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Complex Analysis

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The notion of the complex derivative is the basis of complex function theory. The definition of complex derivative is similar to the derivative of a real function. However, despite a superficial similarity, complex differentiation is a deeply different theory. A complex function f ( z ) 𝑓 ( 𝑧 ) is differentiable at a point z 0 ∈ C 𝑧 0 ∈ 𝐶 if and only if the following limit difference quotient exists Alternatively, letting Δ z = z − z 0 Δ 𝑧 = 𝑧 − 𝑧 0 , we can write We often drop the subscript on z 0 𝑧 0 and introduce the number Δ w = f ( z + Δ z ) − f ( z ) . Δ 𝑤 = 𝑓 ( 𝑧 + Δ 𝑧 ) − 𝑓 ( 𝑧 ) . which denotes the change in the value w = f ( z ) 𝑤 = 𝑓 ( 𝑧 ) corresponding to a change Δ z Δ 𝑧 in the point at which f 𝑓 is evaluated. Then we can write equation ( 2 2 ) as d w d z = lim Δ z → 0 Δ w Δ z . 𝑑 𝑤 𝑑 𝑧 = lim Δ 𝑧 → 0 Δ 𝑤 Δ 𝑧 . Despite the fact that the formula ( 1 1 ) for a derivative is identical in form to that of the derivative of a real-valued f

Complex Differentiation The notion of the complex derivative is the basis of complex function theory. The definition of complex derivative is similar to the derivative of a real function. However, despite a superficial similarity, complex differentiation is a deeply different theory. A complex function $f(z)$ is differentiable at a point $z_0\in \mathbb C$ if and only if the following limit difference quotient exists \begin{eqnarray}\label{diff01} f'(z_0) = \lim_{z \rightarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}. \end{eqnarray} Alternatively, letting $\Delta z = z-z_0,$ we can write \begin{eqnarray

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