*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:

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:

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 the 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.

*Practice Problems*:

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.

_{}^{}
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