pi/2,n], {n, 1, Infinity}]= Pi*(Divide[3,2]*Log[2]- 1) Failure Failure Skip Error
6.16.E1 sin ⁑ x + 1 3 ⁒ sin ⁑ ( 3 ⁒ x ) + 1 5 ⁒ sin ⁑ ( 5 ⁒ x ) + … = { π‘₯ 1 3 3 π‘₯ 1 5 5 π‘₯ … cases {\displaystyle{\displaystyle\sin x+\tfrac{1}{3}\sin\left(3x\right)+\tfrac{1}{5% }\sin\left(5x\right)+\dots=\begin{cases}\frac{1}{4}\pi,&0}}\)% \@add@PDF@RDFa@triples\end{document}\end{cases} sin(x)+(1)/(3)*sin(3*x)+(1)/(5)*sin(5*x)+ .. = Sin[x]+Divide[1,3]*Sin[3*x]+Divide[1,5]*Sin[5*x]+ ... = Error Failure - -
6.16.E1 { cases {\displaystyle{\displaystyle\begin{cases}\frac{1}{4}\pi,&0<x}}\)% \@add@PDF@RDFa@triples\end{document}\end{cases} Error Failure - Error
6.16.E1 x < Ο€ , 0 , & ⁒ x π‘₯ πœ‹ 0 & π‘₯ {\displaystyle{\displaystyle x<\pi,\\ 0,&x}} x < Pi , 0 , x < Pi , 0 , Error Failure - Error
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 𝑑 superscript 𝑒 𝑧 𝑑 1 superscript 𝑑 2 superscript superscript 𝑑 2 1 superscript 𝑑 2 𝑛 𝑑 {\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((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}] Failure Failure Skip Error
6.18#Ex2 B n = ∫ 0 ∞ e - z ⁒ t 1 + t 2 ⁒ ( t 2 1 + t 2 ) n ⁒ d t subscript 𝐡 𝑛 superscript subscript 0 superscript 𝑒 𝑧 𝑑 1 superscript 𝑑 2 superscript superscript 𝑑 2 1 superscript 𝑑 2 𝑛 𝑑 {\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((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}] Failure Failure Skip Error
6.18#Ex3 C n = ∫ 0 ∞ e - z ⁒ t ⁒ ( t 2 1 + t 2 ) n ⁒ d t subscript 𝐢 𝑛 superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript superscript 𝑑 2 1 superscript 𝑑 2 𝑛 𝑑 {\displaystyle{\displaystyle C_{n}=\int_{0}^{\infty}e^{-zt}\left(\frac{t^{2}}{% 1+t^{2}}\right)^{n}\mathrm{d}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}] Failure Failure Skip Error
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