674 Math Conformal Mappings Due October 24, 2005 Name 1. Use conformal mappings to solve the problem and u = 1 on x = 0. u = 0 in the rst quadrant, with u = 0 on y = 0
u=1
u=0
u=0
Hint: Consider = ln z. 2. Use conformal mappings to solve the problem = 0 and u = u0 on = 0 . u = 0 in the sector 0 0 , with u = 0 on
u = u0 u = 0 0 u=0
3. Use conformal mappings to solve the problem u = 0 in the rst quadrant with u = 0 on x = 0, with u = 0 on y = 0 for 0 < x < 1, and u = 1 on y = 0 for x > 1. y
u=0
u=0 1 uy = 0 u=1
Hints: Use the transformation z = sin . To write = + i in terms of z = x + iy, rst prove that y2 x2 1. = cos2 sin2
This is a hyperbola in the xy plane with foci at (1, 0). Recall the geometric property of hyperbolas that the dierence of the distances from a point on the hyperbola to the foci is twice the distance between the vertices. 4. Use conformal mappings to solve the problem u = 0 in the region 0 x /2, y > 0 with the conditions u = 0 for y = 0 or x = /2, and u =...