Math Solutions (Part II) : Cauchy- Riemann Equations

2(b) Given the function: u(x.y) = exp(-x) [x sin y – y cos y]

Find v(x,y) such that f(z) = u + iv is analytic

 

Solution:  

 

a) ¶ u/ ¶ x  =    exp(-x) (sin y) -  x exp(-x) sin y + y (exp(-x) ) cos y =  ¶ v/ ¶ y 

 

b) - ¶ u/ ¶ y  = - x exp (-x) cos y -  y (exp(-x) sin y +  

exp(-x)  cos y  =  ¶ v/ ¶ x 

 

Integrate (a) with respect to y, keeping x constant so that:

v  =  - exp(-x) cos y + x exp (-x) cos y  -  exp(-x)[ y sin y + cos y) + F(x)

 

v =  y exp(-x) sin y + x exp(-x) cos y + F(x)

 

Here, F(x) is an arbitrary real function of x.

 

Substituting the last result for v into the Cauchy equation for
 ¶ v/ ¶ x   we get:

 

y exp(-x) sin y – x exp(-x) cos y + exp (-x) cos y + F’(x)

 

=   - y exp (-x) sin y – x exp (-x) cos y – y exp(-x) sin y

 

Or: F’(x) = 0  and F(x) = c (constant)  Then, from the earlier expression for v:

 

v =   exp (-x) (y sin y + x cos y) + c

 

3) Let f(z) = exp(x) cos(y) + i(exp(x)sin(y) = u(x,y) + iv(x,y)


a) Determine if the function is analytic for both u and v.

b) Determine if the function is harmonic for both u and v

 

Solutions:

 

a)   ¶ u/ ¶ x  =     exp (x) cos y  

 

And: ¶ v/ ¶ y    =    exp (x) cos y  so:  ¶ u/ ¶ x  =    ¶ v/ ¶ y   

 

SO the function is analytic for u.

 

Now,  ¶ v/ ¶ x  =  exp(x) sin y  and :

 

- ¶ u/ ¶ y  =    -  exp (x) [- sin y] =  exp(x) sin y

 

SO:  ¶ v/ ¶ x  =   - ¶ u/ ¶ y 

 

Therefore, the function is also analytic for v.

 

b) ¶ ² u/ ¶ x²    =    ¶ / ¶ x  [ exp (x) cos y ] =  exp (x) cos y

 

¶ ² u/ ¶ y²    =      ¶ / ¶ y  [-  exp (x) sin y ]  =   - exp(x) cos y

 

Then: ¶ ² u/ ¶ x²      +  ¶ ² u/ ¶ y²    =  exp (x) cos y +

 (- exp(x) cos y) = 0

 

So the function is harmonic for u.

 

Looking now at v:

 

¶ ² v/ ¶ x²    =    ¶ / ¶ x  [exp(x) sin y ]  =  exp(x) sin y

 

¶ ² v/ ¶ y²    =   ¶ / ¶ y   [exp (x) cos y ]  =  exp (x) [- sin (y)]

= - exp (x) sin y

 

 Then: ¶ ² v/ ¶ x²      +  ¶ ² v/ ¶ y²   

=  exp(x) sin y +  (- exp (x) sin y) = 0

 

So the function is also harmonic for v.