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)

 

We’ve already shown:

 

¶ 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