A
useful theorem when looking at the Cauchy –Riemann equations is the following:

*A function f(z) = u(x,y) + iv(x,y) is analytic in a domain D if and only if v is a harmonic conjugate of u.*

Consider
the function we saw already in the Oct.
7 blog:

f(z)
= u(x,y) + iv(x,y) = (x

^{2 }– y^{2}) + i2xy^{2 }– y

^{2}) and: v(x,y) = 2x y

Now,
we first check to see if the eqns. are analytic

Take
¶ u/ ¶ x =
2x

And: ¶ v/ ¶ y
= 2x

Since: ¶ u/ ¶ x
= ¶ v/ ¶ y then u(x.y) is analytic

Now
check the other function, v(x.y)

¶ v/ ¶ x = 2y

And: - ¶ u/ ¶ y = -
(-2y) = 2y

So
that

**-****¶****u/****¶****y =****¶****v/****¶****x**and hence v(x,y) is analytic..
Note:
If f(z) is

*analytic everywhere*in the complex plane it is said to be*an entire function*.
Now, to see if it's a harmonic conjugate,
switch u(x,y) with

v(x,y) so that:

v(x,y) so that:

f(z)
= u(x,y) + iv(x,y) = 2xy + i(x

^{2 }– y^{2})
u(x,y)
= 2xy and v(x,y) = (x

^{2 }– y^{2})
We
first check to see if the u, v functions are analytic

Take
¶ u/ ¶ x =
2y

And: ¶ v/ ¶ y
= - 2y

Since: ¶ u/ ¶ x
= ¶ v/ ¶ y

Thus, it holds
only where y = 0, so then f(x) is differentiable only for points

*that lie on the x –axis*and we conclude the function reversed for conjugates is nowhere analytic. The conclusion is thus that while v is a harmonic conjugate of u throughout the z-plane,*v is not a harmonic conjugate of u*.
In
general, and based on this, one is led to conclude that given a function:

f(z)
= u(x,y) + iv(x,y)

then
f(z) is analytic in some domain D if and only if (-if(z) = v(x,y) –iu(x,y) is
also analytic there

Example
(2): Let
f(z) = 3x + y + i(3y – x)

Show
that v is a harmonic conjugate of u and hence the function is analytic in a
domain D when u and v are interchanged for f(z). Is the function also entire?

We
have u(x,y) = 3x + y and v(x,y) = (3y – x)

Check
Cauchy relations:

Take
¶ u/ ¶ x = 3

And: ¶ v/ ¶ y
= 3

Since: ¶ u/ ¶ x
= ¶ v/ ¶ y then u(x.y) is analytic

Now
check the other function, v(x.y):

¶ v/ ¶ x = -1

And: - ¶ u/ ¶ y = -
(1) = -1

So
that

**-****¶****u/****¶****y =****¶****v/****¶****x**and hence v(x,y) is analytic..
Now,
interchange u(x,y) with v(x,y):

f(z)
= 3x - y + i(3y +
x)

We
have u(x,y) = 3x - y and v(x,y) = (3y + x)

Check
Cauchy relations:

Take
¶ u/ ¶ x = ¶ v/ ¶ y

And: ¶ v/ ¶ y
= 3 = ¶ u/ ¶ x

Since: ¶ u/ ¶ x
= ¶ v/ ¶ y then u(x.y) is analytic

Now
check the other function, v(x.y):

¶ v/ ¶ x = +1

And: - ¶ u/ ¶ y = -
(-1) = +1

So
that

**-****¶****u/****¶****y =****¶****v/****¶****x**and hence v(x,y) is analytic..
Since
v is a harmonic conjugate of u then the function is analytic in a domain D when
u and v are interchanged.

If
the function is entire then it also satisfies the LaPlace
equation: Ñ

^{2}u = 0
Then:

¶

^{2}u/ ¶ x^{2}+ ¶^{2}u/ ¶ y^{2}= 0 + 0 = 0
And:

¶

^{2}v/ ¶ x^{2}+ ¶^{2}v/ ¶ y^{2}= 0 + 0 = 0
So
the function is also entire on the complex plane.

Problems
for the Math Maven:

1) Given
the function:

f(z)
= u(x,y) + iv(x,y) = cos x cosh y –
i(sinx sinh y)

a)
Verify
the Cauchy-Riemann equations are satisfied

b)
Are
they also satisfied for the harmonic conjugate, i.e. when u and v are
interchanged?

2)
Let
u(x,y) = (x

^{2 }– y^{2}) + 2x
a)
Show
u(x,y) is a harmonic function

b)
Hence
or otherwise, find the harmonic conjugate v(x,y) of u.

## 1 comment:

nice

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