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### Homework3

Course: MATH 674, Fall 2008
School: Towson
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Word Count: 211

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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...

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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 =...