Exercises for Section 3.2.  The Cauchy-Riemann Equations

Exercise 1.  Use the Cauchy Riemann conditions to determine where the following functions are differentiable, and evaluate the derivatives at those points where they exist.  

1 (a).  [Graphics:Images/CauchyRiemannModHome_gr_1.gif].
Solution 1 (a).


1 (b).  [Graphics:Images/CauchyRiemannModHome_gr_26.gif].
Solution 1 (b).


1 (c).  [Graphics:Images/CauchyRiemannModHome_gr_67.gif].
Solution 1 (c).


1 (d).  [Graphics:Images/CauchyRiemannModHome_gr_102.gif].
Solution 1 (d).


1 (e).  [Graphics:Images/CauchyRiemannModHome_gr_137.gif].  
Solution 1 (e).


1 (f).  [Graphics:Images/CauchyRiemannModHome_gr_166.gif].
Solution 1 (f).


1 (g).  [Graphics:Images/CauchyRiemannModHome_gr_197.gif].  
Solution 1 (g).


1 (h).  [Graphics:Images/CauchyRiemannModHome_gr_231.gif].
Solution 1 (h).


Exercise 2.  Let  [Graphics:Images/CauchyRiemannModHome_gr_260.gif]  be a differentiable function.

Verify the identity  [Graphics:Images/CauchyRiemannModHome_gr_261.gif][Graphics:Images/CauchyRiemannModHome_gr_262.gif].  
Solution 2.


Exercise 3.  Find the constants a and b such that  [Graphics:Images/CauchyRiemannModHome_gr_267.gif]  is differentiable for all  z.  
Solution 3.


Exercise 4.  Let  f(z)   be differentiable at the point  [Graphics:Images/CauchyRiemannModHome_gr_287.gif].  

Let z approach [Graphics:Images/CauchyRiemannModHome_gr_288.gif] along the ray  [Graphics:Images/CauchyRiemannModHome_gr_289.gif]  and use Equation (3-1) to show that Equation (3-14), [Graphics:Images/CauchyRiemannModHome_gr_290.gif]  holds.  
Solution 4.


Exercise 5.  Let  [Graphics:Images/CauchyRiemannModHome_gr_308.gif].  

Show that both  f(z)  and  f'(z)  are differentiable for all  z.  
Solution 5.


Exercise 6.  A vector field  [Graphics:Images/CauchyRiemannModHome_gr_334.gif]  is said to be irrotational if  [Graphics:Images/CauchyRiemannModHome_gr_335.gif].  

It is said to be solenoidal if  [Graphics:Images/CauchyRiemannModHome_gr_336.gif].  

If  f(z)  is an analytic function, show that  [Graphics:Images/CauchyRiemannModHome_gr_337.gif]  is both irrotational and solenoidal.
Solution 6.


Exercise 7.  Use any method to show that the following functions are nowhere differentiable.  

7 (a).    [Graphics:Images/CauchyRiemannModHome_gr_348.gif].   
Solution 7 (a).


7 (b).  [Graphics:Images/CauchyRiemannModHome_gr_386.gif].
Solution 7 (b).


Exercise 8.  Use Theorem 3.5 with regard to the following.  

8 (a).  Let  [Graphics:Images/CauchyRiemannModHome_gr_408.gif],  where  [Graphics:Images/CauchyRiemannModHome_gr_409.gif].

Show that  f(z)  is analytic in the domain indicated and that  [Graphics:Images/CauchyRiemannModHome_gr_410.gif].
Solution 8 (a).


8 (b).  Let  [Graphics:Images/CauchyRiemannModHome_gr_459.gif],  where  [Graphics:Images/CauchyRiemannModHome_gr_460.gif].

Show that  f(z)  is analytic for [Graphics:Images/CauchyRiemannModHome_gr_461.gif],  and find  [Graphics:Images/CauchyRiemannModHome_gr_462.gif].  
Solution 8 (b).


Exercise 9.  Show that the following functions are entire (see Definition 3.2).  

9 (a).  [Graphics:Images/CauchyRiemannModHome_gr_493.gif].
Solution 9 (a).


9 (b).  [Graphics:Images/CauchyRiemannModHome_gr_520.gif].  
Solution 9 (b).


Exercise 10.  To prove Theorem 3.5, the polar form of the Cauchy-Riemann equations,  

10 (a).  Let  [Graphics:Images/CauchyRiemannModHome_gr_547.gif][Graphics:Images/CauchyRiemannModHome_gr_548.gif].  

Use the transformation  [Graphics:Images/CauchyRiemannModHome_gr_549.gif]  (i.e., [Graphics:Images/CauchyRiemannModHome_gr_550.gif]  and the chain rules
          [Graphics:Images/CauchyRiemannModHome_gr_551.gif],    [Graphics:Images/CauchyRiemannModHome_gr_552.gif],   and  

          [Graphics:Images/CauchyRiemannModHome_gr_553.gif],    [Graphics:Images/CauchyRiemannModHome_gr_554.gif],

to prove that

          [Graphics:Images/CauchyRiemannModHome_gr_555.gif],    [Graphics:Images/CauchyRiemannModHome_gr_556.gif],   and  

          [Graphics:Images/CauchyRiemannModHome_gr_557.gif],    [Graphics:Images/CauchyRiemannModHome_gr_558.gif].  
Solution 10 (a).


10 (b).  Use the original Cauchy-Riemann equations for u and v and the results of part 10 (a) to prove that

          [Graphics:Images/CauchyRiemannModHome_gr_563.gif]   and   [Graphics:Images/CauchyRiemannModHome_gr_564.gif],    

thus verifying Equation (3-22)  [Graphics:Images/CauchyRiemannModHome_gr_565.gif]   and   [Graphics:Images/CauchyRiemannModHome_gr_566.gif].  
Solution 10 (b).


10 (c).  Use part 10 (a) and Equations (3-14) and (3-15) to show that the right sides of Equations (3-23) and (3-24) simplify to  [Graphics:Images/CauchyRiemannModHome_gr_575.gif].  
Solution 10 (c).


Exercise 11.  Determine where the following functions are differentiable and where they are analytic.  Explain!  

11 (a).  [Graphics:Images/CauchyRiemannModHome_gr_591.gif].
Solution 11 (a).


11 (b).  [Graphics:Images/CauchyRiemannModHome_gr_614.gif].
Solution 11 (b).


11 (c).  [Graphics:Images/CauchyRiemannModHome_gr_639.gif].
Solution 11 (c).


Exercise 12.  Let  f(z)  and  g(z)  be analytic functions in the domain D.  

If  [Graphics:Images/CauchyRiemannModHome_gr_691.gif]  for all z in D,  then show that  [Graphics:Images/CauchyRiemannModHome_gr_692.gif],  where C is a complex constant.  
Solution 12.


Exercise 13.  Explain how the limit definition for the derivative in complex analysis and the limit definition for the derivative in calculus are different.  How are they similar?  
Solution 13.


Exercise 14.  Let  f(z)  be an analytic function in the domain D.  

Show that if   [Graphics:Images/CauchyRiemannModHome_gr_700.gif]   at all points in D,  then  f(z)  is constant in D.  
Solution 14.


Exercise 15.  Let  f(z)  be a nonconstant analytic function in the domain D.  

Show that the function  [Graphics:Images/CauchyRiemannModHome_gr_711.gif]  is not analytic in D.  
Solution 15.


Exercise 16.  The complex form of the Cauchy-Riemann equations.

Recall that, for  [Graphics:Images/CauchyRiemannModHome_gr_737.gif],  we have the substitutions   [Graphics:Images/CauchyRiemannModHome_gr_738.gif]  and  [Graphics:Images/CauchyRiemannModHome_gr_739.gif].  

16 (a).  Temporarily, think of  [Graphics:Images/CauchyRiemannModHome_gr_740.gif]  as dummy symbols for variables.  

With this perspective, x and y can be viewed as functions of z and [Graphics:Images/CauchyRiemannModHome_gr_741.gif].  

Use the chain rule for a function  [Graphics:Images/CauchyRiemannModHome_gr_742.gif]  of two variables to show that

Solution 16 (a).


16 (b).  Now define the differentiation operator  [Graphics:Images/CauchyRiemannModHome_gr_753.gif]  that is suggested by the equation in part (a).  

With this construct, show that if  [Graphics:Images/CauchyRiemannModHome_gr_754.gif]  is differentiable at  [Graphics:Images/CauchyRiemannModHome_gr_755.gif],  then,


Equating real and imaginary parts thus gives the complex form of the

                    Cauchy-Riemann equations:  [Graphics:Images/CauchyRiemannModHome_gr_759.gif].
Solution 16 (b).























(c) 2008 John H. Mathews, Russell W. Howell