Gamma Function - 5.13 Integrals

From testwiki
Revision as of 16:43, 25 May 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.13.E1 1 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( s + a ) ⁢ Γ ⁡ ( b - s ) ⁢ z - s ⁢ d s = Γ ⁡ ( a + b ) ⁢ z a ( 1 + z ) a + b 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑠 𝑎 Euler-Gamma 𝑏 𝑠 superscript 𝑧 𝑠 𝑠 Euler-Gamma 𝑎 𝑏 superscript 𝑧 𝑎 superscript 1 𝑧 𝑎 𝑏 {\displaystyle{\displaystyle\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma% \left(s+a\right)\Gamma\left(b-s\right)z^{-s}\mathrm{d}s=\frac{\Gamma\left(a+b% \right)z^{a}}{(1+z)^{a+b}}}}
\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+a}\EulerGamma@{b-s}z^{-s}\diff{s} = \frac{\EulerGamma@{a+b}z^{a}}{(1+z)^{a+b}}		 ℜ ⁡ ( a + b ) > 0 , - ℜ ⁡ a < c , c < ℜ ⁡ b , | ph ⁡ z | < π , ℜ ⁡ ( s + a ) > 0 , ℜ ⁡ ( b - s ) > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 𝑏 0 formulae-sequence 𝑎 𝑐 formulae-sequence 𝑐 𝑏 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑠 𝑎 0 formulae-sequence 𝑏 𝑠 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re\left(a+b\right)>0,-\Re a<c,c<\Re b,|% \operatorname{ph}z|<\pi,\Re(s+a)>0,\Re(b-s)>0,\Re(a+b)>0}} 
(1)/(2*Pi*I)*int(GAMMA(s + a)*GAMMA(b - s)*(z)^(- s), s = c - I*infinity..c + I*infinity) = (GAMMA(a + b)*(z)^(a))/((1 + z)^(a + b))

Divide[1,2*Pi*I]*Integrate[Gamma[s + a]*Gamma[b - s]*(z)^(- s), {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None] == Divide[Gamma[a + b]*(z)^(a),(1 + z)^(a + b)]
Skipped - Unable to analyze test case: Null Aborted - Skipped - Because timed out
5.13.E2 1 2 ⁢ π ⁢ ∫ - ∞ ∞ | Γ ⁡ ( a + i ⁢ t ) | 2 ⁢ e ( 2 ⁢ b - π ) ⁢ t ⁢ d t = Γ ⁡ ( 2 ⁢ a ) ( 2 ⁢ sin ⁡ b ) 2 ⁢ a 1 2 𝜋 superscript subscript superscript Euler-Gamma 𝑎 𝑖 𝑡 2 superscript 𝑒 2 𝑏 𝜋 𝑡 𝑡 Euler-Gamma 2 𝑎 superscript 2 𝑏 2 𝑎 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}|\Gamma\left(% a+it\right)|^{2}e^{(2b-\pi)t}\mathrm{d}t=\frac{\Gamma\left(2a\right)}{(2\sin b% )^{2a}}}}
\frac{1}{2\pi}\int_{-\infty}^{\infty}|\EulerGamma@{a+it}|^{2}e^{(2b-\pi)t}\diff{t} = \frac{\EulerGamma@{2a}}{(2\sin@@{b})^{2a}}		 a > 0 , 0 < b , b < π , ℜ ⁡ ( a + i ⁢ t ) > 0 , ℜ ⁡ ( 2 ⁢ a ) > 0 formulae-sequence 𝑎 0 formulae-sequence 0 𝑏 formulae-sequence 𝑏 𝜋 formulae-sequence 𝑎 imaginary-unit 𝑡 0 2 𝑎 0 {\displaystyle{\displaystyle a>0,0<b,b<\pi,\Re(a+\mathrm{i}t)>0,\Re(2a)>0}} 
(1)/(2*Pi)*int((abs(GAMMA(a + I*t)))^(2)* exp((2*b - Pi)*t), t = - infinity..infinity) = (GAMMA(2*a))/((2*sin(b))^(2*a))

Divide[1,2*Pi]*Integrate[(Abs[Gamma[a + I*t]])^(2)* Exp[(2*b - Pi)*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[2*a],(2*Sin[b])^(2*a)]
Skipped - Unable to analyze test case: Null Aborted - Skipped - Because timed out
5.13.E3 1 2 ⁢ π ⁢ ∫ - ∞ ∞ Γ ⁡ ( a + i ⁢ t ) ⁢ Γ ⁡ ( b + i ⁢ t ) ⁢ Γ ⁡ ( c - i ⁢ t ) ⁢ Γ ⁡ ( d - i ⁢ t ) ⁢ d t = Γ ⁡ ( a + c ) ⁢ Γ ⁡ ( a + d ) ⁢ Γ ⁡ ( b + c ) ⁢ Γ ⁡ ( b + d ) Γ ⁡ ( a + b + c + d ) 1 2 𝜋 superscript subscript Euler-Gamma 𝑎 𝑖 𝑡 Euler-Gamma 𝑏 𝑖 𝑡 Euler-Gamma 𝑐 𝑖 𝑡 Euler-Gamma 𝑑 𝑖 𝑡 𝑡 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑎 𝑑 Euler-Gamma 𝑏 𝑐 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma\left(a% +it\right)\Gamma\left(b+it\right)\Gamma\left(c-it\right)\Gamma\left(d-it\right% )\mathrm{d}t=\frac{\Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c% \right)\Gamma\left(b+d\right)}{\Gamma\left(a+b+c+d\right)}}}
\frac{1}{2\pi}\int_{-\infty}^{\infty}\EulerGamma@{a+it}\EulerGamma@{b+it}\EulerGamma@{c-it}\EulerGamma@{d-it}\diff{t} = \frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}}		 ℜ ⁡ ( a + i ⁢ t ) > 0 , ℜ ⁡ ( b + i ⁢ t ) > 0 , ℜ ⁡ ( c - i ⁢ t ) > 0 , ℜ ⁡ ( d - i ⁢ t ) > 0 , ℜ ⁡ ( a + c ) > 0 , ℜ ⁡ ( a + d ) > 0 , ℜ ⁡ ( b + c ) > 0 , ℜ ⁡ ( b + d ) > 0 , ℜ ⁡ ( a + b + c + d ) > 0 formulae-sequence 𝑎 imaginary-unit 𝑡 0 formulae-sequence 𝑏 imaginary-unit 𝑡 0 formulae-sequence 𝑐 imaginary-unit 𝑡 0 formulae-sequence 𝑑 imaginary-unit 𝑡 0 formulae-sequence 𝑎 𝑐 0 formulae-sequence 𝑎 𝑑 0 formulae-sequence 𝑏 𝑐 0 formulae-sequence 𝑏 𝑑 0 𝑎 𝑏 𝑐 𝑑 0 {\displaystyle{\displaystyle\Re(a+\mathrm{i}t)>0,\Re(b+\mathrm{i}t)>0,\Re(c-% \mathrm{i}t)>0,\Re(d-\mathrm{i}t)>0,\Re(a+c)>0,\Re(a+d)>0,\Re(b+c)>0,\Re(b+d)>% 0,\Re(a+b+c+d)>0}} 
(1)/(2*Pi)*int(GAMMA(a + I*t)*GAMMA(b + I*t)*GAMMA(c - I*t)*GAMMA(d - I*t), t = - infinity..infinity) = (GAMMA(a + c)*GAMMA(a + d)*GAMMA(b + c)*GAMMA(b + d))/(GAMMA(a + b + c + d))

Divide[1,2*Pi]*Integrate[Gamma[a + I*t]*Gamma[b + I*t]*Gamma[c - I*t]*Gamma[d - I*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[a + c]*Gamma[a + d]*Gamma[b + c]*Gamma[b + d],Gamma[a + b + c + d]]
Error Aborted - Skipped - Because timed out
5.13.E4 ∫ - ∞ ∞ d t Γ ⁡ ( a + t ) ⁢ Γ ⁡ ( b + t ) ⁢ Γ ⁡ ( c - t ) ⁢ Γ ⁡ ( d - t ) = Γ ⁡ ( a + b + c + d - 3 ) Γ ⁡ ( a + c - 1 ) ⁢ Γ ⁡ ( a + d - 1 ) ⁢ Γ ⁡ ( b + c - 1 ) ⁢ Γ ⁡ ( b + d - 1 ) superscript subscript 𝑡 Euler-Gamma 𝑎 𝑡 Euler-Gamma 𝑏 𝑡 Euler-Gamma 𝑐 𝑡 Euler-Gamma 𝑑 𝑡 Euler-Gamma 𝑎 𝑏 𝑐 𝑑 3 Euler-Gamma 𝑎 𝑐 1 Euler-Gamma 𝑎 𝑑 1 Euler-Gamma 𝑏 𝑐 1 Euler-Gamma 𝑏 𝑑 1 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{\mathrm{d}t}{\Gamma% \left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(c-t\right)\Gamma\left(d-t% \right)}=\frac{\Gamma\left(a+b+c+d-3\right)}{\Gamma\left(a+c-1\right)\Gamma% \left(a+d-1\right)\Gamma\left(b+c-1\right)\Gamma\left(b+d-1\right)}}}
\int_{-\infty}^{\infty}\frac{\diff{t}}{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-t}\EulerGamma@{d-t}} = \frac{\EulerGamma@{a+b+c+d-3}}{\EulerGamma@{a+c-1}\EulerGamma@{a+d-1}\EulerGamma@{b+c-1}\EulerGamma@{b+d-1}}		 ℜ ⁡ ( a + b + c + d ) > 3 , ℜ ⁡ ( a + t ) > 0 , ℜ ⁡ ( b + t ) > 0 , ℜ ⁡ ( c - t ) > 0 , ℜ ⁡ ( d - t ) > 0 , ℜ ⁡ ( a + b + c + d - 3 ) > 0 , ℜ ⁡ ( a + c - 1 ) > 0 , ℜ ⁡ ( a + d - 1 ) > 0 , ℜ ⁡ ( b + c - 1 ) > 0 , ℜ ⁡ ( b + d - 1 ) > 0 formulae-sequence 𝑎 𝑏 𝑐 𝑑 3 formulae-sequence 𝑎 𝑡 0 formulae-sequence 𝑏 𝑡 0 formulae-sequence 𝑐 𝑡 0 formulae-sequence 𝑑 𝑡 0 formulae-sequence 𝑎 𝑏 𝑐 𝑑 3 0 formulae-sequence 𝑎 𝑐 1 0 formulae-sequence 𝑎 𝑑 1 0 formulae-sequence 𝑏 𝑐 1 0 𝑏 𝑑 1 0 {\displaystyle{\displaystyle\Re\left(a+b+c+d\right)>3,\Re(a+t)>0,\Re(b+t)>0,% \Re(c-t)>0,\Re(d-t)>0,\Re(a+b+c+d-3)>0,\Re(a+c-1)>0,\Re(a+d-1)>0,\Re(b+c-1)>0,% \Re(b+d-1)>0}} 
int((1)/(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - t)*GAMMA(d - t)), t = - infinity..infinity) = (GAMMA(a + b + c + d - 3))/(GAMMA(a + c - 1)*GAMMA(a + d - 1)*GAMMA(b + c - 1)*GAMMA(b + d - 1))

Integrate[Divide[1,Gamma[a + t]*Gamma[b + t]*Gamma[c - t]*Gamma[d - t]], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b + c + d - 3],Gamma[a + c - 1]*Gamma[a + d - 1]*Gamma[b + c - 1]*Gamma[b + d - 1]]
Failure Aborted Manual Skip! Skipped - Because timed out
5.13.E5 1 4 ⁢ π ⁢ ∫ - ∞ ∞ ∏ k = 1 4 Γ ⁡ ( a k + i ⁢ t ) ⁢ Γ ⁡ ( a k - i ⁢ t ) Γ ⁡ ( 2 ⁢ i ⁢ t ) ⁢ Γ ⁡ ( - 2 ⁢ i ⁢ t ) ⁢ d t = ∏ 1 ≤ j < k ≤ 4 Γ ⁡ ( a j + a k ) Γ ⁡ ( a 1 + a 2 + a 3 + a 4 ) 1 4 𝜋 superscript subscript superscript subscript product 𝑘 1 4 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 𝑡 subscript product 1 𝑗 𝑘 4 Euler-Gamma subscript 𝑎 𝑗 subscript 𝑎 𝑘 Euler-Gamma subscript 𝑎 1 subscript 𝑎 2 subscript 𝑎 3 subscript 𝑎 4 {\displaystyle{\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{% k=1}^{4}\Gamma\left(a_{k}+it\right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2% it\right)\Gamma\left(-2it\right)}\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}% \Gamma\left(a_{j}+a_{k}\right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)}}}
\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\EulerGamma@{a_{k}+it}\EulerGamma@{a_{k}-it}}{\EulerGamma@{2it}\EulerGamma@{-2it}}\diff{t} = \frac{\prod_{1\leq j<k\leq 4}\EulerGamma@{a_{j}+a_{k}}}{\EulerGamma@{a_{1}+a_{2}+a_{3}+a_{4}}}		 ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re\left(a_{k}\right)>0}} 
(1)/(4*Pi)*int((product(GAMMA(a[k]+ I*t)*GAMMA(a[k]- I*t), k = 1..4))/(GAMMA(2*I*t)*GAMMA(- 2*I*t)), t = - infinity..infinity) = (product(product(GAMMA(a[j]+ a[k]), k = j + 1..4), j = 1..k - 1))/(GAMMA(a[1]+ a[2]+ a[3]+ a[4]))

Divide[1,4*Pi]*Integrate[Divide[Product[Gamma[Subscript[a, k]+ I*t]*Gamma[Subscript[a, k]- I*t], {k, 1, 4}, GenerateConditions->None],Gamma[2*I*t]*Gamma[- 2*I*t]], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Product[Product[Gamma[Subscript[a, j]+ Subscript[a, k]], {k, j + 1, 4}, GenerateConditions->None], {j, 1, k - 1}, GenerateConditions->None],Gamma[Subscript[a, 1]+ Subscript[a, 2]+ Subscript[a, 3]+ Subscript[a, 4]]]
Error Aborted - Skipped - Because timed out
Retrieved from "https://lct.wmflabs.org/w/index.php?title=5.13&oldid=475"