Holomorphic function
Holomorphic functions [ edit ] Main article: Holomorphic function Complex functions that are differentiable at every point of an open subset Ω of the complex plane are said to be holomorphic on Ω . In the context of complex analysis, the derivative of � at � 0 is defined to be [1] � ′ ( � 0 ) = lim � → � 0 � ( � ) − � ( � 0 ) � − � 0 . Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach � 0 in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are...