My Math Forum Multivalued functions and branch cuts

 Complex Analysis Complex Analysis Math Forum

 October 22nd, 2010, 11:49 PM #1 Senior Member   Joined: Sep 2009 Posts: 251 Thanks: 0 Multivalued functions and branch cuts Two questions: 1) A branch point (BP) of a function is a point which has different values when you make a complete circuit around it. You solve this issue by creating a branch cut (BC) between the BPs. If you make a circuit which does not enclose any BP, then the function returns to its original value. What happens if you make a circuit which does not enclose any BP, but does cross the BC? Are you even allowed to do that? 2) I don't quite get why a some functions are considered different values even though the value only differs by 2*Pi, and some functions are not considered different. For example, w(z) = log(z)+i*(t+2*n*Pi), n is integer Is considered multivalued. If z=x is real, positive, then make a circuit about z=0; initially, t=0, w(z) = w(x*e^{I*0}) = log(x). After the circuit, t=2*PI, w(z)=w(x*e^{i*2*Pi})=log(x)+2*pi*i But w(z)=z is not considered multivalued. If z=x (x is real, positive), then t=0, w(z) = w(x*e^{i*0}) = x. After the circuit, t=2*Pi, z=x*e^{i*2*Pi}. Yet w(z)=z is not considered multivalued when w(z)=log(z) is. What's the difference? Thanks for your help. -J

 Tags branch, cuts, functions, multivalued

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post aaron-math Complex Analysis 1 September 24th, 2013 11:31 AM Few_But_Ripe Complex Analysis 2 December 5th, 2011 09:00 AM Dundero Math Books 2 February 14th, 2011 08:49 PM aynjell Complex Analysis 0 October 25th, 2010 02:25 PM STV Complex Analysis 1 August 13th, 2008 08:17 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top