Check if the complex function is differentiableby Fabio010 Tags: check, complex, differentiable, function 

#1
Jan2614, 09:41 AM

P: 83

The question is to check where the following complex function is differentiable.
[tex]w=z \left z\right[/tex] [tex]w=\sqrt{x^2+y^2} (x+i y)[/tex] [tex]u = x\sqrt{x^2+y^2}[/tex] [tex]v = y\sqrt{x^2+y^2}[/tex] Using the Cauchy Riemann equations [tex]\frac{\partial }{\partial x}u=\frac{\partial }{\partial y}v[/tex] [tex]\frac{\partial }{\partial y}u=\frac{\partial }{\partial x}v[/tex] my results: [tex]\frac{x^2}{\sqrt{x^2+y^2}}=\frac{y^2}{\sqrt{x^2+y^2}}[/tex] [tex]\frac{x y}{\sqrt{x^2+y^2}}=0[/tex] solutions says that it's differentiable at (0,0). But doesn't it blow at (0,0)? 



#2
Jan2614, 09:58 AM

P: 428

If you just plug in ##y=0## and ##x=0## you will get an indeterminate form which is meaningless. If you evaluate the limits, I think that you get all expressions equal to ##0##, but double check that.




#3
Jan2614, 07:34 PM

P: 170

Division by zero is not allowed in complex analysis, so your final equations are not defined at x=y=0. They are not equal.




#4
Jan2714, 10:00 AM

P: 428

Check if the complex function is differentiable[itex] \displaystyle\lim_{h\rightarrow0}\displaystyle\frac{(0+0i+h)\left(0+0i +h)\right}{h}=0 [/itex] [itex] \displaystyle\lim_{h\rightarrow0}\displaystyle\frac{(0+0i+ih)\left(0+0 i+ih)\right}{ih}=0 [/itex] So the function is differentiable at ##0##. I don't remember enough from my complex analysis course (which had a number of students who had not taken real analysis, so it was a bit less rigorous than some courses) to reconcile this. My recollection is that the limits of the CauchyRiemann equations could be evaluated, but a quick look online showed that my recollection was incorrect. Perhaps, since the partial derivatives are undefined at 0 the CauchyRiemann equations are not applicable? 



#5
Jan2714, 10:20 AM

Sci Advisor
HW Helper
Thanks
P: 25,175





#6
Jan2714, 06:39 PM

P: 428

Yes, but ##w(0)=0##, so I left it out.




#7
Jan2714, 06:59 PM

Sci Advisor
HW Helper
Thanks
P: 25,175




Register to reply 
Related Discussions  
how to check if function is differentiable at a point  Topology and Analysis  1  
Complex Analysis, Complex Differentiable Question  Calculus & Beyond Homework  1  
Find where a complex function is differentiable  Topology and Analysis  1  
Real/complex differentiable function  Calculus & Beyond Homework  4  
Determing where function is differentiable (Complex Analysis)  Calculus & Beyond Homework  8 