## Wednesday, May 3, 2017

### 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.