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Unformatted text preview: Extra Problems Sheet 4 Stephen Taylor May 6, 2005 1. Let f ( z ) = x 2 + y 2 + i 2 xy (a) Using the definition of differentiability, determine where f is differentiable. We note that the definition of differentiability states Definition 1. If G is an open set in C and f : G C , then f is differentiable at a point z in G if lim h f ( z + h )- f ( z ) h exists. If f is differentiable at every point of G we say that f is differentiable on G . We let z C and consider the above limit for our specific function in terms of z f ( z ) = 1 2 ( z 2 + 2 z z- z 2 ) the limit becomes 1 2 lim h ( z + h ) 2 + 2( z + h ) ( z + h )- ( z + h ) 2- z 2- 2 z z + z 2 h = 1 2 lim h 2 z + 2 z h h + 2 z- 2 z h h + 2 h + h- h 2 h If this limit exists, it must be the same for any direction h approaches the origin in the complex plane. If we approach the limit from the real axis, we note h = h and the above reduces to 4 z + 2 h 4 z . If we approach the limit from the imaginary axis, we find h =- h and the above limit reduces to 4 z- 2 h 4 z ....
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
- Three '09