7. Find a conformal map of the infinite strip 0 < y < π onto the semi-infinite strip 7. Find a conformal map of the infinite strip 0
7. (a) Find the harmonic function in the semi-infinite strip {0 < x < π, 0sy oo) that satisfies the "boundary conditions": u(r, y) (b) What would go awry if we omitted the condition at infinity?
7. (a) Find the harmonic function in the semi-infinite strip {0
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0<x<a, y>0, that satisfies the boundary conditions u(0, y)-0 u(a, y)-0, y > 0, and the additional condition that u(x, y) -0 as yoo, etnyla sin nTX where Cn X where Cn- NTX) where Cn = u(x, y) - -Ttny/a sin(where Cn u(x, y) n=1 u(x, y) - (b) Find the solution if f(x) = x(a-x) V(x)- (c) Let a9. Find the smallest value of yo for...
Q8 [10] Find the image of the semi-infinite strip in the 2 - plane under the Transformations given by 1) to (1) by interpreting both the regions graphically. The semi infinite strip is given by, {2 < Real z < 5, Imag. z <3, ZEC ill) w =
(6 (12 pts). Suppose f is a conformal map of the upper half place one-to-one onto itself with f(-1) = 0, f(0) = 2, and f(1) = 8. Find f(i).
Exercise 4. (5 points) Find a conformal mapping (a 1-1 analytic map) from the complement of the non-negative reals, C \[0,00), onto the unit disc [z< 1.
Problem 1. Make a sketch of the infinite strip 0< < 1. Then, find and sketch its image under the transformation w-12. În all these problems, z x + iy.
7. Prove that the function N4-i/z is conformal in its entire domain Ω-C\ {0}. Find the image wo of the point v3 i under this function, as well as the rotation angle and the stretching/contracting factor of tangent vectors at this point. Find the images Vi and V2 of the vectors vi-1-(1,0) and v2 (0,1) under this map, and check that the angle between the images is the same as the angle between vi and v2
7. Prove that the...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
4. Find a conformal equivalence from the open unit disc to the set W = {z : 0 < arg(z) < π/4)
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...