7.18

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
7.18#Ex1	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right)=\frac{2}{\sqrt{\pi}}e^{-z^{2}}}}$ \repinterfc{-1}@{z} = \frac{2}{\sqrt{\pi}}e^{-z^{2}}		erfc(- 1, z) = (2)/(sqrt(Pi))*exp(- (z)^(2))	I^(- 1)*Erfc[z] == Divide[2,Sqrt[Pi]]*Exp[- (z)^(2)]	Successful	Failure	-	Failed [7 / 7] Result: Complex[-0.6965576261018753, 0.4234600295072003] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.0623272173358496, -3.394891496894652] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
7.18#Ex2	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{0}\mathrm{erfc}}\left(z\right)=\operatorname{erfc}z}}$ \repinterfc{0}@{z} = \erfc@@{z}		erfc(0, z) = erfc(z)	I^(0)*Erfc[z] == Erfc[z]	Successful	Successful	-	Successful [Tested: 7]
7.18.E2	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\int_{z}^{\infty}\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(t\right)\mathrm{d}t}}$ \repinterfc{n}@{z} = \int_{z}^{\infty}\repinterfc{n-1}@{t}\diff{t}		erfc(n, z) = int(erfc(n - 1, t), t = z..infinity)	I^(n)*Erfc[z] == Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [12 / 21] Result: Float(undefined)-.9036864554e-1*I Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 1} Result: Float(undefined)-.2674601677e-1*I Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[0.24282268468866475, 0.18825452738900728] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.18825452738900728, 0.24282268468866475] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
7.18.E2	${\displaystyle{\displaystyle\int_{z}^{\infty}\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(t\right)\mathrm{d}t=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}\frac{(t-z)^{n}}{n!}e^{-t^{2}}\mathrm{d}t}}$ \int_{z}^{\infty}\repinterfc{n-1}@{t}\diff{t} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}\frac{(t-z)^{n}}{n!}e^{-t^{2}}\diff{t}		int(erfc(n - 1, t), t = z..infinity) = (2)/(sqrt(Pi))*int(((t - z)^(n))/(factorial(n))*exp(- (t)^(2)), t = z..infinity)	Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}, GenerateConditions->None] == Divide[2,Sqrt[Pi]]*Integrate[Divide[(t - z)^(n),(n)!]*Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [12 / 21] Result: Float(undefined) Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 1} Result: Float(undefined) Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [13 / 21] Result: Complex[0.09296765524307439, 0.0370882508190411] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.008358539694265255, 0.09727600825382138] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
7.18.E3	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=-\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(z\right)}}$ \deriv{}{z}\repinterfc{n}@{z} = -\repinterfc{n-1}@{z}		diff(erfc(n, z), z) = - erfc(n - 1, z)	D[I^(n)*Erfc[z], z] == - I^(n - 1)*Erfc[z]	Successful	Failure	-	Failed [7 / 7] Result: Complex[0.4234600295072003, 0.6965576261018753] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-3.394891496894652, 2.0623272173358496] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
7.18.E4	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{z^{2}}\operatorname{erfc}z\right)=(-1)^{n}2^{n}n!e^{z^{2}}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)}}$ \deriv[n]{}{z}\left(e^{z^{2}}\erfc@@{z}\right) = (-1)^{n}2^{n}n!e^{z^{2}}\repinterfc{n}@{z}		diff(exp((z)^(2))*erfc(z), [z$(n)]) = (- 1)^(n)* (2)^(n)* factorial(n)*exp((z)^(2))*erfc(n, z)	D[Exp[(z)^(2)]*Erfc[z], {z, n}] == (- 1)^(n)* (2)^(n)* (n)!*Exp[(z)^(2)]*I^(n)*Erfc[z]	Failure	Failure	Skipped - Because timed out	Failed [7 / 7] Result: Complex[-7.3936292130611685, -19.806900214215183] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-48.524741574815884, -12.92653708276189] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
7.18.E5	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}+2z\frac{\mathrm{d}W}{\mathrm{d}z}-2nW=0}}$ \deriv[2]{W}{z}+2z\deriv{W}{z}-2nW = 0	${\displaystyle{\displaystyle W(z)=A\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)+B\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(-z\right)}}$	diff(W, [z$(2)])+ 2*z*diff(W, z)- 2*n*W = 0	D[W, {z, 2}]+ 2*z*D[W, z]- 2*n*W == 0	Failure	Failure	Failed [210 / 210] Result: -1.732050808-1.000000000*I Test Values: {W = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1} Result: -3.464101616-2.*I Test Values: {W = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [210 / 210] Result: Complex[-1.7320508075688774, -0.9999999999999999] Test Values: {Rule[n, 1], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-3.464101615137755, -1.9999999999999998] Test Values: {Rule[n, 2], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
7.18.E6	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{k}}{2^{n-k}k!\Gamma\left(1+\frac{1}{2}(n-k)\right)}}}$ \repinterfc{n}@{z} = \sum_{k=0}^{\infty}\frac{(-1)^{k}z^{k}}{2^{n-k}k!\EulerGamma@{1+\frac{1}{2}(n-k)}}	${\displaystyle{\displaystyle\Re(1+\frac{1}{2}(n-k))>0}}$	erfc(n, z) = sum(((- 1)^(k)* (z)^(k))/((2)^(n - k)* factorial(k)*GAMMA(1 +(1)/(2)*(n - k))), k = 0..infinity)	I^(n)*Erfc[z] == Sum[Divide[(- 1)^(k)* (z)^(k),(2)^(n - k)* (k)!*Gamma[1 +Divide[1,2]*(n - k)]], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [21 / 21] Result: -.8660254034-.4999999991*I Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 1} Result: .4999999999+.4330127014*I Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[0.2428226846886648, 0.18825452738900733] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.09528687214593286, 0.27991093550770596] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
7.18.E7	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=-\frac{z}{n}\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(z\right)+\frac{1}{2n}\mathop{\mathrm{i}^{n-2}\mathrm{erfc}}\left(z\right)}}$ \repinterfc{n}@{z} = -\frac{z}{n}\repinterfc{n-1}@{z}+\frac{1}{2n}\repinterfc{n-2}@{z}		erfc(n, z) = -(z)/(n)*erfc(n - 1, z)+(1)/(2*n)*erfc(n - 2, z)	I^(n)*Erfc[z] == -Divide[z,n]*I^(n - 1)*Erfc[z]+Divide[1,2*n]*I^(n - 2)*Erfc[z]	Successful	Failure	-	Failed [7 / 7] Result: Complex[-0.36581044505750443, -0.05743209207542904] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.9176954586385406, -3.02111135172986] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
7.18.E8	${\displaystyle{\displaystyle(-1)^{n}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)+\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(-z\right)=\frac{i^{-n}}{2^{n-1}n!}H_{n}\left(iz\right)}}$ (-1)^{n}\repinterfc{n}@{z}+\repinterfc{n}@{-z} = \frac{i^{-n}}{2^{n-1}n!}\HermitepolyH{n}@{iz}		(- 1)^(n)* erfc(n, z)+ erfc(n, - z) = ((I)^(- n))/((2)^(n - 1)* factorial(n))*HermiteH(n, I*z)	(- 1)^(n)* I^(n)*Erfc[z]+ I^(n)*Erfc[- z] == Divide[(I)^(- n),(2)^(n - 1)* (n)!]*HermiteH[n, I*z]	Failure	Failure	Successful [Tested: 21]	Failed [21 / 21] Result: Complex[-2.2383806450017882, 0.8042282364091201] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-3.0, -0.8660254037844386] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
7.18.E9	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=e^{-z^{2}}\left(\frac{1}{2^{n}\Gamma\left(\tfrac{1}{2}n+1\right)}M\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)-\frac{z}{2^{n-1}\Gamma\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)}M\left(\tfrac{1}{2}n+1,\tfrac{3}{2},z^{2}\right)\right)}}$ \repinterfc{n}@{z} = e^{-z^{2}}\left(\frac{1}{2^{n}\EulerGamma@{\tfrac{1}{2}n+1}}\KummerconfhyperM@{\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}}-\frac{z}{2^{n-1}\EulerGamma@{\tfrac{1}{2}n+\tfrac{1}{2}}}\KummerconfhyperM@{\tfrac{1}{2}n+1}{\tfrac{3}{2}}{z^{2}}\right)	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}n+1)>0,\Re(\tfrac{1}{2}n+\tfrac{1}{2})>0}}$	erfc(n, z) = exp(- (z)^(2))*((1)/((2)^(n)* GAMMA((1)/(2)*n + 1))*KummerM((1)/(2)*n +(1)/(2), (1)/(2), (z)^(2))-(z)/((2)^(n - 1)* GAMMA((1)/(2)*n +(1)/(2)))*KummerM((1)/(2)*n + 1, (3)/(2), (z)^(2)))	I^(n)*Erfc[z] == Exp[- (z)^(2)]*(Divide[1,(2)^(n)* Gamma[Divide[1,2]*n + 1]]*Hypergeometric1F1[Divide[1,2]*n +Divide[1,2], Divide[1,2], (z)^(2)]-Divide[z,(2)^(n - 1)* Gamma[Divide[1,2]*n +Divide[1,2]]]*Hypergeometric1F1[Divide[1,2]*n + 1, Divide[3,2], (z)^(2)])	Failure	Failure	Successful [Tested: 21]	Failed [21 / 21] Result: Complex[0.24282268468866477, 0.18825452738900755] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.09528687214593284, 0.27991093550770596] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
7.18.E10	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{e^{-z^{2}}}{2^{n}\sqrt{\pi}}U\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)}}$ \repinterfc{n}@{z} = \frac{e^{-z^{2}}}{2^{n}\sqrt{\pi}}\KummerconfhyperU@{\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}}		erfc(n, z) = (exp(- (z)^(2)))/((2)^(n)*sqrt(Pi))*KummerU((1)/(2)*n +(1)/(2), (1)/(2), (z)^(2))	I^(n)*Erfc[z] == Divide[Exp[- (z)^(2)],(2)^(n)*Sqrt[Pi]]*HypergeometricU[Divide[1,2]*n +Divide[1,2], Divide[1,2], (z)^(2)]	Failure	Failure	Failed [6 / 21] Result: 1.000000000-1.732050808*I Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1} Result: .1727305880-1.014340238*I Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[0.24282268468866502, 0.1882545273890069] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.09528687214593298, 0.2799109355077059] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
7.18.E11	${\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{e^{-z^{2}/2}}{\sqrt{2^{n-1}\pi}}U\left(n+\tfrac{1}{2},z\sqrt{2}\right)}}$ \repinterfc{n}@{z} = \frac{e^{-z^{2}/2}}{\sqrt{2^{n-1}\pi}}\paraU@{n+\tfrac{1}{2}}{z\sqrt{2}}		erfc(n, z) = (exp(- (z)^(2)/2))/(sqrt((2)^(n - 1)* Pi))*CylinderU(n +(1)/(2), z*sqrt(2))	I^(n)*Erfc[z] == Divide[Exp[- (z)^(2)/2],Sqrt[(2)^(n - 1)* Pi]]*ParabolicCylinderD[- 1/2 -(n +Divide[1,2]), z*Sqrt[2]]	Failure	Failure	Successful [Tested: 21]	Failed [21 / 21] Result: Complex[0.24282268468866486, 0.1882545273890072] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.09528687214593298, 0.2799109355077059] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data