Dirichlet Triple Integral

# I
∭

x

α
−
1

y

β
−
1

z

γ
−
1

d
x
d
y
d
z

{\displaystyle I = \iiint { { x }^{ \alpha -1 }{ y }^{ \beta -1 }{ z }^{ \gamma -1 }dxdydz } }

 over the curve

(

x
a

)

p

(

y
b

)

q

(

z
c

)

r

=
1

{\displaystyle {\left(\frac{x}{a}\right)}^{p} +{\left(\frac{y}{b}\right)}^{q} + {\left(\frac{z}{c}\right)}^{r} = 1 }

.

# u

(

x
a

)

p

⇒
# x
a

u

1
p

{\displaystyle u = {\left(\frac{x}{a}\right)}^{p} \Rightarrow x = a{u}^{\frac{1}{p}} }

# v

(

y
b

)

q

⇒
# y
b

v

1
q

{\displaystyle v = {\left(\frac{y}{b}\right)}^{q} \Rightarrow y = b{v}^{\frac{1}{q}} }

# w

(

z
c

)

r

⇒
# z
c

w

1
r

{\displaystyle w = {\left(\frac{z}{c}\right)}^{r} \Rightarrow z = c{w}^{\frac{1}{r}} }

and

d
# x

a
p

u

1
p

−
1

d
u

{\displaystyle dx = \frac{a}{p}{u}^{\frac{1}{p}-1}du }

d
# y

b
q

v

1
q

−
1

d
v

{\displaystyle dy = \frac{b}{q}{v}^{\frac{1}{q}-1}dv }

d
# z

c
r

w

1
r

−
1

d
w

{\displaystyle dz = \frac{c}{r}{w}^{\frac{1}{r}-1}dw }

After these substitutions, the integral becomes much lighter:

# I

a

α

b

β

c

γ

p
q
r

∭

u

α
p

−
1

v

β
q

−
1

w

γ
r

−
1

d
w
d
v
d
u

{\displaystyle 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

{\displaystyle u + v + w = 1 }

.

# I

a

α

b

β

c

γ

p
q
r

∫

0

1

∫

0

1
−
u

∫

0

1
−
u
−
v

u

α
p

−
1

v

β
q

−
1

w

γ
r

−
1

d
w
d
v
d
u

{\displaystyle 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

a

α

b

β

c

γ

p
q
r

r
γ

∫

0

1

∫

0

1
−
u

u

α
p

−
1

v

β
q

−
1

(
1
−
u
−
v
)

γ
r

d
v
d
u

{\displaystyle 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

{\displaystyle v = (1-u)t }

 and

d
# v
(
1
−
u
)
d
t

{\displaystyle dv = (1 - u)dt }

:

# I

a

α

b

β

c

γ

p
q
r

r
γ

∫

0

1

u

α
p

−
1

(
1
−
u
)

(

β
q

γ
r

)

∫

0

1

t

β
q

−
1

(
1
−
t
)

γ
r

d
t

{\displaystyle 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

a

α

b

β

c

γ

p
q
r

r
γ

Γ

(

α
p

)

Γ

(

β
q

γ
r

1

)

Γ

(

α
p

β
q

γ
r

1

)

Γ

(

β
q

)

Γ

(

γ
r

1

)

Γ

(

β
q

γ
r

1

)

{\displaystyle 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

Γ
(
n
1
# )
n
Γ
(
n
)

{\displaystyle \Gamma(n+1)=n\Gamma(n)}

, we arrive at the result:

# I

a

α

b

β

c

γ

p
q
r

Γ

(

α
p

)

Γ

(

β
q

)

Γ

(

γ
r

)

Γ

(

α
p

β
q

γ
r

1

)

{\displaystyle 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) } }