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Session 1: Differentiation

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This figure consists of three copies of the complex plane arranged side by side, from left to right. The axes are not labelled. Each of the three diagrams has a different region, bounded by broken lines or curves, and shaded inside. The left-hand diagram shows a horseshoe-shaped region symmetrical about the vertical axis; the boundary of the shape is drawn as a broken line. The arms of the horseshoe are pointing down and they straddle the horizontal axis. The open end of the horseshoe is at the bottom, so that the origin is not inside the region. The middle diagram shows an open annulus, and both of its boundaries are drawn as broken lines. It consists of the area between two concentric circles, centred at the origin. The right-hand diagram shows a square region centred on the origin, with two vertical and two horizontal sides; its boundaries are drawn as broken lines. It is symmetrical about both axes. The origin itself is marked with a hollow dot.