II 2 3 Show from the definition that the functions r Rez and y Im...

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II 2 3 Show from the definition that the functions r Rez and y Im z are not complex differentiable at any point Solution Differentiation f 2 x Rez from the definition z A f z lim A 0 lim A 0 lim A 0 g z Az g z Re z Az Re 2 A 0 Az Differentiation g 2 y Imz from the definition lim lim A 0 Re Az Im Az lim A 0 Az Im z Az Im 2 Az Jo i 1 0 The functions r Rez and y Im z are not complex point because the functions have different limit as Az imaginary axis if Az Ar if Az iAy if Az Ar if Az iAy differentiable at any 0 through real and

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II 2 3 Show from the definition that the functions r Rez and y Im z are not complex differentiable at any point Solution Differentiation f 2 x Rez from the definition z A f z lim A 0 lim A 0 lim A 0 g z Az g z Re z Az Re 2 A 0 Az Differentiation g 2 y Imz from the definition lim lim A 0 Re Az Im Az lim A 0 Az Im z Az Im 2 Az Jo i 1 0 The functions r Rez and y Im z are not complex point because the functions have different limit as Az imaginary axis if Az Ar if Az iAy if Az Ar if Az iAy differentiable at any 0 through real and