7.7

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
7.7.E1	${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{2}{\pi}e^{-z^{2}}\int_{0}^{\infty}\frac{e^{-z^{2}t^{2}}}{t^{2}+1}\mathrm{d}t}}$ \erfc@@{z} = \frac{2}{\pi}e^{-z^{2}}\int_{0}^{\infty}\frac{e^{-z^{2}t^{2}}}{t^{2}+1}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\frac{1}{4}\pi}}$	erfc(z) = (2)/(Pi)*exp(- (z)^(2))*int((exp(- (z)^(2)* (t)^(2)))/((t)^(2)+ 1), t = 0..infinity)	Erfc[z] == Divide[2,Pi]*Exp[- (z)^(2)]*Integrate[Divide[Exp[- (z)^(2)* (t)^(2)],(t)^(2)+ 1], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 4]
7.7.E2	${\displaystyle{\displaystyle\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\mathrm{d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\mathrm{d}t}{t^{2}-z^{2}}}}$ \frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\diff{t}}{t-z} = \frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\diff{t}}{t^{2}-z^{2}}		(1)/(Pi*I)*int((exp(- (t)^(2)))/(t - z), t = - infinity..infinity) = (2*z)/(Pi*I)*int((exp(- (t)^(2)))/((t)^(2)- (z)^(2)), t = 0..infinity)	Divide[1,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],t - z], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[2*z,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],(t)^(2)- (z)^(2)], {t, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [7 / 7] Result: .2137917882+.3702982391*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = I} Result: .572853371e-1-.2137917880*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = I} ... skip entries to safe data	Successful [Tested: 1]
7.7.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at^{2}+2izt}\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-z^{2}/a}+\frac{i}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right)}}$ \int_{0}^{\infty}e^{-at^{2}+2izt}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-z^{2}/a}+\frac{i}{\sqrt{a}}\DawsonsintF@{\frac{z}{\sqrt{a}}}	${\displaystyle{\displaystyle\Re a>0}}$	int(exp(- a*(t)^(2)+ 2*I*z*t), t = 0..infinity) = (1)/(2)*sqrt((Pi)/(a))*exp(- (z)^(2)/a)+(I)/(sqrt(a))*dawson((z)/(sqrt(a)))	Integrate[Exp[- a*(t)^(2)+ 2*I*z*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[- (z)^(2)/a]+Divide[I,Sqrt[a]]*DawsonF[Divide[z,Sqrt[a]]]	Failure	Successful	Successful [Tested: 21]	Successful [Tested: 21]
7.7.E4	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}\mathrm{d}t=\sqrt{\frac{\pi}{a}}e^{az^{2}}\operatorname{erfc}\left(\sqrt{a}z\right)}}$ \int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}\diff{t} = \sqrt{\frac{\pi}{a}}e^{az^{2}}\erfc@{\sqrt{a}z}	${\displaystyle{\displaystyle\Re a>0,\Re z>0}}$	int((exp(- a*t))/(sqrt(t + (z)^(2))), t = 0..infinity) = sqrt((Pi)/(a))*exp(a*(z)^(2))*erfc(sqrt(a)*z)	Integrate[Divide[Exp[- a*t],Sqrt[t + (z)^(2)]], {t, 0, Infinity}, GenerateConditions->None] == Sqrt[Divide[Pi,a]]*Exp[a*(z)^(2)]*Erfc[Sqrt[a]*z]	Successful	Successful	-	Successful [Tested: 15]
7.7.E5	${\displaystyle{\displaystyle\int_{0}^{1}\frac{e^{-at^{2}}}{t^{2}+1}\mathrm{d}t=\frac{\pi}{4}e^{a}\left(1-(\operatorname{erf}\sqrt{a})^{2}\right)}}$ \int_{0}^{1}\frac{e^{-at^{2}}}{t^{2}+1}\diff{t} = \frac{\pi}{4}e^{a}\left(1-(\erf@@{\sqrt{a}})^{2}\right)	${\displaystyle{\displaystyle\Re a>0}}$	int((exp(- a*(t)^(2)))/((t)^(2)+ 1), t = 0..1) = (Pi)/(4)*exp(a)*(1 -(erf(sqrt(a)))^(2))	Integrate[Divide[Exp[- a*(t)^(2)],(t)^(2)+ 1], {t, 0, 1}, GenerateConditions->None] == Divide[Pi,4]*Exp[a]*(1 -(Erf[Sqrt[a]])^(2))	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
7.7.E6	${\displaystyle{\displaystyle\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{2}-ac)/a}\operatorname{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right)}}$ \int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{2}-ac)/a}\erfc@{\sqrt{a}x+\frac{b}{\sqrt{a}}}	${\displaystyle{\displaystyle\Re a>0}}$	int(exp(-(a*(t)^(2)+ 2*b*t + c)), t = x..infinity) = (1)/(2)*sqrt((Pi)/(a))*exp(((b)^(2)- a*c)/a)*erfc(sqrt(a)*x +(b)/(sqrt(a)))	Integrate[Exp[-(a*(t)^(2)+ 2*b*t + c)], {t, x, Infinity}, GenerateConditions->None] == Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[((b)^(2)- a*c)/a]*Erfc[Sqrt[a]*x +Divide[b,Sqrt[a]]]	Failure	Successful	Successful [Tested: 300]	Successful [Tested: 300]
7.7.E7	${\displaystyle{\displaystyle\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\mathrm{d}t=\frac{\sqrt{\pi}}{4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x)\right)+e^{-2ab}\operatorname{erfc}\left(ax-(b/x)\right)\right)}}$ \int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\diff{t} = \frac{\sqrt{\pi}}{4a}\left(e^{2ab}\erfc@{ax+(b/x)}+e^{-2ab}\erfc@{ax-(b/x)}\right)	${\displaystyle{\displaystyle x>0,|\operatorname{ph}a|<\tfrac{1}{4}\pi}}$	int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/(t)^(2))), t = x..infinity) = (sqrt(Pi))/(4*a)*(exp(2*a*b)*erfc(a*x +(b/x))+ exp(- 2*a*b)*erfc(a*x -(b/x)))	Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/(t)^(2))], {t, x, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],4*a]*(Exp[2*a*b]*Erfc[a*x +(b/x)]+ Exp[- 2*a*b]*Erfc[a*x -(b/x)])	Failure	Aborted	Successful [Tested: 54]	Skipped - Because timed out
7.7.E8	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\mathrm{d}t=\frac{\sqrt{\pi}}{2a}e^{-2ab}}}$ \int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\diff{t} = \frac{\sqrt{\pi}}{2a}e^{-2ab}	${\displaystyle{\displaystyle|\operatorname{ph}a|<\tfrac{1}{4}\pi,|\operatorname{ph}b|<\tfrac{1}{4}\pi}}$	int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/(t)^(2))), t = 0..infinity) = (sqrt(Pi))/(2*a)*exp(- 2*a*b)	Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/(t)^(2))], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],2*a]*Exp[- 2*a*b]	Successful	Successful	-	Successful [Tested: 9]
7.7.E9	${\displaystyle{\displaystyle\int_{0}^{x}\operatorname{erf}t\mathrm{d}t=x\operatorname{erf}x+\frac{1}{\sqrt{\pi}}\left(e^{-x^{2}}-1\right)}}$ \int_{0}^{x}\erf@@{t}\diff{t} = x\erf@@{x}+\frac{1}{\sqrt{\pi}}\left(e^{-x^{2}}-1\right)		int(erf(t), t = 0..x) = x*erf(x)+(1)/(sqrt(Pi))*(exp(- (x)^(2))- 1)	Integrate[Erf[t], {t, 0, x}, GenerateConditions->None] == x*Erf[x]+Divide[1,Sqrt[Pi]]*(Exp[- (x)^(2)]- 1)	Successful	Successful	-	Successful [Tested: 3]
7.7.E10	${\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\pi z^{2}t/2}}{\sqrt{t}(t^{2}+1)}\mathrm{d}t}}$ \auxFresnelf@{z} = \frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\pi z^{2}t/2}}{\sqrt{t}(t^{2}+1)}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\frac{1}{4}\pi}}$	Fresnelf(z) = (1)/(Pi*sqrt(2))*int((exp(- Pi*(z)^(2)* t/2))/(sqrt(t)*((t)^(2)+ 1)), t = 0..infinity)	FresnelF[z] == Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Exp[- Pi*(z)^(2)* t/2],Sqrt[t]*((t)^(2)+ 1)], {t, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 4]	Successful [Tested: 4]
7.7.E11	${\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2}t/2}}{t^{2}+1}\mathrm{d}t}}$ \auxFresnelg@{z} = \frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2}t/2}}{t^{2}+1}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\frac{1}{4}\pi}}$	Fresnelg(z) = (1)/(Pi*sqrt(2))*int((sqrt(t)*exp(- Pi*(z)^(2)* t/2))/((t)^(2)+ 1), t = 0..infinity)	FresnelG[z] == Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Sqrt[t]*Exp[- Pi*(z)^(2)* t/2],(t)^(2)+ 1], {t, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 4]	Successful [Tested: 4]
7.7.E12	${\displaystyle{\displaystyle\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-\pi iz^{2}/2}\int_{z}^{\infty}e^{\pi it^{2}/2}\mathrm{d}t}}$ \auxFresnelg@{z}+i\auxFresnelf@{z} = e^{-\pi iz^{2}/2}\int_{z}^{\infty}e^{\pi it^{2}/2}\diff{t}		Fresnelg(z)+ I*Fresnelf(z) = exp(- Pi*I*(z)^(2)/2)*int(exp(Pi*I*(t)^(2)/2), t = z..infinity)	FresnelG[z]+ I*FresnelF[z] == Exp[- Pi*I*(z)^(2)/2]*Integrate[Exp[Pi*I*(t)^(2)/2], {t, z, Infinity}, GenerateConditions->None]	Successful	Failure	-	Failed [2 / 7] Result: Complex[0.1740270274183789, -0.23657015577401255] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.17402702741837872, 0.2365701557740125] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}
7.7.E13	${\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma\left(s+\tfrac{3}{4}\right)\Gamma\left(\tfrac{1}{4}-s\right)\mathrm{d}s}}$ \auxFresnelf@{z} = \frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\EulerGamma@{s}\EulerGamma@{s+\tfrac{1}{2}}\*\EulerGamma@{s+\tfrac{3}{4}}\EulerGamma@{\tfrac{1}{4}-s}\diff{s}	${\displaystyle{\displaystyle\Re s>0,\Re(s+\tfrac{1}{2})>0,\Re(s+\tfrac{3}{4})>0,\Re(\tfrac{1}{4}-s)>0}}$	Fresnelf(z) = ((2*Pi)^(- 3/2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(3)/(4))*GAMMA((1)/(4)- s), s = c - I*infinity..c + I*infinity)	FresnelF[z] == Divide[(2*Pi)^(- 3/2),2*Pi*I]*Integrate[\[Zeta]^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[3,4]]*Gamma[Divide[1,4]- s], {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [300 / 300] Result: .2811902531-.108667706*I Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I} Result: .2811902531-.108667706*I Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
7.7.E14	${\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma\left(s+\tfrac{1}{4}\right)\Gamma\left(\tfrac{3}{4}-s\right)\mathrm{d}s}}$ \auxFresnelg@{z} = \frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\EulerGamma@{s}\EulerGamma@{s+\tfrac{1}{2}}\*\EulerGamma@{s+\tfrac{1}{4}}\EulerGamma@{\tfrac{3}{4}-s}\diff{s}	${\displaystyle{\displaystyle\Re s>0,\Re(s+\tfrac{1}{2})>0,\Re(s+\tfrac{1}{4})>0,\Re(\tfrac{3}{4}-s)>0}}$	Fresnelg(z) = ((2*Pi)^(- 3/2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(1)/(4))*GAMMA((3)/(4)- s), s = c - I*infinity..c + I*infinity)	FresnelG[z] == Divide[(2*Pi)^(- 3/2),2*Pi*I]*Integrate[\[Zeta]^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[1,4]]*Gamma[Divide[3,4]- s], {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [300 / 300] Result: .39257720e-1-.645221857e-1*I Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I} Result: .39257720e-1-.645221857e-1*I Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
7.7.E15	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\cos\left(t^{2}\right)\mathrm{d}t=\sqrt{\frac{\pi}{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi}}\right)}}$ \int_{0}^{\infty}e^{-at}\cos@{t^{2}}\diff{t} = \sqrt{\frac{\pi}{2}}\auxFresnelf@{\frac{a}{\sqrt{2\pi}}}	${\displaystyle{\displaystyle\Re a>0}}$	int(exp(- a*t)*cos((t)^(2)), t = 0..infinity) = sqrt((Pi)/(2))*Fresnelf((a)/(sqrt(2*Pi)))	Integrate[Exp[- a*t]*Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Sqrt[Divide[Pi,2]]*FresnelF[Divide[a,Sqrt[2*Pi]]]	Successful	Aborted	-	Successful [Tested: 3]
7.7.E16	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\sin\left(t^{2}\right)\mathrm{d}t=\sqrt{\frac{\pi}{2}}\mathrm{g}\left(\frac{a}{\sqrt{2\pi}}\right)}}$ \int_{0}^{\infty}e^{-at}\sin@{t^{2}}\diff{t} = \sqrt{\frac{\pi}{2}}\auxFresnelg@{\frac{a}{\sqrt{2\pi}}}	${\displaystyle{\displaystyle\Re a>0}}$	int(exp(- a*t)*sin((t)^(2)), t = 0..infinity) = sqrt((Pi)/(2))*Fresnelg((a)/(sqrt(2*Pi)))	Integrate[Exp[- a*t]*Sin[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Sqrt[Divide[Pi,2]]*FresnelG[Divide[a,Sqrt[2*Pi]]]	Successful	Aborted	-	Successful [Tested: 3]