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# Step-by-Step Math Answer

Express
*
x
*
and
*
y
*
in terms of
*
r
*
and
*
θ
*
.

We express
*
x
*
and
*
y
*
in terms of
*
r
*
and
*
θ
*
:

*
x
*
=
*
r
*
(cos
*
θ
*
),

*
y
*
=
*
r
*
(sin
*
θ
*
).

Find ∂
*
x
*
/ ∂
*
r
*
, ∂
*
x
*
/ ∂
*
θ
*
, ∂
*
y
*
/ ∂
*
r
*
, and ∂
*
y
*
/ ∂
*
θ
*
.

We find ∂
*
x
*
/ ∂
*
r
*
, ∂
*
x
*
/ ∂
*
θ
*
, ∂
*
y
*
/ ∂
*
r
*
, and ∂
*
y
*
/ ∂
*
θ
*
:

Now find ∂
*
u
*
/ ∂
*
r
*
, ∂
*
u
*
/ ∂
*
θ
*
, ∂
*
v
*
/ ∂
*
r
*
, and ∂
*
v
*
/ ∂
*
θ
*
.

Now we find ∂
*
u
*
/ ∂
*
r
*
, ∂
*
u
*
/ ∂
*
θ
*
, ∂
*
v
*
/ ∂
*
r
*
, and ∂
*
v
*
/ ∂
*
θ
*
:

Similarly,

Assume that the Cartesian form of the Cauchy–Riemann equations holds, and show that the polar form holds.

Assuming that the Cartesian form of the Cauchy–Riemann equations holds, we show that the first equation in polar form holds:

Replace ∂
*
v
*
/ ∂
*
x
*
with –(∂
*
u
*
/ ∂
*
y
*
), and ∂
*
v
*
/ ∂
*
y
*
with ∂
*
u
*
/ ∂
*
x
*
.

Replacing ∂
*
v
*
/ ∂
*
x
*
with –(∂
*
u
*
/ ∂
*
y
*
), and ∂
*
v
*
/ ∂
*
y
*
with ∂
*
u
*
/ ∂
*
x
*
, we have

so the first one holds.

Now show that the second equation in polar form holds.

Now we show that the second equation in polar form holds:

Replace ∂
*
u
*
/ ∂
*
x
*
with ∂
*
v
*
/ ∂
*
y
*
, and ∂
*
u
*
/ ∂
*
y
*
with –(∂
*
v
*
/ ∂
*
x
*
).

Replacing ∂
*
u
*
/ ∂
*
x
*
with ∂
*
v
*
/ ∂
*
y
*
, and ∂
*
u
*
/ ∂
*
y
*
with –(∂
*
v
*
/ ∂
*
x
*
), we have

so the second one holds.