MATH 508-010, Fall 2022 First Assignment
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MATH 508-010, Fall 2022
First Assignment
1. Write the function f(z) = in the form f(z) = u(x,y) + iv(x,y), and
find its domain.
2. Show that the inversion mapping w = maps the circle |z| = r onto the circle
|w| = 1/r ; and the circle |z − 1| = 1 onto the vertical line x = 1/2.
3. For the function f(z) = ez , describe the image of the vertical line Re z = −1, and the image of the line Imz = 3π/4.
4. Show that the Joukowski mapping w = J(z) = (z + ) maps the unit
circle |z| = 1 onto the real interval [−1, 1]; and it maps the circle |z| = r (r > 0, r 1) onto the ellipse
u2 v2
[ (r + )] 2 [ (r − )] 2
with foci at ±1.
5. Show that if r and θ are polar coordinates, then the functions rn cosnθ and rn sinnθ, where n is an integer, are harmonic as functions of x and y .
2022-09-23