The document provides a complex analysis problem asking to find the image of the region outside the unit circle |z| = 1 and between the rays arg(z) = pi/4 and arg(z) = 3pi/4 under the mapping w = Ln z. The solution expresses z as x + iy where x^2 + y^2 > 1 and tan(y/x) lies between -1 and +1, and expresses ln(z) = ln(x + iy) for variables x and y that satisfy the given conditions on x^2 + y^2 and tan(y/x).

1. Complex Analysis
Find the image of (the region outside the unite circle |z| = 1 and between the ray arg(z) = pi/4 and
arg(z) = 3pi/4 ) under the mapping w = Ln z
Solution
w= ln z
pi/41
z = x+iy where x^2+y^2 >1 andtan y/x lies between -1 and +1
ln z = ln(x+iy) for x,y such tht x^2+y^2 >1 andtan y/x lies between -1 and +1
x = r cost, y =rsint, r >1 and pi/4