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extra_4 - Extra Problems Sheet 4 Stephen Taylor May 6 2005...

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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 0 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 0 ( z + h ) 2 + 2( z + h ) ( z + h ) - ( z + h ) 2 - z 2 - 2 z ¯ z + ¯ z 2 h = 1 2 lim h 0 2 z + 2 z ¯ h h + 2¯ z - 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 z . So 4 z = 4¯ z which implies z = ¯ z . Since this condition must hold everywhere f is differentiable, we define a new function f * taking it into account.
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