Understand why the integral around a closed loop is zero if no "sources" or "sinks" (singularities) are inside. It’s like a fluid flow with no holes in the pipe. Recommended Resources for Visual Learners Visual Complex Analysis - MAA.org
Notice that "analytic" functions preserve angles. If you have a grid of small squares, an analytic function might turn them into curved "squares," but the 90∘90 raised to the composed with power corners remain 90∘90 raised to the composed with power Visual Complex Analysis
Multiplying by a complex number is a simultaneous "stretch" (amplitude) and "rotation" (phase). Euler's Formula Geometrically: Understand eiθe raised to the i theta power Understand why the integral around a closed loop