Integral of z^m

This is really an interactive version of the illustrations at the pages 396 and 397 of the textbook.

In this applet we are calculating the integral of a power function z^m for m between −3 and 2 over a contour given by an arc L (green) with center at the origin and radius r (between 0 and 2).

The number w (shown in red) is the Riemann sum:

w = ∑ ζ_k^m•d_k

where ζ_k is a number on L half way between Z_k-1 and Z_k.

You can change the radius of the arc L using the slider r, and the power m.

Suggestions for exploring:

• Play with the m slider to see the Riemann sums for different powers. Notice the significance of the value m=−1.
• For different values of m, play with the radius of the arc L. As you change the radius, what happens to the w? How is its modulus affected? What happend to its argument? Can you explain what's going on?
• Set the m to −1. Drag the r slider to change the radius of L. What happens to w? Can you explain?

Jan Hlavacek, Created with GeoGebra