limz→z0z2−z20z−z0=limz→z0(z−z0)(z+z0)z−z0=2z0 \[\begin{align*} \lim z \rightar...limz→z0z2−z20z−z0=limz→z0(z−z0)(z+z0)z−z0=2z0limz→z0ez−ez0z−z0=ez0limz→z0ez−z0−1z−z0=ez0limz→z0∞∑n=0(z−z0)n(n+1)!=ez0
In this case Δz=iΔy and \[\begin{align} f^{\prime}(z) &=\lim _{\Delta z \rightarrow 0} \frac{f(z+\Delta z)-f(z)}{\Delta z}\nonumber \\ &=\lim _{\Delta y \rightarrow 0} \frac{u(x, y+\D...In this case Δz=iΔy and f′(z)=limΔz→0f(z+Δz)−f(z)Δz=limΔy→0u(x,y+Δy)+iv(x,y+Δy)−(u(x,y)+iv(x,y))iΔy=limΔy→0u(x,y+Δy)−u(x,y)iΔy+limΔy→0v(x,y+Δy)−v(x,y)Δy
