Gamma Function - 5.12 Beta Function

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DLMF Formula Constraints Maple Mathematica Symbolic
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5.12.E1 B ⁡ ( a , b ) = ∫ 0 1 t a - 1 ⁢ ( 1 - t ) b - 1 ⁢ d t Euler-Beta 𝑎 𝑏 superscript subscript 0 1 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 𝑡 {\displaystyle{\displaystyle\mathrm{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t% )^{b-1}\mathrm{d}t}}
\EulerBeta@{a}{b} = \int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t}		 

Beta(a, b) = int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)

Beta[a, b] == Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}, GenerateConditions->None]
Failure Successful Error
Failed [11 / 36]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -2]}


Result: Indeterminate
Test Values: {Rule[a, 1.5], Rule[b, -2]}

... skip entries to safe data
5.12.E1 ∫ 0 1 t a - 1 ⁢ ( 1 - t ) b - 1 ⁢ d t = Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) Γ ⁡ ( a + b ) superscript subscript 0 1 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 𝑡 Euler-Gamma 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{1}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{% \Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}}}
\int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}		 ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re b>0,\Re(a+b)>0}} 
int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1) = (GAMMA(a)*GAMMA(b))/(GAMMA(a + b))

Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}, GenerateConditions->None] == Divide[Gamma[a]*Gamma[b],Gamma[a + b]]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 9]
5.12.E2 ∫ 0 π / 2 sin 2 ⁢ a - 1 ⁡ θ ⁢ cos 2 ⁢ b - 1 ⁡ θ ⁢ d θ = 1 2 ⁢ B ⁡ ( a , b ) superscript subscript 0 𝜋 2 2 𝑎 1 𝜃 2 𝑏 1 𝜃 𝜃 1 2 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\pi/2}{\sin^{2a-1}}\theta{\cos^{2b-1}}% \theta\mathrm{d}\theta=\tfrac{1}{2}\mathrm{B}\left(a,b\right)}}
\int_{0}^{\pi/2}\sin^{2a-1}@@{\theta}\cos^{2b-1}@@{\theta}\diff{\theta} = \tfrac{1}{2}\EulerBeta@{a}{b}		 ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re b>0,\Re(a+b)>0}} 
int((sin(theta))^(2*a - 1)* (cos(theta))^(2*b - 1), theta = 0..Pi/2) = (1)/(2)*Beta(a, b)

Integrate[(Sin[\[Theta]])^(2*a - 1)* (Cos[\[Theta]])^(2*b - 1), {\[Theta], 0, Pi/2}, GenerateConditions->None] == Divide[1,2]*Beta[a, b]
Failure Successful Error Successful [Tested: 9]
5.12.E3 ∫ 0 ∞ t a - 1 ⁢ d t ( 1 + t ) a + b = B ⁡ ( a , b ) superscript subscript 0 superscript 𝑡 𝑎 1 𝑡 superscript 1 𝑡 𝑎 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{a-1}\mathrm{d}t}{(1+t)^{% a+b}}=\mathrm{B}\left(a,b\right)}}
\int_{0}^{\infty}\frac{t^{a-1}\diff{t}}{(1+t)^{a+b}} = \EulerBeta@{a}{b}		 ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re b>0,\Re(a+b)>0}} 
int(((t)^(a - 1))/((1 + t)^(a + b)), t = 0..infinity) = Beta(a, b)

Integrate[Divide[(t)^(a - 1),(1 + t)^(a + b)], {t, 0, Infinity}, GenerateConditions->None] == Beta[a, b]
Failure Successful Error Successful [Tested: 9]
5.12.E4 ∫ 0 1 t a - 1 ⁢ ( 1 - t ) b - 1 ( t + z ) a + b ⁢ d t = B ⁡ ( a , b ) ⁢ ( 1 + z ) - a ⁢ z - b superscript subscript 0 1 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 superscript 𝑡 𝑧 𝑎 𝑏 𝑡 Euler-Beta 𝑎 𝑏 superscript 1 𝑧 𝑎 superscript 𝑧 𝑏 {\displaystyle{\displaystyle\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}% \mathrm{d}t=\mathrm{B}\left(a,b\right)(1+z)^{-a}z^{-b}}}
\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\diff{t} = \EulerBeta@{a}{b}(1+z)^{-a}z^{-b}		 | ph ⁡ z | < π , ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,\Re a>0,\Re b>0,\Re(a+b)>% 0}} 
int(((t)^(a - 1)*(1 - t)^(b - 1))/((t + z)^(a + b)), t = 0..1) = Beta(a, b)*(1 + z)^(- a)* (z)^(- b)

Integrate[Divide[(t)^(a - 1)*(1 - t)^(b - 1),(t + z)^(a + b)], {t, 0, 1}, GenerateConditions->None] == Beta[a, b]*(1 + z)^(- a)* (z)^(- b)
Failure Failure Skipped - Because timed out
Failed [77 / 252]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


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

... skip entries to safe data
5.12.E5 ∫ 0 π / 2 ( cos ⁡ t ) a - 1 ⁢ cos ⁡ ( b ⁢ t ) ⁢ d t = π 2 a ⁢ 1 a ⁢ B ⁡ ( 1 2 ⁢ ( a + b + 1 ) , 1 2 ⁢ ( a - b + 1 ) ) superscript subscript 0 𝜋 2 superscript 𝑡 𝑎 1 𝑏 𝑡 𝑡 𝜋 superscript 2 𝑎 1 𝑎 Euler-Beta 1 2 𝑎 𝑏 1 1 2 𝑎 𝑏 1 {\displaystyle{\displaystyle\int_{0}^{\pi/2}(\cos t)^{a-1}\cos\left(bt\right)% \mathrm{d}t=\frac{\pi}{2^{a}}\frac{1}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),% \frac{1}{2}(a-b+1)\right)}}}
\int_{0}^{\pi/2}(\cos@@{t})^{a-1}\cos@{bt}\diff{t} = \frac{\pi}{2^{a}}\frac{1}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}		 ℜ ⁡ a > 0 , ℜ ⁡ ( 1 2 ⁢ ( a + b + 1 ) ) > 0 , ℜ ⁡ ( 1 2 ⁢ ( a - b + 1 ) ) > 0 , ℜ ⁡ ( ( 1 2 ⁢ ( a + b + 1 ) ) + b ) > 0 , ℜ ⁡ ( a + ( 1 2 ⁢ ( a - b + 1 ) ) ) > 0 formulae-sequence 𝑎 0 formulae-sequence 1 2 𝑎 𝑏 1 0 formulae-sequence 1 2 𝑎 𝑏 1 0 formulae-sequence 1 2 𝑎 𝑏 1 𝑏 0 𝑎 1 2 𝑎 𝑏 1 0 {\displaystyle{\displaystyle\Re a>0,\Re(\frac{1}{2}(a+b+1))>0,\Re(\frac{1}{2}(% a-b+1))>0,\Re((\frac{1}{2}(a+b+1))+b)>0,\Re(a+(\frac{1}{2}(a-b+1)))>0}} 
int((cos(t))^(a - 1)* cos(b*t), t = 0..Pi/2) = (Pi)/((2)^(a))*(1)/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))

Integrate[(Cos[t])^(a - 1)* Cos[b*t], {t, 0, Pi/2}, GenerateConditions->None] == Divide[Pi,(2)^(a)]*Divide[1,a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]
Failure Aborted Successful [Tested: 18] Skipped - Because timed out
5.12.E6 ∫ 0 π ( sin ⁡ t ) a - 1 ⁢ e i ⁢ b ⁢ t ⁢ d t = π 2 a - 1 ⁢ e i ⁢ π ⁢ b / 2 a ⁢ B ⁡ ( 1 2 ⁢ ( a + b + 1 ) , 1 2 ⁢ ( a - b + 1 ) ) superscript subscript 0 𝜋 superscript 𝑡 𝑎 1 superscript 𝑒 𝑖 𝑏 𝑡 𝑡 𝜋 superscript 2 𝑎 1 superscript 𝑒 𝑖 𝜋 𝑏 2 𝑎 Euler-Beta 1 2 𝑎 𝑏 1 1 2 𝑎 𝑏 1 {\displaystyle{\displaystyle\int_{0}^{\pi}(\sin t)^{a-1}e^{ibt}\mathrm{d}t=% \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),% \frac{1}{2}(a-b+1)\right)}}}
\int_{0}^{\pi}(\sin@@{t})^{a-1}e^{ibt}\diff{t} = \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}		 ℜ ⁡ a > 0 , ℜ ⁡ ( 1 2 ⁢ ( a + b + 1 ) ) > 0 , ℜ ⁡ ( 1 2 ⁢ ( a - b + 1 ) ) > 0 , ℜ ⁡ ( ( 1 2 ⁢ ( a + b + 1 ) ) + b ) > 0 , ℜ ⁡ ( a + ( 1 2 ⁢ ( a - b + 1 ) ) ) > 0 formulae-sequence 𝑎 0 formulae-sequence 1 2 𝑎 𝑏 1 0 formulae-sequence 1 2 𝑎 𝑏 1 0 formulae-sequence 1 2 𝑎 𝑏 1 𝑏 0 𝑎 1 2 𝑎 𝑏 1 0 {\displaystyle{\displaystyle\Re a>0,\Re(\frac{1}{2}(a+b+1))>0,\Re(\frac{1}{2}(% a-b+1))>0,\Re((\frac{1}{2}(a+b+1))+b)>0,\Re(a+(\frac{1}{2}(a-b+1)))>0}} 
int((sin(t))^(a - 1)* exp(I*b*t), t = 0..Pi) = (Pi)/((2)^(a - 1))*(exp(I*Pi*b/2))/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))

Integrate[(Sin[t])^(a - 1)* Exp[I*b*t], {t, 0, Pi}, GenerateConditions->None] == Divide[Pi,(2)^(a - 1)]*Divide[Exp[I*Pi*b/2],a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]
Failure Aborted Successful [Tested: 18]
Failed [9 / 18]
Result: DirectedInfinity[]
Test Values: {Rule[a, 1.5], Rule[b, -1.5]}


Result: DirectedInfinity[]
Test Values: {Rule[a, 1.5], Rule[b, 0.5]}

... skip entries to safe data
5.12.E7 ∫ 0 ∞ cosh ⁡ ( 2 ⁢ b ⁢ t ) ( cosh ⁡ t ) 2 ⁢ a ⁢ d t = 4 a - 1 ⁢ B ⁡ ( a + b , a - b ) superscript subscript 0 2 𝑏 𝑡 superscript 𝑡 2 𝑎 𝑡 superscript 4 𝑎 1 Euler-Beta 𝑎 𝑏 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\cosh\left(2bt\right)}{(% \cosh t)^{2a}}\mathrm{d}t=4^{a-1}\mathrm{B}\left(a+b,a-b\right)}}
\int_{0}^{\infty}\frac{\cosh@{2bt}}{(\cosh@@{t})^{2a}}\diff{t} = 4^{a-1}\EulerBeta@{a+b}{a-b}		 ℜ ⁡ a > | ℜ ⁡ b | , ℜ ⁡ ( a + b ) > 0 , ℜ ⁡ ( a - b ) > 0 , ℜ ⁡ ( ( a + b ) + b ) > 0 , ℜ ⁡ ( a + ( a - b ) ) > 0 formulae-sequence 𝑎 𝑏 formulae-sequence 𝑎 𝑏 0 formulae-sequence 𝑎 𝑏 0 formulae-sequence 𝑎 𝑏 𝑏 0 𝑎 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re a>|\Re b|,\Re(a+b)>0,\Re(a-b)>0,\Re((a+b)+b)>0% ,\Re(a+(a-b))>0}} 
int((cosh(2*b*t))/((cosh(t))^(2*a)), t = 0..infinity) = (4)^(a - 1)* Beta(a + b, a - b)

Integrate[Divide[Cosh[2*b*t],(Cosh[t])^(2*a)], {t, 0, Infinity}, GenerateConditions->None] == (4)^(a - 1)* Beta[a + b, a - b]
Failure Failure Successful [Tested: 6] Successful [Tested: 6]
5.12.E8 1 2 ⁢ π ⁢ ∫ - ∞ ∞ d t ( w + i ⁢ t ) a ⁢ ( z - i ⁢ t ) b = ( w + z ) 1 - a - b ( a + b - 1 ) ⁢ B ⁡ ( a , b ) 1 2 𝜋 superscript subscript 𝑡 superscript 𝑤 𝑖 𝑡 𝑎 superscript 𝑧 𝑖 𝑡 𝑏 superscript 𝑤 𝑧 1 𝑎 𝑏 𝑎 𝑏 1 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\mathrm% {d}t}{(w+it)^{a}(z-it)^{b}}=\frac{(w+z)^{1-a-b}}{(a+b-1)\mathrm{B}\left(a,b% \right)}}}
\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\diff{t}}{(w+it)^{a}(z-it)^{b}} = \frac{(w+z)^{1-a-b}}{(a+b-1)\EulerBeta@{a}{b}}		 ℜ ⁡ ( a + b ) > 1 , ℜ ⁡ w > 0 , ℜ ⁡ z > 0 , ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 𝑏 1 formulae-sequence 𝑤 0 formulae-sequence 𝑧 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re\left(a+b\right)>1,\Re w>0,\Re z>0,\Re a>0,\Re b% >0,\Re(a+b)>0}} 
(1)/(2*Pi)*int((1)/((w + I*t)^(a)*(z - I*t)^(b)), t = - infinity..infinity) = ((w + z)^(1 - a - b))/((a + b - 1)*Beta(a, b))

Divide[1,2*Pi]*Integrate[Divide[1,(w + I*t)^(a)*(z - I*t)^(b)], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(w + z)^(1 - a - b),(a + b - 1)*Beta[a, b]]
Skipped - Unable to analyze test case: Null Failure - Successful [Tested: 250]
5.12.E9 1 2 ⁢ π ⁢ i ⁢ ∫ c - ∞ ⁢ i c + ∞ ⁢ i t - a ⁢ ( 1 - t ) - 1 - b ⁢ d t = 1 b ⁢ B ⁡ ( a , b ) 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝑡 𝑎 superscript 1 𝑡 1 𝑏 𝑡 1 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{-% a}(1-t)^{-1-b}\mathrm{d}t=\frac{1}{b\mathrm{B}\left(a,b\right)}}}
\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{-a}(1-t)^{-1-b}\diff{t} = \frac{1}{b\EulerBeta@{a}{b}}		 0 < c , c < 1 , ℜ ⁡ ( a + b ) > 0 , ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 0 𝑐 formulae-sequence 𝑐 1 formulae-sequence 𝑎 𝑏 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle 0<c,c<1,\Re\left(a+b\right)>0,\Re a>0,\Re b>0,\Re% (a+b)>0}} 
(1)/(2*Pi*I)*int((t)^(- a)*(1 - t)^(- 1 - b), t = c - infinity*I..c + infinity*I) = (1)/(b*Beta(a, b))

Divide[1,2*Pi*I]*Integrate[(t)^(- a)*(1 - t)^(- 1 - b), {t, c - Infinity*I, c + Infinity*I}, GenerateConditions->None] == Divide[1,b*Beta[a, b]]
Skipped - Unable to analyze test case: Null Aborted - Skipped - Because timed out
5.12.E10 1 2 ⁢ π ⁢ i ⁢ ∫ 0 ( 1 + ) t a - 1 ⁢ ( t - 1 ) b - 1 ⁢ d t = sin ⁡ ( π ⁢ b ) π ⁢ B ⁡ ( a , b ) 1 2 𝜋 𝑖 superscript subscript 0 limit-from 1 superscript 𝑡 𝑎 1 superscript 𝑡 1 𝑏 1 𝑡 𝜋 𝑏 𝜋 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}% \mathrm{d}t=\frac{\sin\left(\pi b\right)}{\pi}\mathrm{B}\left(a,b\right)}}
\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}\diff{t} = \frac{\sin@{\pi b}}{\pi}\EulerBeta@{a}{b}		 ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re b>0,\Re(a+b)>0}} 
(1)/(2*Pi*I)*int((t)^(a - 1)*(t - 1)^(b - 1), t = 0..(1 +)) = (sin(Pi*b))/(Pi)*Beta(a, b)

Divide[1,2*Pi*I]*Integrate[(t)^(a - 1)*(t - 1)^(b - 1), {t, 0, (1 +)}, GenerateConditions->None] == Divide[Sin[Pi*b],Pi]*Beta[a, b]
Skipped - Unable to analyze test case: Null Failure - Error
5.12.E11 1 e 2 ⁢ π ⁢ i ⁢ a - 1 ⁢ ∫ ∞ ( 0 + ) t a - 1 ⁢ ( 1 + t ) - a - b ⁢ d t = B ⁡ ( a , b ) 1 superscript 𝑒 2 𝜋 𝑖 𝑎 1 superscript subscript limit-from 0 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑎 𝑏 𝑡 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}% (1+t)^{-a-b}\mathrm{d}t=\mathrm{B}\left(a,b\right)}}
\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\diff{t} = \EulerBeta@{a}{b}		 ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re b>0,\Re(a+b)>0}} 
(1)/(exp(2*Pi*I*a)- 1)*int((t)^(a - 1)*(1 + t)^(- a - b), t = infinity..(0 +)) = Beta(a, b)

Divide[1,Exp[2*Pi*I*a]- 1]*Integrate[(t)^(a - 1)*(1 + t)^(- a - b), {t, Infinity, (0 +)}, GenerateConditions->None] == Beta[a, b]
Error Failure - Error
5.12.E12 ∫ P ( 1 + , 0 + , 1 - , 0 - ) t a - 1 ⁢ ( 1 - t ) b - 1 ⁢ d t = - 4 ⁢ e π ⁢ i ⁢ ( a + b ) ⁢ sin ⁡ ( π ⁢ a ) ⁢ sin ⁡ ( π ⁢ b ) ⁢ B ⁡ ( a , b ) superscript subscript 𝑃 limit-from 1 limit-from 0 limit-from 1 limit-from 0 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 𝑡 4 superscript 𝑒 𝜋 𝑖 𝑎 𝑏 𝜋 𝑎 𝜋 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\mathrm{% d}t=-4e^{\pi i(a+b)}\sin\left(\pi a\right)\sin\left(\pi b\right)\mathrm{B}% \left(a,b\right)}}
\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\diff{t} = -4e^{\pi i(a+b)}\sin@{\pi a}\sin@{\pi b}\EulerBeta@{a}{b}		 ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( a + b ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re a>0,\Re b>0,\Re(a+b)>0}} 
int((t)^(a - 1)*(1 - t)^(b - 1), t = P..(1 + , 0 + , 1 - , 0 -)) = - 4*exp(Pi*I*(a + b))*sin(Pi*a)*sin(Pi*b)*Beta(a, b)

Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, P, (1 + , 0 + , 1 - , 0 -)}, GenerateConditions->None] == - 4*Exp[Pi*I*(a + b)]*Sin[Pi*a]*Sin[Pi*b]*Beta[a, b]
Error Failure - Error
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