u=(x-1)^3-3xy^2+3y^2 functions are harmonic and find its analytic functions
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Step-by-step explanation:
In this case, the pair of functions are:
u
(
x
,
y
)
=
x
3
−
3
x
y
2
v
(
x
,
y
)
=
3
x
2
y
−
y
3
By applying partial derivatives we get:
u
(
x
,
y
)
=
x
3
−
3
x
y
2
⇒
u
x
=
∂
∂
x
(
x
3
−
3
x
y
2
)
=
(
3
x
2
−
3
y
2
)
[
Applying Partial derivative with respect to
x
]
u
(
x
,
y
)
=
x
3
−
3
x
y
2
⇒
u
y
=
∂
∂
y
(
x
3
−
3
x
y
2
)
=
(
−
6
x
y
)
[
Applying Partial derivative with respect to
y
]
v
(
x
,
y
)
=
3
x
2
y
−
y
3
⇒
v
x
=
∂
∂
x
(
3
x
2
y
−
y
3
)
=
(
6
x
y
)
[
Applying Partial derivative with respect to
x
]
v
(
x
,
y
)
=
3
x
2
y
−
y
3
⇒
v
y
=
∂
∂
y
(
3
x
2
y
−
y
3
)
=
(
3
x
2
−
3
y
2
)
[
Applying Partial derivative with respect to
y
]
Which means obviously:
∂
u
∂
x
=
∂
v
∂
y
∂
u
∂
y
=
−
∂
v
∂
x
Hence
u
and
v
are Harmonic Conjugate to each other.
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