Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class 15 (Wednesday February 18)

SUMMARY Mappings by elementary functions CURRENT READING Brown & Curchill, pages 75-81 NEXT READING Brown & Curchill pages 81-85

Consider the mapping consisting of the exponential function. Let’s think about where w = e^{z} would map horizontal and vertical lines. \underline{Examples} Let the vertical line be given by Re z = α, in that case, the image of this line is w = e^{α} · e^{iy}, where y can be any real value. What does this set of points look like? Sketch how the point w changes as the pre-image point changes with y [Hint: does |w| depend on the size of y?]

Now, �gure out where w = e^{z} maps horizontal lines and sketch them on the axes above.

\underline{Groupwork} Putting this information together we can see that the set 0 < y < π ∩ x ≤ 0 gets mapped to the interior of the semi-circle (|w| < 1 ∩ Im w > 0 under what mapping w = f(z)?

How would we have to change our pre-image set so that w = e^{z} would map the pre-image to the entire upper half-plane?

More mappings Now we know that the exponential function maps vertical and horizontal line segments to portions of circles (called sectors) and rays. The logarithm function is the inverse of the exponential function so we can use this information to determine its mapping properties. As usual, we have to determine which branch of the logarithm we are talking about. For most purposes, we will just be interested in the principal branch, Log z. I usually think the best way to think about the logarithm mapping is to try and think about the reverse of what the exponential function would do. Here is an example of a useful mapping using Log z

We say that a mapping is conformal when it preserves the angle and sense between two directed curves as it maps them from the z plane to the w plane. If the magnitude of the angle but not the sense is preserved, the mapping is called isogonal

Any analytic function which is nonconstant on a domain maps an open set to an open set. We will look at mappings by other analytic functions.