Math Revisited: Harmonic Conjugates And Cauchy-Riemann Equations

Harmonic conjugates and the Cauchy-Riemann equations are among the most important topics in advanced calculus. 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:

f(z) = u(x,y) + iv(x,y) =  (x² – y²) + i2xy

 

So that: u(x,y) =  (x² – y²)     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(x² – y²)

 

u(x,y) = 2xy and v(x,y) =  (x² – y²)

 

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 ²u = 0

 

Then:

¶ ² u/ ¶ x²  +  ¶ ² u/ ¶ y²       = 0 + 0 = 0

 

And: 

 

¶ ² v/ ¶ x²  +  ¶ ² v/ ¶ y²    =  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) =  (x² – y²) +  2x

 

a)     Show u(x,y) is a harmonic function

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