## Tuesday, October 29, 2013

### MATH SOLUTIONS- Harmonic Conjugates

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

We have to verify the Cauchy- Riemann equations. We note that:

u(x,y) =  cos x cosh y  and v(x,y) = - sinx sinh y

Then:

u/ x  = - sinx cosh y    =   v/ y

And:

v/ x  = - cos x sinh y =  - u/ y

So the Cauchy_Riemann equations are satisfied

b2)     Let u(x,y) =  (x2 – y2) +  2x

Show u(x,y) is a harmonic function

Solution: If it’s harmonic then we must have:

2 u/ x2  +  2 u/ y2     =  0

Take the 1st, 2nd partials:

u/ x  = 2x + 2   and  u/ y = -2y

2 u/ x2   =  2  and  2 u/ y2     =   =  -2

Therefore:

2 u/ x2  +  2 u/ y2     =   (2)  + (-2)  = 0

SO that u(x,y) is a harmonic function.

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

u/ x  = 2x + 2   and  u/ y = -2y

By the Cauchy –Riemann equations:

v/ x  =  -  u/ y = 2y

And:

u/ x    =  v/ y  = 2x + 2

Take the differential using the chain rule:

dv  = ( v/ x  ) dx  + ( v/ y ) dy

Substitute from Cauchy-Riemann equations:

dv = 2y (dx) + (2x + 2) dy

Integrating:

v = 2 xy  + 2xy + 2y

Or:

v = 4xy + 2y =   2y (2x + 1) + C