Problem 1.Use Schwarz - Christoffel Formula Theorem to describe the image of the upper half-plane...
(Complex analysis)
Exercise 6 a) Show that the image of the half-plane y > c (c = const) in the z-plane 1 under the inversion mapping w--s the interior of a circle provided that C0 the inversion mapping w hen0? the inversion mapping w = z when c < 0? b) What is the image of the half-plane y > c (c -const) in the z-plane under c) What is the image of the half-plane y > c (cconst) in...
14. What is the image of the upper half-plane under a mapping of the form az + b a, b, c, d real; ad - bc < 0?
14. What is the image of the upper half-plane under a mapping of the form az + b a, b, c, d real; ad - bc
2. Find the image of the upper half-plane by the mapping 1-χα 1 za where 0< α< 1 and 20 has its principal value.
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0 slit along the segment from 0 to i, a nonpolygonal region. (Use the principal square root throughout.) Hint: The desired non-polygonal region can be obtained as a "limit" of a sequence of polygonal regions.)
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0...
5. Prove that f(z) = (2+1/2) is a conformal map from the half-disc {z = x +iy : 2< 1, y >0} to the upper half-plane. (Hint: The equation f(z) = w reduces to the quadratic equation z2 + 2wz +1 = 0, which has two distinct roots in C whenever w # £1. This is certainly the case if WE H.
(5). This problem involves the mapping w(z)-,(z + z") between the z-plane and the w-plane. The two parts can be solved independently. 2 (a). Identify all of the values of z for which the mapping w(z) fails to be conformal. In each case, explain why the mapping is not conformal at that value of z. (b). Find the image in the w-plane of the unit circle Iz1, Graph it, label the axes, and label the w-plane points that correspond to...
Solve the following questions using confomal mapping from
complex analysis
7.1 Compute the images of the real and imaginary axes and (a) the lower half-plane under the map f(z) = (2+2)/(z-i), (b) the right half-plane under the map f(z) (z 1)/(z +1), (c) the left half-plane under the map f(z) = (z+ 1)/(2-1)
7.1 Compute the images of the real and imaginary axes and (a) the lower half-plane under the map f(z) = (2+2)/(z-i), (b) the right half-plane under the...
1. Consider the problem of steady state heat flow in the half-plane, 22T a2T + ar2 =0 for ER and y>0, ay2 subject to the boundary condition T(3,0) = g(x), and T +0 as yo. You will solve the problem using the Fourier transform in 2, with T(w,y) = ZELT(, y)e-iw= ds 2 (a) Derive an ODE for T. You can assume T +0 as|a . (b) Derive conditions for I at y = 0 and as y. You can...
please solve these two questions completely with steps thank you!
2. Find the image of a horizontal line under the mapping w e Problem 5. Evaluate the following integrals, justifying your procedures. 1. e z, where C is the circle with radius, Centre 1,positively oriented. 2. Let CRbe the circle ll R(R> 1), described in the counterclockwise direction. Show that Log Problem 6. The function g(z) = Vre2 (r > 0,-r < θπ) is analytic in its domain of definition,...
(Complex Analysis)
The linear mapping wFUz+p, where α, β e C maps the point ZFI+1 to the point wi-i, and the poin to the point w2-1i a) Determine α and β. b) Find the region in the w-plane corresponding to the upper half-plane Im(z) 20 in 9. the z-plane. Sketch the region in the w-plane. c) Find the region in the w-plane corresponding to the disk Iz 2 in the z-plane d) Find the fixed points of the mapping
The...