Unformatted text preview: Log z 3 = ln | z 3 | +Arg z 3 is not continuous when the argument of z is equal to-π ,-π/ 3 or π/ 3 , or if z = 0 . At all other points it is continuous and moreover analytic since it is as a composition of two analytic functions. To be more precise recall that by the chain rule [ f ( g ( z ))] = f ( g ( z )) g ( z ) . In our case f ( w ) = Log w and g ( z ) = z 3 . Recall that derivarive of Log w exists when Arg w 6 =-π and w 6 = 0 . Hence the f ( z 3 ) exists if Arg z 3 6 =-π and z 6 = 0 . From the above we see that Arg z 3 =-π if Arg z =-π , π/ 3 or π/ 3 . So Log z 3 is analitic on the set of complex numbers that are different from zero and whose argument is not equal to-π ,-π/ 3 , or π/ 3 ....
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- Spring '09
- Derivative, arg, continuous partial derivatives