Complex Analysis: Theory and Applications · Problem 5.1.13

Complex functions are differentiable using the same power rules and chain rules you know from real calculus—the key is that $i$ is just a constant, and exponentials always differentiate to themselves (times the derivative of the exponent). Treat each function mechanically: pull down the exponent, apply the chain rule where the argument is composite, and remember that $\frac{d}{dz}e^u = e^u \cdot \frac{du}{dz}$.

Complex functions are differentiable using the same power rules and chain rules you know from real calculus—the key is that $i$ is just a constant, and exponentials always differentiate to themselves (times the derivative of the exponent). Treat each function mechanically: pull down the exponent, apply the chain rule where the argument is composite, and remember that $\frac{d}{dz}e^u = e^u \cdot \frac{du}{dz}$.