Results of Incomplete Gamma and Related Functions

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DLMF Formula Maple Mathematica Symbolic
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Mathematica
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8.2.E1 GAMMA(a)-GAMMA(a, z)= int((t)^(a - 1)* exp(- t), t = 0..z) Gamma[a, 0, z]= Integrate[(t)^(a - 1)* Exp[- t], {t, 0, z}] Failure Successful Skip -
8.2.E2 GAMMA(a, z)= int((t)^(a - 1)* exp(- t), t = z..infinity) Gamma[a, z]= Integrate[(t)^(a - 1)* Exp[- t], {t, z, Infinity}] Failure Failure Skip Successful
8.2.E3 GAMMA(a)-GAMMA(a, z)+ GAMMA(a, z)= GAMMA(a) Gamma[a, 0, z]+ Gamma[a, z]= Gamma[a] Successful Successful - -
8.2#Ex1 (GAMMA(a)-GAMMA(a, z))/GAMMA(a)=(GAMMA(a)-GAMMA(a, z))/(GAMMA(a)) GammaRegularized[a, 0, z]=Divide[Gamma[a, 0, z],Gamma[a]] Successful Successful - -
8.2#Ex2 GAMMA(a, z)/GAMMA(a)=(GAMMA(a, z))/(GAMMA(a)) GammaRegularized[a, z]=Divide[Gamma[a, z],Gamma[a]] Successful Successful - -
8.2.E5 (GAMMA(a)-GAMMA(a, z))/GAMMA(a)+ GAMMA(a, z)/GAMMA(a)= 1 GammaRegularized[a, 0, z]+ GammaRegularized[a, z]= 1 Successful Successful - -
8.2.E6 (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)= (z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a) Error Successful Error - -
8.2.E6 (z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a)=((z)^(- a))/(GAMMA(a))*GAMMA(a)-GAMMA(a, z) (z)^(- a)* GammaRegularized[a, 0, z]=Divide[(z)^(- a),Gamma[a]]*Gamma[a, 0, z] Failure Successful
Fail
.3504429851+.4826856014*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-.4474572306+.2704599710*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
23.62700226+82.69161801*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
3.420707652-13.57627439*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
-
8.2.E7 (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)=(1)/(GAMMA(a))*int((t)^(a - 1)* exp(- z*t), t = 0..1) Error Failure Error Skip -
8.2.E8 GAMMA(a)-GAMMA(a, z*exp(2*Pi*m*I))= exp(2*Pi*m*I*a)*GAMMA(a)-GAMMA(a, z) Gamma[a, 0, z*Exp[2*Pi*m*I]]= Exp[2*Pi*m*I*a]*Gamma[a, 0, z] Failure Failure Successful Successful
8.2.E9 GAMMA(a, z*exp(2*Pi*m*I))= exp(2*Pi*m*I*a)*GAMMA(a, z)+(1 - exp(2*Pi*m*I*a))* GAMMA(a) Gamma[a, z*Exp[2*Pi*m*I]]= Exp[2*Pi*m*I*a]*Gamma[a, z]+(1 - Exp[2*Pi*m*I*a])* Gamma[a] Failure Failure
Fail
-.2249049111-.4410511843e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
-.2248750758-.4411585330e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-.2248750795-.4411584875e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-1.005323136+.3326243216*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1}
... skip entries to safe data
Fail
Complex[-0.22490491118791595, -0.04410511845656586] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.2248750764783257, -0.044115852492705915] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.22487507925834865, -0.04411584909968558] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.0053231382729926, 0.33262432134470665] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.2.E10 exp(- Pi*I*a)*GAMMA(a, z*exp(Pi*I))- exp(Pi*I*a)*GAMMA(a, z*exp(- Pi*I))= -(2*Pi*I)/(GAMMA(1 - a)) Exp[- Pi*I*a]*Gamma[a, z*Exp[Pi*I]]- Exp[Pi*I*a]*Gamma[a, z*Exp[- Pi*I]]= -Divide[2*Pi*I,Gamma[1 - a]] Failure Failure
Fail
-7167.292469-174.9289096*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
2.16987973+12.77160007*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
8.705606105-17.43270949*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-4.50134822-89.91653387*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-7167.2924809060105, -174.9289096706231] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.169879706441371, 12.771600034859095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[8.70560609871773, -17.43270953363519] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.501348191090425, -89.91653394957189] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.2.E11 GAMMA(a, z*exp(+ Pi*I))= GAMMA(a)*(1 - (z)^(a)* exp(+ Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a)) Error Failure Error
Fail
20.46249972+81.80630504*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1.005323138+.3326243220*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
1095.761010-111.2868863*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1231.554386+1108.053849*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
-
8.2.E11 GAMMA(a, z*exp(- Pi*I))= GAMMA(a)*(1 - (z)^(a)* exp(- Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a)) Error Failure Error
Fail
1095.761010+111.2868863*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1231.554386-1108.053849*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
20.46249972-81.80630504*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.005323138-.3326243220*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
... skip entries to safe data
-
8.2.E12 diff(w, [z$(2)])+(1 +(1 - a)/(z))* diff(w, z)= 0 D[w, {z, 2}]+(1 +Divide[1 - a,z])* D[w, z]= 0 Successful Successful - -
8.2.E13 diff(w, [z$(2)])-(1 +(1 - a)/(z))* diff(w, z)+(1 - a)/((z)^(2))*w = 0 D[w, {z, 2}]-(1 +Divide[1 - a,z])* D[w, z]+Divide[1 - a,(z)^(2)]*w = 0 Failure Failure
Fail
-.6464466093-.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.6464466093+.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-.6464466093-.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
.6464466093+.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.6464466094067263, -0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.6464466094067263, 0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.6464466094067263, -0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.6464466094067263, 0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.2.E14 z*diff((+)^(-(z))*(GAMMA(z)-GAMMA(z, +))/GAMMA(z), [(a + 1 + z)*$(2)])*diff((+)^(-(z))*(GAMMA(z)-GAMMA(z, +))/GAMMA(z), a)*(0)^(-(=))*(GAMMA(=)-GAMMA(=, 0))/GAMMA(=) Error Error Error - -
8.4.E1 GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2))= 2*int(exp(- (t)^(2)), t = 0..z) Gamma[Divide[1,2], 0, (z)^(2)]= 2*Integrate[Exp[- (t)^(2)], {t, 0, z}] Failure Failure Skip
Fail
Complex[3.581461769189045, -0.9710415344467407] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.581461769189045, 0.9710415344467407] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
8.4.E1 2*int(exp(- (t)^(2)), t = 0..z)=sqrt(Pi)*erf(z) 2*Integrate[Exp[- (t)^(2)], {t, 0, z}]=Sqrt[Pi]*Erf[z] Successful Successful - -
8.4.E2 (0)^(-(a))*(GAMMA(a)-GAMMA(a, 0))/GAMMA(a)=(1)/(GAMMA(a + 1)) Error Failure Error
Fail
-.6493698774+1.106937485*I <- {a = 2^(1/2)+I*2^(1/2)}
-.6493698774-1.106937485*I <- {a = 2^(1/2)-I*2^(1/2)}
4.564263782+2.639434666*I <- {a = -2^(1/2)-I*2^(1/2)}
4.564263782-2.639434666*I <- {a = -2^(1/2)+I*2^(1/2)}
-
8.4.E3 (- (z)^(2))^(-((1)/(2)))*(GAMMA((1)/(2))-GAMMA((1)/(2), - (z)^(2)))/GAMMA((1)/(2))=(2*exp((z)^(2)))/(z*sqrt(Pi))*dawson(z) Error Successful Error - -
8.4.E4 GAMMA(0, z)= int((t)^(- 1)* exp(- t), t = z..infinity) Gamma[0, z]= Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}] Successful Failure - Successful
8.4.E4 int((t)^(- 1)* exp(- t), t = z..infinity)= Ei(z) Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}]= -ExpIntegralEi[-(z)] Failure Failure Skip Successful
8.4.E5 GAMMA(1, z)= exp(- z) Gamma[1, z]= Exp[- z] Successful Successful - -
8.4.E6 GAMMA((1)/(2), (z)^(2))= 2*int(exp(- (t)^(2)), t = z..infinity) Gamma[Divide[1,2], (z)^(2)]= 2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}] Failure Failure Skip
Fail
Complex[-3.581461769189044, 0.9710415344467407] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.581461769189044, -0.9710415344467407] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
8.4.E6 2*int(exp(- (t)^(2)), t = z..infinity)=sqrt(Pi)*erfc(z) 2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}]=Sqrt[Pi]*Erfc[z] Successful Successful - -
8.4.E7 GAMMA(n + 1)-GAMMA(n + 1, z)= factorial(n)*(1 - exp(- z)*exp(1)[n]*(z)) Gamma[n + 1, 0, z]= (n)!*(1 - Exp[- z]*Subscript[E, n]*(z)) Failure Failure Error Successful
8.4.E8 GAMMA(n + 1, z)= factorial(n)*exp(- z)*exp(1)[n]*(z) Gamma[n + 1, z]= (n)!*Exp[- z]*Subscript[E, n]*(z) Failure Failure Error Successful
8.4.E9 (GAMMA(n + 1)-GAMMA(n + 1, z))/GAMMA(n + 1)= 1 - exp(- z)*exp(1)[n]*(z) GammaRegularized[n + 1, 0, z]= 1 - Exp[- z]*Subscript[E, n]*(z) Failure Failure Error Successful
8.4.E10 GAMMA(n + 1, z)/GAMMA(n + 1)= exp(- z)*exp(1)[n]*(z) GammaRegularized[n + 1, z]= Exp[- z]*Subscript[E, n]*(z) Failure Failure Error Successful
8.4.E12 (z)^(-(- n))*(GAMMA(- n)-GAMMA(- n, z))/GAMMA(- n)= (z)^(n) Error Failure Error
Fail
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)+I*2^(1/2), n = 1}
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)+I*2^(1/2), n = 2}
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)+I*2^(1/2), n = 3}
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
-
8.4.E13 GAMMA(1 - n, z)= (z)^(1 - n)* Ei(n, z) Gamma[1 - n, z]= (z)^(1 - n)* ExpIntegralE[n, z] Successful Successful - -
8.4.E14 GAMMA(n +(1)/(2), (z)^(2))/GAMMA(n +(1)/(2))= erfc(z)+(exp(- (z)^(2)))/(sqrt(Pi))*sum(((z)^(2*k - 1))/(pochhammer((1)/(2), k)), k = 1..n) GammaRegularized[n +Divide[1,2], (z)^(2)]= Erfc[z]+Divide[Exp[- (z)^(2)],Sqrt[Pi]]*Sum[Divide[(z)^(2*k - 1),Pochhammer[Divide[1,2], k]], {k, 1, n}] Failure Failure Skip
Fail
Complex[-6.522116143801526, 0.8770870118427658] <- {Rule[n, 1], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-7.400077458243353, -11.126893574158686] <- {Rule[n, 2], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[11.80629147935897, -12.531631677265604] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-6.522116143801526, -0.8770870118427658] <- {Rule[n, 1], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.4.E15 GAMMA(- n, z)=((- 1)^(n))/(factorial(n))*(Ei(z)- exp(- z)*sum(((- 1)^(k)* factorial(k))/((z)^(k + 1)), k = 0..n - 1)) Gamma[- n, z]=Divide[(- 1)^(n),(n)!]*(-ExpIntegralEi[-(z)]- Exp[- z]*Sum[Divide[(- 1)^(k)* (k)!,(z)^(k + 1)], {k, 0, n - 1}]) Failure Failure Skip
Fail
Complex[1.3877787807814457*^-17, 3.141592653589793] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.734723475976807*^-18, -1.5707963267948966] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-4.3368086899420177*^-19, 0.5235987755982987] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.3877787807814457*^-17, -3.141592653589793] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.5.E1 GAMMA(a)-GAMMA(a, z)= (a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z) Gamma[a, 0, z]= (a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z] Successful Successful - -
8.5.E1 (a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z)= (a)^(- 1)* (z)^(a)* KummerM(a, 1 + a, - z) (a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z]= (a)^(- 1)* (z)^(a)* Hypergeometric1F1[a, 1 + a, - z] Successful Successful - -
8.5.E2 (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)= exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a) Error Successful Error - -
8.5.E2 exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a)= KummerM(a, 1 + a, - z)/GAMMA(1 + a) Exp[- z]*Hypergeometric1F1Regularized[1, 1 + a, z]= Hypergeometric1F1Regularized[a, 1 + a, - z] Successful Successful - -
8.5.E3 GAMMA(a, z)= exp(- z)*KummerU(1 - a, 1 - a, z) Gamma[a, z]= Exp[- z]*HypergeometricU[1 - a, 1 - a, z] Successful Successful - -
8.5.E3 exp(- z)*KummerU(1 - a, 1 - a, z)= (z)^(a)* exp(- z)*KummerU(1, 1 + a, z) Exp[- z]*HypergeometricU[1 - a, 1 - a, z]= (z)^(a)* Exp[- z]*HypergeometricU[1, 1 + a, z] Successful Successful - -
8.5.E4 GAMMA(a)-GAMMA(a, z)= (a)^(- 1)* (z)^((1)/(2)*a -(1)/(2))* exp(-(1)/(2)*z)*WhittakerM((1)/(2)*a -(1)/(2), (1)/(2)*a, z) Gamma[a, 0, z]= (a)^(- 1)* (z)^(Divide[1,2]*a -Divide[1,2])* Exp[-Divide[1,2]*z]*WhittakerM[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z] Successful Successful - -
8.5.E5 GAMMA(a, z)= exp(-(1)/(2)*z)*(z)^((1)/(2)*a -(1)/(2))* WhittakerW((1)/(2)*a -(1)/(2), (1)/(2)*a, z) Gamma[a, z]= Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]*a -Divide[1,2])* WhittakerW[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z] Successful Successful - -
8.6.E1 GAMMA(a)-GAMMA(a, z)=((z)^(a))/(sin(Pi*a))*int(exp(z*cos(t))*cos(a*t + z*sin(t)), t = 0..Pi) Gamma[a, 0, z]=Divide[(z)^(a),Sin[Pi*a]]*Integrate[Exp[z*Cos[t]]*Cos[a*t + z*Sin[t]], {t, 0, Pi}] Failure Failure Skip Error
8.6.E2 GAMMA(a)-GAMMA(a, z)= (z)^((1)/(2)*a)* int(exp(- t)*(t)^((1)/(2)*a - 1)* BesselJ(a, 2*sqrt(z*t)), t = 0..infinity) Gamma[a, 0, z]= (z)^(Divide[1,2]*a)* Integrate[Exp[- t]*(t)^(Divide[1,2]*a - 1)* BesselJ[a, 2*Sqrt[z*t]], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E3 GAMMA(a)-GAMMA(a, z)= (z)^(a)* int(exp(- a*t - z*exp(- t)), t = 0..infinity) Gamma[a, 0, z]= (z)^(a)* Integrate[Exp[- a*t - z*Exp[- t]], {t, 0, Infinity}] Failure Failure Skip Successful
8.6.E4 GAMMA(a, z)=((z)^(a)* exp(- z))/(GAMMA(1 - a))*int(((t)^(- a)* exp(- t))/(z + t), t = 0..infinity) Gamma[a, z]=Divide[(z)^(a)* Exp[- z],Gamma[1 - a]]*Integrate[Divide[(t)^(- a)* Exp[- t],z + t], {t, 0, Infinity}] Failure Failure Skip Successful
8.6.E5 GAMMA(a, z)= (z)^(a)* exp(- z)*int((exp(- z*t))/((1 + t)^(1 - a)), t = 0..infinity) Gamma[a, z]= (z)^(a)* Exp[- z]*Integrate[Divide[Exp[- z*t],(1 + t)^(1 - a)], {t, 0, Infinity}] Successful Failure - Error
8.6.E6 GAMMA(a, z)=(2*(z)^((1)/(2)*a)* exp(- z))/(GAMMA(1 - a))*int(exp(- t)*(t)^(-(1)/(2)*a)* BesselK(a, 2*sqrt(z*t)), t = 0..infinity) Gamma[a, z]=Divide[2*(z)^(Divide[1,2]*a)* Exp[- z],Gamma[1 - a]]*Integrate[Exp[- t]*(t)^(-Divide[1,2]*a)* BesselK[a, 2*Sqrt[z*t]], {t, 0, Infinity}] Successful Failure - Error
8.6.E7 GAMMA(a, z)= (z)^(a)* int(exp(a*t - z*exp(t)), t = 0..infinity) Gamma[a, z]= (z)^(a)* Integrate[Exp[a*t - z*Exp[t]], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E8 GAMMA(a)-GAMMA(a, z)=(- I*(z)^(a))/(2*sin(Pi*a))*int((t)^(a - 1)* exp(z*t), t = - 1..(0 +)) Gamma[a, 0, z]=Divide[- I*(z)^(a),2*Sin[Pi*a]]*Integrate[(t)^(a - 1)* Exp[z*t], {t, - 1, (0 +)}] Error Failure - Error
8.6.E9 GAMMA(- a, z*exp(+ Pi*I))=(exp(z)*exp(- Pi*I*a))/(GAMMA(1 + a))*int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity) Gamma[- a, z*Exp[+ Pi*I]]=Divide[Exp[z]*Exp[- Pi*I*a],Gamma[1 + a]]*Integrate[Divide[(t)^(a)* Exp[- z*t],t - 1], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E9 GAMMA(- a, z*exp(- Pi*I))=(exp(z)*exp(+ Pi*I*a))/(GAMMA(1 + a))*int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity) Gamma[- a, z*Exp[- Pi*I]]=Divide[Exp[z]*Exp[+ Pi*I*a],Gamma[1 + a]]*Integrate[Divide[(t)^(a)* Exp[- z*t],t - 1], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E10 GAMMA(a)-GAMMA(a, z)=(1)/(2*Pi*I)*int((GAMMA(s))/(a - s)*(z)^(a - s), s = c - I*infinity..c + I*infinity) Gamma[a, 0, z]=Divide[1,2*Pi*I]*Integrate[Divide[Gamma[s],a - s]*(z)^(a - s), {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
8.6.E11 GAMMA(a, z)=(1)/(2*Pi*I)*int(GAMMA(s + a)*((z)^(- s))/(s), s = c - I*infinity..c + I*infinity) Gamma[a, z]=Divide[1,2*Pi*I]*Integrate[Gamma[s + a]*Divide[(z)^(- s),s], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
8.6.E12 GAMMA(a, z)= -((z)^(a - 1)* exp(- z))/(GAMMA(1 - a))*(1)/(2*Pi*I)*int(GAMMA(s + 1 - a)*(Pi*(z)^(- s))/(sin(Pi*s)), s = c - I*infinity..c + I*infinity) Gamma[a, z]= -Divide[(z)^(a - 1)* Exp[- z],Gamma[1 - a]]*Divide[1,2*Pi*I]*Integrate[Gamma[s + 1 - a]*Divide[Pi*(z)^(- s),Sin[Pi*s]], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
8.7.E1 (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)= exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity) Error Successful Error - -
8.7.E1 exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity)=(1)/(GAMMA(a))*sum(((- z)^(k))/(factorial(k)*(a + k)), k = 0..infinity) Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}]=Divide[1,Gamma[a]]*Sum[Divide[(- z)^(k),(k)!*(a + k)], {k, 0, Infinity}] Successful Successful - -
8.7.E3 GAMMA(a, z)= GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity) Gamma[a, z]= Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}] Successful Successful - -
8.7.E3 GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity)= GAMMA(a)*(1 - (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity)) Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}]= Gamma[a]*(1 - (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}]) Successful Successful - -
8.8.E1 GAMMA(a + 1)-GAMMA(a + 1, z)= a*GAMMA(a)-GAMMA(a, z)- (z)^(a)* exp(- z) Gamma[a + 1, 0, z]= a*Gamma[a, 0, z]- (z)^(a)* Exp[- z] Failure Successful
Fail
.135004907e-1-.2375774782*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.8693672828+.710002389*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
107.1902160-63.3824277*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-.1657436948-.7422690683*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
-
8.8.E2 GAMMA(a + 1, z)= a*GAMMA(a, z)+ (z)^(a)* exp(- z) Gamma[a + 1, z]= a*Gamma[a, z]+ (z)^(a)* Exp[- z] Failure Successful Successful -
8.8.E4 z*(z)^(-(a + 1))*(GAMMA(a + 1)-GAMMA(a + 1, z))/GAMMA(a + 1)= (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)-(exp(- z))/(GAMMA(a + 1)) Error Failure Error Successful -
8.8.E5 (GAMMA(a + 1)-GAMMA(a + 1, z))/GAMMA(a + 1)= (GAMMA(a)-GAMMA(a, z))/GAMMA(a)-((z)^(a)* exp(- z))/(GAMMA(a + 1)) GammaRegularized[a + 1, 0, z]= GammaRegularized[a, 0, z]-Divide[(z)^(a)* Exp[- z],Gamma[a + 1]] Failure Successful Successful -
8.8.E6 GAMMA(a + 1, z)/GAMMA(a + 1)= GAMMA(a, z)/GAMMA(a)+((z)^(a)* exp(- z))/(GAMMA(a + 1)) GammaRegularized[a + 1, z]= GammaRegularized[a, z]+Divide[(z)^(a)* Exp[- z],Gamma[a + 1]] Failure Successful Successful -
8.8.E7 GAMMA(a + n)-GAMMA(a + n, z)= pochhammer(a, n)*GAMMA(a)-GAMMA(a, z)- (z)^(a)* exp(- z)*sum((GAMMA(a + n))/(GAMMA(a + k + 1))*(z)^(k), k = 0..n - 1) Gamma[a + n, 0, z]= Pochhammer[a, n]*Gamma[a, 0, z]- (z)^(a)* Exp[- z]*Sum[Divide[Gamma[a + n],Gamma[a + k + 1]]*(z)^(k), {k, 0, n - 1}] Failure Successful Skip -
8.8.E8 GAMMA(a)-GAMMA(a, z)=(GAMMA(a))/(GAMMA(a - n))*GAMMA(a - n)-GAMMA(a - n, z)- (z)^(a - 1)* exp(- z)*sum((GAMMA(a))/(GAMMA(a - k))*(z)^(- k), k = 0..n - 1) Gamma[a, 0, z]=Divide[Gamma[a],Gamma[a - n]]*Gamma[a - n, 0, z]- (z)^(a - 1)* Exp[- z]*Sum[Divide[Gamma[a],Gamma[a - k]]*(z)^(- k), {k, 0, n - 1}] Failure Successful Skip -
8.8.E9 GAMMA(a + n, z)= pochhammer(a, n)*GAMMA(a, z)+ (z)^(a)* exp(- z)*sum((GAMMA(a + n))/(GAMMA(a + k + 1))*(z)^(k), k = 0..n - 1) Gamma[a + n, z]= Pochhammer[a, n]*Gamma[a, z]+ (z)^(a)* Exp[- z]*Sum[Divide[Gamma[a + n],Gamma[a + k + 1]]*(z)^(k), {k, 0, n - 1}] Successful Successful - -
8.8.E10 GAMMA(a, z)=(GAMMA(a))/(GAMMA(a - n))*GAMMA(a - n, z)+ (z)^(a - 1)* exp(- z)*sum((GAMMA(a))/(GAMMA(a - k))*(z)^(- k), k = 0..n - 1) Gamma[a, z]=Divide[Gamma[a],Gamma[a - n]]*Gamma[a - n, z]+ (z)^(a - 1)* Exp[- z]*Sum[Divide[Gamma[a],Gamma[a - k]]*(z)^(- k), {k, 0, n - 1}] Failure Successful Skip -
8.8.E11 (GAMMA(a + n)-GAMMA(a + n, z))/GAMMA(a + n)= (GAMMA(a)-GAMMA(a, z))/GAMMA(a)- (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..n - 1) GammaRegularized[a + n, 0, z]= GammaRegularized[a, 0, z]- (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, n - 1}] Successful Successful - -
8.8.E12 GAMMA(a + n, z)/GAMMA(a + n)= GAMMA(a, z)/GAMMA(a)+ (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..n - 1) GammaRegularized[a + n, z]= GammaRegularized[a, z]+ (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, n - 1}] Successful Successful - -
8.8.E13 diff(GAMMA(a)-GAMMA(a, z), z)= - diff(GAMMA(a, z), z) D[Gamma[a, 0, z], z]= - D[Gamma[a, z], z] Successful Successful - -
8.8.E13 - diff(GAMMA(a, z), z)= (z)^(a - 1)* exp(- z) - D[Gamma[a, z], z]= (z)^(a - 1)* Exp[- z] Successful Successful - -
8.8.E15 diff((z)^(- a)* GAMMA(a)-GAMMA(a, z), [z$(n)])=(- 1)^(n)* (z)^(- a - n)* GAMMA(a + n)-GAMMA(a + n, z) D[(z)^(- a)* Gamma[a, 0, z], {z, n}]=(- 1)^(n)* (z)^(- a - n)* Gamma[a + n, 0, z] Failure Failure Skip Skip
8.8.E16 diff((z)^(- a)* GAMMA(a, z), [z$(n)])=(- 1)^(n)* (z)^(- a - n)* GAMMA(a + n, z) D[(z)^(- a)* Gamma[a, z], {z, n}]=(- 1)^(n)* (z)^(- a - n)* Gamma[a + n, z] Failure Failure Skip Skip
8.8.E17 diff(exp(z)*GAMMA(a)-GAMMA(a, z), [z$(n)])=(- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n)-GAMMA(a - n, z) D[Exp[z]*Gamma[a, 0, z], {z, n}]=(- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, 0, z] Failure Failure Skip Successful
8.8.E18 diff((z)^(a)* exp(z)*(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a), [z$(n)])= (z)^(a - n)* exp(z)*(z)^(-(a - n))*(GAMMA(a - n)-GAMMA(a - n, z))/GAMMA(a - n) Error Failure Error Skip -
8.8.E19 diff(exp(z)*GAMMA(a, z), [z$(n)])=(- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n, z) D[Exp[z]*Gamma[a, z], {z, n}]=(- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, z] Failure Failure Skip Skip
8.10.E1 (x)^(1 - a)* exp(x)*GAMMA(a, x)< = 1 (x)^(1 - a)* Exp[x]*Gamma[a, x]< = 1 Failure Failure Skip Successful
8.10.E2 GAMMA(a)-GAMMA(a, x)> =((x)^(a - 1))/(a)*(1 - exp(- x)) Gamma[a, 0, x]> =Divide[(x)^(a - 1),a]*(1 - Exp[- x]) Failure Failure Skip Successful
8.10.E3 (x)^(1 - a)* exp(x)*GAMMA(a, x)= 1 +(a - 1)/(x)*vartheta (x)^(1 - a)* Exp[x]*Gamma[a, x]= 1 +Divide[a - 1,x]*\[CurlyTheta] Failure Failure
Fail
1.052938223-1.733408016*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 1}
.6195824495-.7346525318*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 2}
.4531580595-.4544327802*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 3}
-2.947061775-.5618351419*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[1.0529382235611282, -1.733408017034722] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.6195824493248067, -0.7346525326366091] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.4531580595377106, -0.4544327806624232] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.947061776438873, -0.5618351417809119] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ϑ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.10.E5 A[n]< (x)^(1 - a)* exp(x)*GAMMA(a, x) Subscript[A, n]< (x)^(1 - a)* Exp[x]*Gamma[a, x] Failure Failure Successful Successful
8.10.E5 (x)^(1 - a)* exp(x)*GAMMA(a, x)< B[n] (x)^(1 - a)* Exp[x]*Gamma[a, x]< Subscript[B, n] Failure Failure Successful Successful
8.10.E7 I = int((t)^(a - 1)* exp(t), t = 0..x) I = Integrate[(t)^(a - 1)* Exp[t], {t, 0, x}] Failure Failure Skip
Fail
Complex[-2.925303491814363, 1.0] <- {Rule[a, Rational[1, 2]], Rule[x, 1]}
Complex[-6.687685525621974, 1.0000000000000002] <- {Rule[a, Rational[1, 2]], Rule[x, 2]}
Complex[-14.626171384019093, 1.0000000000000007] <- {Rule[a, Rational[1, 2]], Rule[x, 3]}
8.10.E7 int((t)^(a - 1)* exp(t), t = 0..x)= GAMMA(a)*(x)^(a)* (- x)^(-(a))*(GAMMA(a)-GAMMA(a, - x))/GAMMA(a) Error Failure Error Skip -
8.10#Ex5 c[a]=(GAMMA(1 + a))^(1/(a - 1)) Subscript[c, a]=(Gamma[1 + a])^(1/(a - 1)) Failure Failure
Fail
-.342222950+.7512982152*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = 2^(1/2)+I*2^(1/2)}
-.342222950-2.077128909*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = 2^(1/2)-I*2^(1/2)}
-3.170650074-2.077128909*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = -2^(1/2)-I*2^(1/2)}
-3.170650074+.7512982152*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Successful
8.10#Ex6 d[a]=(GAMMA(1 + a))^(- 1/ a) Subscript[d, a]=(Gamma[1 + a])^(- 1/ a) Failure Failure
Fail
.7353701374+1.747162536*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = 2^(1/2)+I*2^(1/2)}
.7353701374-1.081264588*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = 2^(1/2)-I*2^(1/2)}
-2.093056987-1.081264588*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = -2^(1/2)-I*2^(1/2)}
-2.093056987+1.747162536*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Successful
8.10.E10 (x)/(2*a)*((1 +(2)/(x))^(a)- 1)< (x)^(1 - a)* exp(x)*GAMMA(a, x) Divide[x,2*a]*((1 +Divide[2,x])^(a)- 1)< (x)^(1 - a)* Exp[x]*Gamma[a, x] Failure Failure Successful Successful
8.10.E10 (x)^(1 - a)* exp(x)*GAMMA(a, x)< =(x)/(a*c[a])*((1 +(c[a])/(x))^(a)- 1) (x)^(1 - a)* Exp[x]*Gamma[a, x]< =Divide[x,a*Subscript[c, a]]*((1 +Divide[Subscript[c, a],x])^(a)- 1) Failure Failure Successful Successful
8.10.E11 (1 - exp(- alpha[a]*x))^(a)< = (GAMMA(a)-GAMMA(a, x))/GAMMA(a) (1 - Exp[- Subscript[\[Alpha], a]*x])^(a)< = GammaRegularized[a, 0, x] Failure Failure Successful Successful
8.10.E11 (GAMMA(a)-GAMMA(a, x))/GAMMA(a)< =(1 - exp(- beta[a]*x))^(a) GammaRegularized[a, 0, x]< =(1 - Exp[- Subscript[\[Beta], a]*x])^(a) Failure Failure Successful Successful
8.10.E13 (GAMMA(n, n))/(GAMMA(n))<(1)/(2) Divide[Gamma[n, n],Gamma[n]]<Divide[1,2] Failure Failure Successful Successful
8.10.E13 (1)/(2)<(GAMMA(n, n - 1))/(GAMMA(n)) Divide[1,2]<Divide[Gamma[n, n - 1],Gamma[n]] Failure Failure Successful Successful
8.11.E2 GAMMA(a, z)= (z)^(a - 1)* exp(- z)*(sum((u[k])/((z)^(k)), k = 0..n - 1)+ R[n]*(a , z)) Gamma[a, z]= (z)^(a - 1)* Exp[- z]*(Sum[Divide[Subscript[u, k],(z)^(k)], {k, 0, n - 1}]+ Subscript[R, n]*(a , z)) Failure Failure Skip Error
8.11.E4 GAMMA(a)-GAMMA(a, z)= (z)^(a)* exp(- z)*sum(((z)^(k))/(pochhammer(a, k + 1)), k = 0..infinity) Gamma[a, 0, z]= (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Pochhammer[a, k + 1]], {k, 0, Infinity}] Successful Successful - -
8.11.E15 S[n]*(x)=(GAMMA(n + 1)-GAMMA(n + 1, n*x))/((n*x)^(n)* exp(- n*x)) Subscript[S, n]*(x)=Divide[Gamma[n + 1, 0, n*x],(n*x)^(n)* Exp[- n*x]] Failure Failure
Fail
.6959317335+1.414213562*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
.633899074+2.828427124*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
-1.119204955+4.242640686*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
.219685512+1.414213562*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
Successful
8.12.E3 (GAMMA(a)-GAMMA(a, z))/GAMMA(a)=(1)/(2)*erfc(- eta*sqrt(a/ 2))- S*(a , eta) GammaRegularized[a, 0, z]=Divide[1,2]*Erfc[- \[Eta]*Sqrt[a/ 2]]- S*(a , \[Eta]) Failure Failure Error Error
8.12.E4 GAMMA(a, z)/GAMMA(a)=(1)/(2)*erfc(eta*sqrt(a/ 2))+ S*(a , eta) GammaRegularized[a, z]=Divide[1,2]*Erfc[\[Eta]*Sqrt[a/ 2]]+ S*(a , \[Eta]) Failure Failure Error Error
8.12.E5 (exp(+ Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(+ Pi*I))/GAMMA(- a)= +(1)/(2)*erfc(+ I*eta*sqrt(a/ 2))- I*T*(a , eta) Divide[Exp[+ Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[+ Pi*I]]= +Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/ 2]]- I*T*(a , \[Eta]) Failure Failure Error Error
8.12.E5 (exp(- Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(- Pi*I))/GAMMA(- a)= -(1)/(2)*erfc(- I*eta*sqrt(a/ 2))- I*T*(a , eta) Divide[Exp[- Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[- Pi*I]]= -Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/ 2]]- I*T*(a , \[Eta]) Failure Failure Error Error
8.12#Ex5 GAMMA(a + 1)*(exp(+ Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(+ Pi*I))= -(1)/(2)*erfc(+ I*eta*sqrt(a/ 2))+ I*T*(a , eta) Gamma[a + 1]*Divide[Exp[+ Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[+ Pi*I]]= -Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/ 2]]+ I*T*(a , \[Eta]) Failure Failure Error Error
8.12#Ex5 GAMMA(a + 1)*(exp(- Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(- Pi*I))= +(1)/(2)*erfc(- I*eta*sqrt(a/ 2))+ I*T*(a , eta) Gamma[a + 1]*Divide[Exp[- Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[- Pi*I]]= +Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/ 2]]+ I*T*(a , \[Eta]) Failure Failure Error Error
8.12.E6 (z)^(- a)* (- z)^(-(- a))*(GAMMA(- a)-GAMMA(- a, - z))/GAMMA(- a)= cos(Pi*a)- 2*sin(Pi*a)*((exp((1)/(2)*a*(eta)^(2)))/(sqrt(Pi))*dawson(eta*sqrt(a/ 2))+ T*(a , eta)) Error Failure Error Error -
8.12.E10 c[k]*(eta)=(1)/(eta)*diff(c[k - 1]*(eta), eta)+(- 1)^(k)*(g[k])/(mu) Subscript[c, k]*(\[Eta])=Divide[1,\[Eta]]*D[Subscript[c, k - 1]*(\[Eta]), \[Eta]]+(- 1)^(k)*Divide[Subscript[g, k],\[Mu]] Failure Failure Skip Skip
8.12#Ex23 d*(+ chi)=sqrt((1)/(2)*Pi)*exp((chi)^(2)/ 2)*erfc(+ chi/sqrt(2)) d*(+ \[Chi])=Sqrt[Divide[1,2]*Pi]*Exp[(\[Chi])^(2)/ 2]*Erfc[+ \[Chi]/Sqrt[2]] Failure Failure
Fail
-.3819402210+4.260963736*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)+I*2^(1/2)}
3.618059777+.2609637385*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)-I*2^(1/2)}
-.3819402210-3.739036260*I <- {chi = 2^(1/2)+I*2^(1/2), d = -2^(1/2)-I*2^(1/2)}
-4.381940219+.2609637385*I <- {chi = 2^(1/2)+I*2^(1/2), d = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.3819402207134648, 4.260963738906431] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.6180597792865354, -0.26096373890643143] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.425065647600777, -6.540234379036898] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.574934352399223, 2.5402343790368977] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.12#Ex23 d*(- chi)=sqrt((1)/(2)*Pi)*exp((chi)^(2)/ 2)*erfc(- chi/sqrt(2)) d*(- \[Chi])=Sqrt[Divide[1,2]*Pi]*Exp[(\[Chi])^(2)/ 2]*Erfc[- \[Chi]/Sqrt[2]] Failure Failure
Fail
1.425065646-6.540234377*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)+I*2^(1/2)}
-2.574934352-2.540234379*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)-I*2^(1/2)}
1.425065646+1.459765619*I <- {chi = 2^(1/2)+I*2^(1/2), d = -2^(1/2)-I*2^(1/2)}
5.425065644-2.540234379*I <- {chi = 2^(1/2)+I*2^(1/2), d = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.425065647600777, -6.540234379036898] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.574934352399223, 2.5402343790368977] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.3819402207134648, 4.260963738906431] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.6180597792865354, -0.26096373890643143] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.12.E21 GAMMA(a, x)/GAMMA(a)= q GammaRegularized[a, x]= q Failure Failure
Fail
-.6276752047-.7874152397*I <- {a = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), x = 1}
-1.269609688-.9406490460*I <- {a = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), x = 2}
-1.440152063-1.201678512*I <- {a = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), x = 3}
-.6276752047+2.041011884*I <- {a = 2^(1/2)+I*2^(1/2), q = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[-0.6276752046971461, -0.7874152400294763] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[-1.2696096879275383, -0.9406490461902074] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[-1.4401520638257446, -1.2016785120794473] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[-0.6276752046971461, 2.0410118847167142] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
... skip entries to safe data
8.13.E1 1 + (a)^(- 1)< x[-]*(a) 1 + (a)^(- 1)< Subscript[x, -]*(a) Error Failure - Error
8.13.E1 x[-]*(a)< ln(abs(a)) Subscript[x, -]*(a)< Log[Abs[a]] Error Failure - Error
8.14.E1 int(exp(- a*x)*(GAMMA(b)-GAMMA(b, x))/(GAMMA(b)), x = 0..infinity)=((1 + a)^(- b))/(a) Integrate[Exp[- a*x]*Divide[Gamma[b, 0, x],Gamma[b]], {x, 0, Infinity}]=Divide[(1 + a)^(- b),a] Successful Failure - Error
8.14.E2 int(exp(- a*x)*GAMMA(b, x), x = 0..infinity)= GAMMA(b)*(1 -(1 + a)^(- b))/(a) Integrate[Exp[- a*x]*Gamma[b, x], {x, 0, Infinity}]= Gamma[b]*Divide[1 -(1 + a)^(- b),a] Failure Failure Skip Error
8.14.E3