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)

 
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

 

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