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Dirichlet Triple Integral

Evaluate the triple integral \[I = \iiint { { x }^{ \alpha -1 }{ y }^{ \beta -1 }{ z }^{ \gamma -1 }dxdydz } \] over the curve \({\left(\frac{x}{a}\right)}^{p} +{\left(\frac{y}{b}\right)}^{q} + {\left(\frac{z}{c}\right)}^{r} = 1\).

Solution

Let \[u = {\left(\frac{x}{a}\right)}^{p} \Rightarrow x = a{u}^{\frac{1}{p}}\] \[v = {\left(\frac{y}{b}\right)}^{q} \Rightarrow y = b{v}^{\frac{1}{q}}\] \[w = {\left(\frac{z}{c}\right)}^{r} \Rightarrow z = c{w}^{\frac{1}{r}}\]

and

\[dx = \frac{a}{p}{u}^{\frac{1}{p}-1}du\] \[dy = \frac{b}{q}{v}^{\frac{1}{q}-1}dv\] \[dz = \frac{c}{r}{w}^{\frac{1}{r}-1}dw\]

After these substitutions, the integral becomes much lighter: \[I = \frac{{a}^{\alpha}{b}^{\beta}{c}^{\gamma}}{pqr} \iiint { { u }^{ \frac{\alpha}{p} -1 }{ v }^{ \frac{\beta}{q} -1 }{ w }^{ \frac{\gamma}{r} -1 }dwdvdu } \] evaluated over the curve \(u + v + w = 1\).

\[I=\frac { { a }^{ \alpha }{ b }^{ \beta }{ c }^{ \gamma } }{ pqr } \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1-u }{ \int _{ 0 }^{ 1-u-v }{ { u }^{ \frac { \alpha }{ p } -1 }{ v }^{ \frac { \beta }{ q } -1 }{ w }^{ \frac { \gamma }{ r } -1 }dwdvdu } } } \]

\[I=\frac { { a }^{ \alpha }{ b }^{ \beta }{ c }^{ \gamma } }{ pqr } \frac{r}{\gamma} \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1-u }{ { u }^{ \frac { \alpha }{ p } -1 }{ v }^{ \frac { \beta }{ q } -1 }{ (1-u-v) }^{ \frac { \gamma }{ r } }dvdu } } \]

Substitute \(v = (1-u)t\) and \(dv = (1 - u)dt\).

\[I=\frac { { a }^{ \alpha }{ b }^{ \beta }{ c }^{ \gamma } }{ pqr } \frac { r }{ \gamma } \int _{ 0 }^{ 1 }{ { u }^{ \frac { \alpha }{ p } -1 }{ (1-u) }^{ \left( \frac { \beta }{ q } +\frac { \gamma }{ r } \right) }\int _{ 0 }^{ 1 }{ { t }^{ \frac { \beta }{ q } -1 }{ (1-t) }^{ \frac { \gamma }{ r } }dt } } .\]

Here we apply two beta functions

\[I=\frac { { a }^{ \alpha }{ b }^{ \beta }{ c }^{ \gamma } }{ pqr } \frac { r }{ \gamma } \frac { \Gamma \left( \frac { \alpha }{ p } \right) \Gamma \left( \frac { \beta }{ q } +\frac { \gamma }{ r } +1 \right) }{ \Gamma \left( \frac { \alpha }{ p } +\frac { \beta }{ q } +\frac { \gamma }{ r } +1 \right) } \frac { \Gamma \left( \frac { \beta }{ q } \right) \Gamma \left( \frac { \gamma }{ r } +1 \right) }{ \Gamma \left( \frac { \beta }{ q } +\frac { \gamma }{ r } +1 \right) } .\]

Applying the gamma function property \(\Gamma(n+1) = n\Gamma(n)\), we arrive at the result: \[I=\frac { { a }^{ \alpha }{ b }^{ \beta }{ c }^{ \gamma } }{ pqr } \frac { \Gamma \left( \frac { \alpha }{ p } \right) \Gamma \left( \frac { \beta }{ q } \right) \Gamma \left( \frac { \gamma }{ r } \right) }{ \Gamma \left( \frac { \alpha }{ p } +\frac { \beta }{ q } +\frac { \gamma }{ r } +1 \right) } . \]

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
2 years, 5 months ago

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