Introduction to complex analysis

Completion requirements

The figure shows Cartesian axes labelled x and y. It is focused on the upper-right quadrant. A smooth curve that looks like part of the graph of a quadratic function with positive coefficient of x-squared starts in the upper-left quadrant a little above the x-axis and a little to the left of the y-axis. At this point it is nearly horizontal. As x increases the curve slopes upwards with steadily increasing gradient. A line with positive gradient also starts in the upper-left quadrant, below the curve, and slopes upwards into the upper-right quadrant. The curve and the line intersect at two points in the upper-right quadrant, both of which are marked with solid dots. The coordinates of these points are illustrated by vertical and horizontal broken line segments joining each point to the x and y axes. From the lower point of intersection, the vertical broken line segment crosses the x-axis at a point marked c. The horizontal broken line segment crosses the y-axis at a point marked f of c. From the upper point of intersection, the vertical broken line segment crosses the x-axis at a point marked x. The horizontal broken line segment crosses the y-axis at a point marked f of x. From the lower point of intersection a horizontal line segment is drawn to the right. It meets a vertical line segment that extends downwards from the upper point of intersection. They form a right-angled triangle with a segment of the original sloping line as its hypotenuse. The length of the horizontal side of this triangle is marked x minus c. The length of its vertical side is marked f of x minus f of c.