Conformal Mapping and Bilinear Transformations

Determine the conformal map \(f\) that takes the region \(D\) exterior to two circular cylinders of radius \(R\) to the region outside a circular disc of unit radius centered at the origin (see figure 1). Use the following steps:

1. Show how the map \(\omega =i-2R/z\) takes the region \(D\) to the infinite strip \(0<\operatorname {Im}\omega <2\).
2. Show how the map \(\xi =e^{\pi \omega /2}\) takes the infinite strip to the upper-half complex \(\xi \)-plane.
3. Show how the map \(\eta =(\xi +i)/(\xi -i)\) takes the upper-half complex \(\xi \)-plane to the region outside the unit disc, that is, \(|\eta |>1\).
4. Find \(f\).

Figure 1 description: The map \(f\) sends the \(z\)-plane region exterior to two vertically tangent circular cylinders of radius \(R\) to the \(\eta \)-plane region exterior to the unit disc. The original figure shows the two tangent circles on the left, an arrow labeled \(f\), and a unit circle in the \(\eta \)-plane on the right.

Figure 1: Find the conformal map \(f\)

1. Show that the transformation \(w=2z+1/z\) maps the unit circle conformally onto the ellipse \[ \left (\frac {u}{3}\right )^2+v^2=1. \] Here \(w=u+iv\).
2. Find the Möbius transformation that takes the points \(z_1=0\), \(z_2=1\), \(z_3=-i\) into \(w_1=1\), \(w_2=0\), \(w_3=i\).

1. Consider a 2-dimensional map \((x,y)\to (u,v)\).

   Explain: If the map is given by an analytic function [\(u+iv=f(x+iy)\)] and at some point \(z_0=x_0+iy_0\) the derivative \(f'(z_0)\ne 0\), then this map preserves small shapes in the vicinity of \(z_0\).

2. Does there exist a conformal transformation \(u+iv=f(x+iy)\) that maps triangle \[ \Delta =\{(x,y):0<x<1,\ 0<y<1,\ x+y<1\} \] to circle \[ D=\{(u,v):u^2+v^2<1\}\,? \]

Let \((X,d)\) be a complete metric space and \(f\) a contraction on \(X\) with contraction constant \(\alpha <1\). Let \(x_f\) be the unique fixed point of \(f\). Fix \(x_0\in X\) and consider the sequence of points defined by \(x_n=f(x_{n-1})\) for \(n\in \mathbb {N}\).

1. Prove that \[ d(x_n,x_f)\le \frac {\alpha ^n}{1-\alpha }d(x_0,x_1). \] Hint: consider \(d(x_m,x_n)\) for \(m>n\).
2. Let \(S=[1,\infty )\) and \(f:S\to S\) be defined by \(f(x)=\frac {1}{2}(x+2/x)\). Prove that \(f\) is a contraction mapping on \(S\) and compute a contraction constant. Determine all fixed points of \(f\) on \(S\). Using part (a), explain why the sequence \((x_n)\) might be useful.

Consider the transformation on the extended complex plane \(\overline {\mathbb {C}}\) \[ f(z)=w=z+\frac {1}{z}. \]

1. Find all possible fixed points.
2. Show that the image of the points in the upper half \(z\)-plane that are exterior to the circle \(\lvert z\rvert =1\) corresponds to the entire upper half \(w\)-plane.
3. Find the image of the unit circle.
4. Show that \(f(z)\) is a two-to-one function.

Prove that a bilinear map takes all circles into circles.

1. Consider a 2-dimensional map \((x,y)\to (u,v)\). Explain: If the map is given by an analytic function [\(u+iv=f(x+iy)\)] and at some point \(z_0=x_0+iy_0\) the derivative \(f'(z_0)\ne 0\), then this map preserves small shapes in the vicinity of \(z_0\).
2. Is a square conformally equivalent to a circle?
3. Explain: A map by analytic function preserves Laplace’s equation.

1. Show that any bilinear map takes circles into circles.
2. Find the image of the triangle \[ \{z=x+iy:\ x\ge 0,\ y\ge 0,\ x+y\le 1\} \] under the mapping \(w=1/z\).

Find the image of the half-strip \[ \{z=x+iy:\ x>0,\ 0<y<1\} \] under the mapping \(w=1/z\).

1. State and prove the Fredholm Alternative Theorem for a linear transformation \(A:\mathbb {R}^n\to \mathbb {R}^m\).
2. Find and give a graphical interpretation of the pseudo-inverse of the matrix \[ A= \begin {pmatrix} 1 & 1\\ 2 & 2 \end {pmatrix}. \]

The fourth quadrant \(x\ge 0\), \(y\le 0\) is occupied by an impermeable solid, around which air flows. Treating the flow as incompressible and irrotational use conformal mapping to map to a simpler problem and give one non-trivial solution for the complex potential \(f(z)\) in the original coordinates. Find the equation in polar coordinates for the streamline passing through \(z=-1\). What happens to the fluid velocity as one approaches the corner?

Consider the flow around a thin airfoil of length \(2a\) that is inclined at an angle \(\alpha \) to the \(x\)-axis. Assume that there is uniform flow at \(\infty \) with velocity \(U_x=U\), \(U_y=0\).

1. Show that the conformal map \[ z=\frac {ae^{-i\alpha }}{2}\left (\omega +\frac {1}{\omega }\right ) \] maps the unit circle in the \(\omega \)-plane to the airfoil in the \(z\)-plane.
2. Consider the complex potential \[ \Gamma (\omega )=\frac {aU}{2}\left (e^{-i\alpha }\omega +\frac {e^{i\alpha }}{\omega }\right ) -\frac {i\gamma }{2\pi }\log \omega . \] Calculate the corresponding potential \(\Omega (z)=\Gamma (\omega (z))\) where \(\omega (z)\) is the inverse of the conformal map in part (a). What does the term in \(\gamma \) represent?
3. Calculate \(\Omega '(z)\) and show that it satisfies the correct asymptotic flow. Determine what happens at the end points of the airfoil, and use this to calculate \(\gamma \).

Consider the bilinear transformation \[ \omega =\frac {az+b}{cz+d},\qquad ad-bc\ne 0. \]

1. Show that this transformation maps circles to circles.
2. Show that the bilinear transformations form a group.
3. In the case \((a,b,c,d)=(1,-i,1,i)\) show that the upper-half plane is mapped on to the unit disc.

Consider the bilinear transformation \[ \omega =\frac {az+b}{cz+d},\qquad ad-bc\ne 0. \]

1. Show that this transformation maps circles to circles.
2. Show that the bilinear transformations form a group.
3. In the case \((a,b,c,d)=(1,-i,1,i)\) show that the upper-half plane is mapped on to the unit disc.

Show that the conformal transformation \[ w=\frac {z-a}{z+a},\qquad a=\sqrt {c^2-\rho ^2} \] where \(0<\rho <c\), maps the domain bounded by the circle \(|z-c|=\rho \) and the imaginary axis onto an annular region centered about the origin with outer radius \(|w|=1\) and inner radius \(|w|=\delta \). Find the radius \(\delta \).

Suppose that \(T\) is a transformation of a complete metric space \(X\), and \(T^n\) (for a certain positive integer \(n\)) is a contraction with fixed point \(a\). Show that

1. \(a\) is a fixed point of \(T\);
2. \(T\) has only one fixed point;
3. the sequence of successive iterations \(u_0\), \(u_1=Tu_0\), \(u_2=Tu_1,\ldots \) tends to \(a\) for any initial element \(u_0\in X\).

Find a conformal transformation of the domain \(D\) between the unit circle \(\Gamma _1=\{z=x+iy: |z|=1\}\) and the vertical line \(\Gamma _2=\{z=x+iy: x=5/3\}\) onto a domain between two concentric circles.

Using conformal mapping find the electrostatic potential \(\phi (x,y)\) in the region between the circumferences \[ \gamma =\{z:\ |z|=1\}\qquad \text {and}\qquad \Gamma =\{z:\ |z-1|=2.5\} \] satisfying the boundary conditions \[ \phi \equiv 0\text { on }\gamma ,\qquad \phi \equiv 1\text { on }\Gamma . \]

Given a set of linearly independent vectors \(u_1,u_2,\ldots ,u_n\)

1. Describe the Gram-Schmidt procedure to produce an orthogonal set of vectors from the original set.
2. Suppose \(U\) is a matrix with linearly independent columns. Describe the \(QR\) factorization of \(U\) using Householder transformations.
3. In what sense are these two procedures equivalent? Prove your answer.