8.21: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/8.21.E3 8.21.E3] || [[Item:Q2724|<math>\int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</syntaxhighlight> || <math>0 < \realpart@@{a}, \realpart@@{a} < 1, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)* exp(+ I*t), t = 0..infinity) = exp(+(1)/(2)*Pi*I*a)*GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)* Exp[+ I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[+Divide[1,2]*Pi*I*a]*Gamma[a]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 1]
| [https://dlmf.nist.gov/8.21.E3 8.21.E3] || <math qid="Q2724">\int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</syntaxhighlight> || <math>0 < \realpart@@{a}, \realpart@@{a} < 1, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)* exp(+ I*t), t = 0..infinity) = exp(+(1)/(2)*Pi*I*a)*GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)* Exp[+ I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[+Divide[1,2]*Pi*I*a]*Gamma[a]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/8.21.E3 8.21.E3] || [[Item:Q2724|<math>\int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</syntaxhighlight> || <math>0 < \realpart@@{a}, \realpart@@{a} < 1, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)* exp(- I*t), t = 0..infinity) = exp(-(1)/(2)*Pi*I*a)*GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)* Exp[- I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[-Divide[1,2]*Pi*I*a]*Gamma[a]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 1]
| [https://dlmf.nist.gov/8.21.E3 8.21.E3] || <math qid="Q2724">\int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</syntaxhighlight> || <math>0 < \realpart@@{a}, \realpart@@{a} < 1, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)* exp(- I*t), t = 0..infinity) = exp(-(1)/(2)*Pi*I*a)*GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)* Exp[- I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[-Divide[1,2]*Pi*I*a]*Gamma[a]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 1]
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Latest revision as of 11:19, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.21.E3 ∫ 0 ∞ t a - 1 ⁢ e + i ⁢ t ⁢ d t = e + 1 2 ⁢ π ⁢ i ⁢ a ⁢ Γ ⁡ ( a ) superscript subscript 0 superscript 𝑡 𝑎 1 superscript 𝑒 imaginary-unit 𝑡 𝑡 superscript 𝑒 1 2 𝜋 imaginary-unit 𝑎 Euler-Gamma 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{+\mathrm{i}t}\mathrm{d}% t=e^{+\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right)}}
\int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}		 0 < ℜ ⁡ a , ℜ ⁡ a < 1 , ℜ ⁡ a > 0 formulae-sequence 0 𝑎 formulae-sequence 𝑎 1 𝑎 0 {\displaystyle{\displaystyle 0<\Re a,\Re a<1,\Re a>0}} 
int((t)^(a - 1)* exp(+ I*t), t = 0..infinity) = exp(+(1)/(2)*Pi*I*a)*GAMMA(a)

Integrate[(t)^(a - 1)* Exp[+ I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[+Divide[1,2]*Pi*I*a]*Gamma[a]
Successful Aborted - Successful [Tested: 1]
8.21.E3 ∫ 0 ∞ t a - 1 ⁢ e - i ⁢ t ⁢ d t = e - 1 2 ⁢ π ⁢ i ⁢ a ⁢ Γ ⁡ ( a ) superscript subscript 0 superscript 𝑡 𝑎 1 superscript 𝑒 imaginary-unit 𝑡 𝑡 superscript 𝑒 1 2 𝜋 imaginary-unit 𝑎 Euler-Gamma 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{-\mathrm{i}t}\mathrm{d}% t=e^{-\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right)}}
\int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}		 0 < ℜ ⁡ a , ℜ ⁡ a < 1 , ℜ ⁡ a > 0 formulae-sequence 0 𝑎 formulae-sequence 𝑎 1 𝑎 0 {\displaystyle{\displaystyle 0<\Re a,\Re a<1,\Re a>0}} 
int((t)^(a - 1)* exp(- I*t), t = 0..infinity) = exp(-(1)/(2)*Pi*I*a)*GAMMA(a)

Integrate[(t)^(a - 1)* Exp[- I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[-Divide[1,2]*Pi*I*a]*Gamma[a]
Successful Aborted - Successful [Tested: 1]
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