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January 18th, 2010, 02:23 PM  #1 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  different branches different derivatives
In complex analysis, certain types of functions are multiplevalued, such as the logarithmic function. To find the derivative of any multivalued function, you need to look at the function on a particular branch. Entire functions, like the exponential function for example, have the same derivative regardless the branch you consider. Is there a complexvalued function that has different derivatives depending on the branch you choose? Can someone give an example of such a function with its derivatives on different branches? Thanks. 
January 19th, 2010, 01:46 PM  #2  
Global Moderator Joined: May 2007 Posts: 6,834 Thanks: 733  Re: different branches different derivatives Quote:
Example: ?z, with derivative .5/?z. On the two branches the function has opposite sign, and so has the derivative.  
January 20th, 2010, 12:35 AM  #3 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  Re: different branches different derivatives
Thanks for your clarification. When I was posting this question, I was really puzzled by the connection among branches, multivalued functions and derivatives. May I ask you one more question to further clarify the idea of having different derivatives on different branches of multivalued functions? I have a multivalued function in my mind right now which is the natural logarithmic function, f(z)=In(z) for example. On the principle branch (pi, pi], its derivative is 1/z. How would you find its derivatives on other branches, such as (0, 2pi], (pi/4, 3*pi/4]? I mean could you please tell me a way to find the derivatives on other branches on this logarithmic function? Thanks. 
January 20th, 2010, 01:16 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,834 Thanks: 733  Re: different branches different derivatives
ln(z) is special in that on the different branches, it differs by a constant value (2?ni), where n is an integer. As a result the derivative (1/z) is the same on all branches.

January 20th, 2010, 02:36 PM  #5 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  Re: different branches different derivatives
Thank you so much for your help, mathman. I know you are very busy with other posters and I should not keep bothering you with the same topic. Could you please answer one more question? I promise that this is really the last one. My next multivalued function is f(z)=z^(1/3). In polar form, f(z)=r^(1/3)*e^[(theta+2*k*pi)i/3] where k=0,1,2 So, its derivatives on the three branches where k=0,1,2 are z^(2/3)/3, z^(2/3)/3 * e^(pi/3)i, and z^(2/3)/3 * e^(2*pi/3)i respectively. Can you tell me whether the derivatives are correct or not? I really appreciate your help, mathman. 
January 20th, 2010, 05:53 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 21,016 Thanks: 2251 
You shouldn't have gone from 2*k*pi to k*pi.

January 20th, 2010, 06:52 PM  #7  
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  Re: Quote:
Could you please explain it?  
January 21st, 2010, 01:40 PM  #8  
Global Moderator Joined: May 2007 Posts: 6,834 Thanks: 733  Re: different branches different derivatives Quote:
 

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