Example 3.10.  Show that, if f is is the principal square root function given by  

            [Graphics:Images/CauchyRiemannMod_gr_269.gif]   

where the domain is restricted to be  [Graphics:Images/CauchyRiemannMod_gr_270.gif],  then the derivative is given by  

            [Graphics:Images/CauchyRiemannMod_gr_271.gif]  

for every point in the domain  [Graphics:Images/CauchyRiemannMod_gr_272.gif].  

Explore Solution 3.10.

Enter  [Graphics:../Images/CauchyRiemannMod_gr_281.gif] and use them to form f[z].  Then verify that polar form of the Cauchy-Riemann equations hold.

[Graphics:../Images/CauchyRiemannMod_gr_282.gif]





[Graphics:../Images/CauchyRiemannMod_gr_283.gif]

[Graphics:../Images/CauchyRiemannMod_gr_284.gif]

 

 

  Second, find the derivative using polar formulae  [Graphics:../Images/CauchyRiemannMod_gr_285.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_286.gif].  

[Graphics:../Images/CauchyRiemannMod_gr_287.gif]





[Graphics:../Images/CauchyRiemannMod_gr_288.gif]

 

We have shown that if  [Graphics:../Images/CauchyRiemannMod_gr_289.gif]  where the domain is restricted to be  [Graphics:../Images/CauchyRiemannMod_gr_290.gif],  then the derivative is given by  [Graphics:../Images/CauchyRiemannMod_gr_291.gif]  where  [Graphics:../Images/CauchyRiemannMod_gr_292.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell