Analyze the flow having potential f (z) = iKa 3Log 2z ia3

In each of parts (a) through (e), find the image of the rectangle under the mapping w = ez . Sketch the rectangle in the z-plane and its image in the w-plane. (a) 0 x , 0 y (b) 1 x 1,/2 y /2 (c) 0 x 1, 0 y /4 (d) 1 x 2, 0 y (e) 1 x 2,/2 y /2

In each of parts (a) through (e), find the image of the rectangle under the mapping w = cos(z). Sketch the rectangle in the z-plane and its image in the w-plane. (a) 0 x 1, 1 y 2 (b) /2 x , 1 y 3 (c) 0 x ,/2 y (d) x 2, 1 y 2 (e) 0 x /2, 0 y 1

In each of parts (a) through (e), find the image of the rectangle under the mapping w = 4 sin(z). Sketch the rectangle in the z-plane and its image in the w-plane. (a) 0 x /2, 0 y /2 (b) /4 x /2, 0 y /2 (c) 0 x 1, 0 y /6 (d) /2 x 3/2, 0 y /2 (e) 1 x 2, 1 y 2

Determine the image of the sector /4 5/4 under the mapping w = z2 . Sketch the sector and its image.

Determine the image of the sector /6 /3 under the mapping w = z3 . Sketch the sector and its image.

Show that the mapping w = 1 2 z + 1 z  maps the circle |z| =r onto an ellipse with foci 1 in the w-plane. Sketch a typical circle and its image.

Show that the mapping of Problem 6 maps a half-line = k onto a hyperbola with foci 1 in the w-plane, assuming that sin(k) = 0 and cos(k) = 0. Sketch a typical half-line and its image.

Let D consist of all z in the rectangle having vertices i and i, with a positive number. (a) Determine the image of D under the mapping w = cos(z). Sketch D and its image. (b) Determine the image of D under the mapping w = sin(z). Sketch this image

Determine the image of D under the mapping w=2z2 . Sketch this image

Determine the image of the infinite strip 0 Im 2 under the mapping w = ez .

In each of Problems 11 through 16, find the image of the given circle or line under the bilinear transformation.w = 2i/z;Re(z) = 4

In each of Problems 11 through 16, find the image of the given circle or line under the bilinear transformation.w = 2i z 4;Re(z) = 5

In each of Problems 11 through 16, find the image of the given circle or line under the bilinear transformation.w = (z i)/i z;(z + z)/2 + (z z)/2i = 4

In each of Problems 11 through 16, find the image of the given circle or line under the bilinear transformation.w = (z 1 + i)/(2z + 1);|z| = 4

In each of Problems 11 through 16, find the image of the given circle or line under the bilinear transformation.w = (2z 5)(z + i);z + z (3/2i)(z z) 5 = 0

In each of Problems 11 through 16, find the image of the given circle or line under the bilinear transformation.w = ((1 + 3i)z 2)/z;|z i| = 1

In each of Problems 17 through 21, find a bilinear transformation taking the given points to the indicated images.11, 2 i, 31 + i

In each of Problems 17 through 21, find a bilinear transformation taking the given points to the indicated images.i i, 1 i, 20

In each of Problems 17 through 21, find a bilinear transformation taking the given points to the indicated images.11 + i, 2i 3 i, 4

In each of Problems 17 through 21, find a bilinear transformation taking the given points to the indicated images.5 + 2i 1, 3i 0,1

In each of Problems 17 through 21, find a bilinear transformation taking the given points to the indicated images.6 + i 2 i,i 3i, 4 i

Prove that the composition of two conformal mappings is conformal.

Show that the mapping w =T (z)=z is not conformal

Suppose T is a bilinear transformation that is not the identity mapping or a translation. Show that T must have either one or two fixed points. Why does this fail for translations?

A point z0 is a fixed point of a bilinear transformation T if T (z0)=z0. Suppose a bilinear transformation T has three fixed points. Show that T must be the identity mapping, sending each z to itself.

Let T and S be bilinear mappings that agree at three points. Show that T = S.

Define the cross ratio of four complex numbers z1,z2,z3, and z4 to be the image of z1 under the bilinear transformation that maps z2 1, z3 0, z4 . Denote this cross ratio [z1,z2,z3,z4]. Suppose T is any bilinear transformation. Show that T preserves this cross product. This means that [z1,z2,z3,z4]=[T (z1), T (z2), T (z3), T (z4)].

Show that the cross ratio [z1,z2,z3,z4] is the image of z1 under the bilinear transformation w = 1 (z3 z4)(z z2) (z3 z2)(z z4) .

Show that a cross ratio [z1,z2,z3,z4] is real if and only if the zjs are on a circle or a line.

In each of Problems 1 through 6, find a bilinear transformation of the first domain onto the second.|z| < 3 onto |w 1 + i| < 6

In each of Problems 1 through 6, find a bilinear transformation of the first domain onto the second.|z| < 3 onto |w 1 + i| > 6

In each of Problems 1 through 6, find a bilinear transformation of the first domain onto the second.|z + 2i| < 1 onto |w 3| > 2

In each of Problems 1 through 6, find a bilinear transformation of the first domain onto the second.Re(z) > 1 onto Im(w) > 1

In each of Problems 1 through 6, find a bilinear transformation of the first domain onto the second.Re(z) < 0 onto |w| < 4

In each of Problems 1 through 6, find a bilinear transformation of the first domain onto the second.Im(z) > 4 onto |w i| > 2

Find a conformal mapping of the upper half-plane onto the wedge 9 < arg(w) < /3

Let w = f (z) = log(z) be defined by restricting the argument of z to lie in [0, 2]. Show that f takes the upper half-plane onto the strip 0 < Re(w) < .

Show that the Schwarz-Christoffel transformation f (z) = 2i z 0 ( + 1) 1/2 ( 1) 1/2 1/2 d maps the upper half-plane onto the rectangle with vertices 0, c, c + ic, and ic where c = (1/2)(1/4)/ (3/4). For this problem, it is necessary to know the integral formula B(x, y) = 1 0 ux1 (1 u) y1 du for the beta function B(x, y), and also to know that, in terms of the gamma function, B(x, y) = (x)(y) (x + y) . Hint: See Section 15.3 (Problems 39 and 40).

Upper half-plane: u(x, 0) = f (x)

Right quarter plane: Re(z) > 0, u(x, 0) = f (x), and u(0, y) = 0.

The disk |z z0| < R if u(x, y) = x y for (x, y) on the boundary

Right half-plane with boundary condition: u(0, y) = 1 for 1 x 1 0 for |y| > 1

The unit disk if u(x, y) = x y for (x, y) on the boundary circle

The unit disk with u(ei ) = 1 for 0 /4 0 for /4 << 2

Solve the Dirichlet problem for the strip 1 < Im(z) < 1, Re(z) > 0 with the boundary conditions u(x, 1) = u(x,1) = 0 for x > 0 and u(0, y) = 1 |y| for 1 y 1.

In each of Problems 1 through 4, analyze the flow having the given complex potential. Sketch some equipotential curves and streamlines and determine the velocity, any stagnation points, and whether the flow has any sources or sinks.f (z) = az with a a nonzero complex number.

In each of Problems 1 through 4, analyze the flow having the given complex potential. Sketch some equipotential curves and streamlines and determine the velocity, any stagnation points, and whether the flow has any sources or sinks.f (z) = z3

In each of Problems 1 through 4, analyze the flow having the given complex potential. Sketch some equipotential curves and streamlines and determine the velocity, any stagnation points, and whether the flow has any sources or sinks.f (z) = cos(z)

In each of Problems 1 through 4, analyze the flow having the given complex potential. Sketch some equipotential curves and streamlines and determine the velocity, any stagnation points, and whether the flow has any sources or sinks.f (z) = z + i z2

f (z) = KLog(z z0) with K as a nonzero real constant and z0 as a complex number. Show that z0 is a source if K > 0 and a sink if K < 0. Sketch some equipotential curves and streamlines. Log(z) is defined here to be the branch of log(z) obtained by restricting arg (z)<.

Let f (z) = KLogz a z b  with K as a nonzero real numbers and a and b as distinct complex numbers. Sketch some equipotential curves and streamlines and determine any stagnation points, sources, and sinks.

Let f (z) = K z + 1 z  with K as a nonzero real number. Sketch some equipotential curves and streamlines. Show that f models flow around the upper half of the unit circle.

Let f (z) = m ik 2 Logz a z b  with m and k as nonzero, real numbers and a and b as distinct, complex numbers. Show that this flow has a source or sink of strength m and a vortex of strength k at both a and b. A point combining properties of a source or sink and a vortex is called a spiral vortex. Sketch some equipotential curves and streamlines for this flow.

Analyze the flow having complex potential f (z) = K z + 1 z  + ib 2 Log(z) with k and b as nonzero, real numbers. Sketch some equipotential curves and streamlines.

Analyze the flow having potential f (z) = iKa 3Log 2z ia3 2z + ia3  with K and a positive constants. Show that f models an irrotational flow around a cylinder 4x 2 + 4(y a)2 = a2 with a flat boundary along the y-axis. Sketch some equipotential curves and streamlines.

Use Blasiuss theorem to show that the force per unit width on the cylinder in Problem 10 has vertical component of 23aK2 with as the constant density of the fluid