Error Functions, Dawson’s and Fresnel Integrals - 8.2 Definitions and Basic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.2.E1 γ ⁡ ( a , z ) = ∫ 0 z t a - 1 ⁢ e - t ⁢ d t incomplete-gamma 𝑎 𝑧 superscript subscript 0 𝑧 superscript 𝑡 𝑎 1 superscript 𝑒 𝑡 𝑡 {\displaystyle{\displaystyle\gamma\left(a,z\right)=\int_{0}^{z}t^{a-1}e^{-t}% \mathrm{d}t}}
\incgamma@{a}{z} = \int_{0}^{z}t^{a-1}e^{-t}\diff{t}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
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}, GenerateConditions->None]
Failure Successful Successful [Tested: 21] Successful [Tested: 21]
8.2.E2 Γ ⁡ ( a , z ) = ∫ z ∞ t a - 1 ⁢ e - t ⁢ d t incomplete-Gamma 𝑎 𝑧 superscript subscript 𝑧 superscript 𝑡 𝑎 1 superscript 𝑒 𝑡 𝑡 {\displaystyle{\displaystyle\Gamma\left(a,z\right)=\int_{z}^{\infty}t^{a-1}e^{% -t}\mathrm{d}t}}
\incGamma@{a}{z} = \int_{z}^{\infty}t^{a-1}e^{-t}\diff{t}		 

GAMMA(a, z) = int((t)^(a - 1)* exp(- t), t = z..infinity)

Gamma[a, z] == Integrate[(t)^(a - 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None]
Failure Successful
Failed [14 / 42]
Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -2, z = 1/2*3^(1/2)+1/2*I}


Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 42]
8.2.E3 γ ⁡ ( a , z ) + Γ ⁡ ( a , z ) = Γ ⁡ ( a ) incomplete-gamma 𝑎 𝑧 incomplete-Gamma 𝑎 𝑧 Euler-Gamma 𝑎 {\displaystyle{\displaystyle\gamma\left(a,z\right)+\Gamma\left(a,z\right)=% \Gamma\left(a\right)}}
\incgamma@{a}{z}+\incGamma@{a}{z} = \EulerGamma@{a}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
GAMMA(a)-GAMMA(a, z)+ GAMMA(a, z) = GAMMA(a)

Gamma[a, 0, z]+ Gamma[a, z] == Gamma[a]
Successful Successful - Successful [Tested: 7]
8.2#Ex1 P ⁡ ( a , z ) = γ ⁡ ( a , z ) Γ ⁡ ( a ) incomplete-gamma-P 𝑎 𝑧 incomplete-gamma 𝑎 𝑧 Euler-Gamma 𝑎 {\displaystyle{\displaystyle P\left(a,z\right)=\frac{\gamma\left(a,z\right)}{% \Gamma\left(a\right)}}}
\normincGammaP@{a}{z} = \frac{\incgamma@{a}{z}}{\EulerGamma@{a}}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
(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 - Successful [Tested: 21]
8.2#Ex2 Q ⁡ ( a , z ) = Γ ⁡ ( a , z ) Γ ⁡ ( a ) incomplete-gamma-Q 𝑎 𝑧 incomplete-Gamma 𝑎 𝑧 Euler-Gamma 𝑎 {\displaystyle{\displaystyle Q\left(a,z\right)=\frac{\Gamma\left(a,z\right)}{% \Gamma\left(a\right)}}}
\normincGammaQ@{a}{z} = \frac{\incGamma@{a}{z}}{\EulerGamma@{a}}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
GAMMA(a, z)/GAMMA(a) = (GAMMA(a, z))/(GAMMA(a))

GammaRegularized[a, z] == Divide[Gamma[a, z],Gamma[a]]
Successful Successful - Successful [Tested: 21]
8.2.E5 P ⁡ ( a , z ) + Q ⁡ ( a , z ) = 1 incomplete-gamma-P 𝑎 𝑧 incomplete-gamma-Q 𝑎 𝑧 1 {\displaystyle{\displaystyle P\left(a,z\right)+Q\left(a,z\right)=1}}
\normincGammaP@{a}{z}+\normincGammaQ@{a}{z} = 1		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
(GAMMA(a)-GAMMA(a, z))/GAMMA(a)+ GAMMA(a, z)/GAMMA(a) = 1

GammaRegularized[a, 0, z]+ GammaRegularized[a, z] == 1
Successful Successful - Successful [Tested: 21]
8.2.E6 γ * ⁡ ( a , z ) = z - a ⁢ P ⁡ ( a , z ) incomplete-gamma-star 𝑎 𝑧 superscript 𝑧 𝑎 incomplete-gamma-P 𝑎 𝑧 {\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=z^{-a}P\left(a,z\right)}}
\scincgamma@{a}{z} = z^{-a}\normincGammaP@{a}{z}		 

(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a)

Error
Successful Missing Macro Error Skip - symbolical successful subtest -
8.2.E6 z - a ⁢ P ⁡ ( a , z ) = z - a Γ ⁡ ( a ) ⁢ γ ⁡ ( a , z ) superscript 𝑧 𝑎 incomplete-gamma-P 𝑎 𝑧 superscript 𝑧 𝑎 Euler-Gamma 𝑎 incomplete-gamma 𝑎 𝑧 {\displaystyle{\displaystyle z^{-a}P\left(a,z\right)=\frac{z^{-a}}{\Gamma\left% (a\right)}\gamma\left(a,z\right)}}
z^{-a}\normincGammaP@{a}{z} = \frac{z^{-a}}{\EulerGamma@{a}}\incgamma@{a}{z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
(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
Failed [21 / 21]
Result: .2669922311+.3869142026*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}


Result: 3.493684189-.9843316111*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 21]
8.2.E7 γ * ⁡ ( a , z ) = 1 Γ ⁡ ( a ) ⁢ ∫ 0 1 t a - 1 ⁢ e - z ⁢ t ⁢ d t incomplete-gamma-star 𝑎 𝑧 1 Euler-Gamma 𝑎 superscript subscript 0 1 superscript 𝑡 𝑎 1 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=\frac{1}{\Gamma\left(a% \right)}\int_{0}^{1}t^{a-1}e^{-zt}\mathrm{d}t}}
\scincgamma@{a}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{1}t^{a-1}e^{-zt}\diff{t}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (1)/(GAMMA(a))*int((t)^(a - 1)* exp(- z*t), t = 0..1)

Error
Failure Missing Macro Error Successful [Tested: 21] -
8.2.E8 γ ⁡ ( a , z ⁢ e 2 ⁢ π ⁢ m ⁢ i ) = e 2 ⁢ π ⁢ m ⁢ i ⁢ a ⁢ γ ⁡ ( a , z ) incomplete-gamma 𝑎 𝑧 superscript 𝑒 2 𝜋 𝑚 𝑖 superscript 𝑒 2 𝜋 𝑚 𝑖 𝑎 incomplete-gamma 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(a,ze^{2\pi mi}\right)=e^{2\pi mia}% \gamma\left(a,z\right)}}
\incgamma@{a}{ze^{2\pi mi}} = e^{2\pi mia}\incgamma@{a}{z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
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
Failed [28 / 63]
Result: 1.772453851+.14e-8*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 1, a = 1}


Result: 1.772453851+.62e-8*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 3, a = 1}

... skip entries to safe data
 Successful [Tested: 21]
8.2.E9 Γ ⁡ ( a , z ⁢ e 2 ⁢ π ⁢ m ⁢ i ) = e 2 ⁢ π ⁢ m ⁢ i ⁢ a ⁢ Γ ⁡ ( a , z ) + ( 1 - e 2 ⁢ π ⁢ m ⁢ i ⁢ a ) ⁢ Γ ⁡ ( a ) incomplete-Gamma 𝑎 𝑧 superscript 𝑒 2 𝜋 𝑚 𝑖 superscript 𝑒 2 𝜋 𝑚 𝑖 𝑎 incomplete-Gamma 𝑎 𝑧 1 superscript 𝑒 2 𝜋 𝑚 𝑖 𝑎 Euler-Gamma 𝑎 {\displaystyle{\displaystyle\Gamma\left(a,ze^{2\pi mi}\right)=e^{2\pi mia}% \Gamma\left(a,z\right)+(1-e^{2\pi mia})\Gamma\left(a\right)}}
\incGamma@{a}{ze^{2\pi mi}} = e^{2\pi mia}\incGamma@{a}{z}+(1-e^{2\pi mia})\EulerGamma@{a}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
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
Failed [28 / 63]
Result: -.701776495-.3980325655*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 1}


Result: -.701776493-.3980325669*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 3}

... skip entries to safe data
Failed [28 / 63]
Result: Complex[-0.7017764946044596, -0.3980325648566406]
Test Values: {Rule[a, 1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7017764946044598, -0.39803256485664035]
Test Values: {Rule[a, 1.5], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
8.2.E10 e - π ⁢ i ⁢ a ⁢ Γ ⁡ ( a , z ⁢ e π ⁢ i ) - e π ⁢ i ⁢ a ⁢ Γ ⁡ ( a , z ⁢ e - π ⁢ i ) = - 2 ⁢ π ⁢ i Γ ⁡ ( 1 - a ) superscript 𝑒 𝜋 𝑖 𝑎 incomplete-Gamma 𝑎 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑒 𝜋 𝑖 𝑎 incomplete-Gamma 𝑎 𝑧 superscript 𝑒 𝜋 𝑖 2 𝜋 𝑖 Euler-Gamma 1 𝑎 {\displaystyle{\displaystyle e^{-\pi ia}\Gamma\left(a,ze^{\pi i}\right)-e^{\pi ia% }\Gamma\left(a,ze^{-\pi i}\right)=-\frac{2\pi i}{\Gamma\left(1-a\right)}}}
e^{-\pi ia}\incGamma@{a}{ze^{\pi i}}-e^{\pi ia}\incGamma@{a}{ze^{-\pi i}} = -\frac{2\pi i}{\EulerGamma@{1-a}}		 ℜ ⁡ ( 1 - a ) > 0 1 𝑎 0 {\displaystyle{\displaystyle\Re(1-a)>0}} 
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
Failed [28 / 28]
Result: -2.710645106+2.690107924*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I}


Result: .1295806364+5.171352915*I
Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [28 / 28]
Result: Complex[-2.710645107283499, 2.690107923521634]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.12958063668571548, 5.171352913200156]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.2.E11 Γ ⁡ ( a , z ⁢ e + π ⁢ i ) = Γ ⁡ ( a ) ⁢ ( 1 - z a ⁢ e + π ⁢ i ⁢ a ⁢ γ * ⁡ ( a , - z ) ) incomplete-Gamma 𝑎 𝑧 superscript 𝑒 𝜋 𝑖 Euler-Gamma 𝑎 1 superscript 𝑧 𝑎 superscript 𝑒 𝜋 𝑖 𝑎 incomplete-gamma-star 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(a,ze^{+\pi i}\right)=\Gamma\left(a% \right)(1-z^{a}e^{+\pi ia}\gamma^{*}\left(a,-z\right))}}
\incGamma@{a}{ze^{+\pi i}} = \EulerGamma@{a}(1-z^{a}e^{+\pi ia}\scincgamma@{a}{-z})		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
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 Missing Macro Error
Failed [4 / 21]
Result: 2.046852240-1.006153525*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}


Result: -.4563214597+.8560373719*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 -
8.2.E11 Γ ⁡ ( a , z ⁢ e - π ⁢ i ) = Γ ⁡ ( a ) ⁢ ( 1 - z a ⁢ e - π ⁢ i ⁢ a ⁢ γ * ⁡ ( a , - z ) ) incomplete-Gamma 𝑎 𝑧 superscript 𝑒 𝜋 𝑖 Euler-Gamma 𝑎 1 superscript 𝑧 𝑎 superscript 𝑒 𝜋 𝑖 𝑎 incomplete-gamma-star 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(a,ze^{-\pi i}\right)=\Gamma\left(a% \right)(1-z^{a}e^{-\pi ia}\gamma^{*}\left(a,-z\right))}}
\incGamma@{a}{ze^{-\pi i}} = \EulerGamma@{a}(1-z^{a}e^{-\pi ia}\scincgamma@{a}{-z})		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
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 Missing Macro Error
Failed [10 / 21]
Result: 1.510498964-.9249588863*I
Test Values: {a = 1.5, z = 1/2-1/2*I*3^(1/2)}


Result: -.7017764949-.3980325647*I
Test Values: {a = 1.5, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
 -
8.2.E12 d 2 w d z 2 + ( 1 + 1 - a z ) ⁢ d w d z = 0 derivative 𝑤 𝑧 2 1 1 𝑎 𝑧 derivative 𝑤 𝑧 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% 1+\frac{1-a}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}=0}}
\deriv[2]{w}{z}+\left(1+\frac{1-a}{z}\right)\deriv{w}{z} = 0		 

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 - Successful [Tested: 300]
8.2.E13 d 2 w d z 2 - ( 1 + 1 - a z ) ⁢ d w d z + 1 - a z 2 ⁢ w = 0 derivative 𝑤 𝑧 2 1 1 𝑎 𝑧 derivative 𝑤 𝑧 1 𝑎 superscript 𝑧 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(% 1+\frac{1-a}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{1-a}{z^{2}}w=0}}
\deriv[2]{w}{z}-\left(1+\frac{1-a}{z}\right)\deriv{w}{z}+\frac{1-a}{z^{2}}w = 0		 

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
Failed [300 / 300]
Result: 2.165063509-1.250000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -2.165063509+1.250000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.1650635094610964, -1.25]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.1650635094610964, 1.2500000000000004]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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