Chapter 20

Use (2) to describe the image of the upper half-plane \(y \geq 0\) under the conformal mapping \(w=f(z)\) that satisfies the given conditions. Do not attempt to find \(f(z)\). \(f^{\prime}(z)=(z-1)^{-1 / 2}, f(1)=0\)

A linear fractional transformation is given. (a) Compute \(T(0), T(1)\), and \(T(\infty)\). (b) Find the images of the circles \(|z|=1\) and \(|z-1|=1\). (c) Find the image of the disk \(|z| \leq 1\). \(T(z)=\frac{i}{z}\)

Determine where the given complex mapping is conformal. \(f(z)=z^{3}-3 z+1\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. \(y=x\) under \(w=1 / z\)

\(f^{\prime}(z)=(z-1)^{-1 / 2}, \quad f(1)=0\)

Verify that div \(\mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) for the given vector field \(\mathbf{F}(x, y)\) by examining the corresponding complex function \(g(z)=P(x, y)-i Q(x, y)\). Find a complex potential for the vector field and sketch the equipotential lines. \(\mathbf{F}(x, y)=-y \mathbf{i}-x \mathbf{j}\)

Use (2) to describe the image of the upper half-plane \(y \geq 0\) under the conformal mapping \(w=f(z)\) that satisfies the given conditions. Do not attempt to find \(f(z)\). \(f^{\prime}(z)=(z+1)^{-1 / 3}, \quad f(-1)=0\)

A linear fractional transformation is given. (a) Compute \(T(0), T(1)\), and \(T(\infty)\). (b) Find the images of the circles \(|z|=1\) and \(|z-1|=1\). (c) Find the image of the disk \(|z| \leq 1\). \(T(z)=\frac{1}{z-1}\)

Determine where the given complex mapping is conformal. \(f(z)=\cos z\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. \(y=1\) under \(w=1 / z\)

\(f^{\prime}(z)=(z+1)^{-1 / 3}, \quad f(-1)=0\)

Verify that div \(\mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) for the given vector field \(\mathbf{F}(x, y)\) by examining the corresponding complex function \(g(z)=P(x, y)-i Q(x, y)\). Find a complex potential for the vector field and sketch the equipotential lines. \(\mathbf{F}(x, y)=\frac{x}{x^{2}+y^{2}} \mathbf{i}+\frac{y}{x^{2}+y^{2}} \mathbf{j}\)

A linear fractional transformation is given. (a) Compute \(T(0), T(1)\), and \(T(\infty)\). (b) Find the images of the circles \(|z|=1\) and \(|z-1|=1\). (c) Find the image of the disk \(|z| \leq 1\). \(T(z)=\frac{z+1}{z-1}\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Hyperbola \(x y=1\) under \(w=z^{2}\)

\(f^{\prime}(z)=(z+1)^{-1 / 2}(z-1)^{1 / 2}, \quad f(-1)=0\)

Verify that div \(\mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) for the given vector field \(\mathbf{F}(x, y)\) by examining the corresponding complex function \(g(z)=P(x, y)-i Q(x, y)\). Find a complex potential for the vector field and sketch the equipotential lines. \(\mathbf{F}(x, y)=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{i}+\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{j}\)

A linear fractional transformation is given. (a) Compute \(T(0), T(1)\), and \(T(\infty)\). (b) Find the images of the circles \(|z|=1\) and \(|z-1|=1\). (c) Find the image of the disk \(|z| \leq 1\). \(T(z)=\frac{z-i}{z}\)

Determine where the given complex mapping is conformal. \(f(z)=z+\operatorname{Ln} z+1\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Hyperbola \(x^{2}-y^{2}=4\) under \(w=z^{2}\)

The analytic function \(f(z)=\cosh z\) is conformal except at \(z=\)

\(f^{\prime}(z)=(z+1)^{-1 / 2}(z-1)^{-3 / 4}, \quad f(-1)=0\)

The potential \(\phi\) on the wedge \(0 \leq \operatorname{Arg} z \leq \pi / 4\) satisfies the boundary conditions \(\phi(x, 0)=0\) and \(\phi(x, x)=1\) for \(x>0\) Determine a complex potential, the equipotential lines, and the corresponding force field \(\mathbf{F}\).

Find the solution of the Dirichlet problem in the upper halfplane that satisfies the boundary condition \(u(x, 0)=x^{2}\) when \(0

Use the matrix method to compute \(S^{-1}(w)\) and \(S^{-1}(T(z))\) for each pair of linear fractional transformations. \(T(z)=\frac{z}{i z-1}\) and \(S(z)=\frac{i z+1}{z-1}\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Semicircle \(|z|=1, y>0\), under \(w=\operatorname{Ln} z\)

If \(w=f(z)\) is an analytic function that maps a domain \(D\) onto the upper half-plane \(v>0\), then the function \(u=\operatorname{Arg}(f(z))\) is harmonic in \(D\).

Use the matrix method to compute \(S^{-1}(w)\) and \(S^{-1}(T(z))\) for each pair of linear fractional transformations. \(T(z)=\frac{i z}{z-2 i}\) and \(S(z)=\frac{2 z+1}{z+1}\)

Determine where the given complex mapping is conformal. \(f(z)=\pi i-\frac{1}{2}[\operatorname{Ln}(z+1)+\operatorname{Ln}(z-1)]\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Ray \(\theta=\pi / 4\) under \(w=\operatorname{Ln} z\)

Is the image of the circle \(|z-1|=1\) under the complex mapping \(T(z)=(z-1) /(z-2)\) a circle or a line?

The potential \(\phi\) on the semicircle \(|z| \leq 1, y \geq 0\), satisfies the boundary conditions \(\phi(x, 0)=0,-1

Use the matrix method to compute \(S^{-1}(w)\) and \(S^{-1}(T(z))\) for each pair of linear fractional transformations. \(T(z)=\frac{2 z-3}{z-3}\) and \(S(z)=\frac{z-2}{z-1}\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Ray \(\theta=\theta_{0}\) under \(w=z^{1 / 2}\)

Use the identity \(\cos z=\sin (\pi / 2-z)\) to find the image of the strip \(0 \leq x \leq \pi\) under the complex mapping \(w=\cos z\). What is the image of a horizontal line in the strip?

Use the matrix method to compute \(S^{-1}(w)\) and \(S^{-1}(T(z))\) for each pair of linear fractional transformations. \(T(z)=\frac{z-1+i}{i z-2} \quad\) and \(\quad S(z)=\frac{(2-i) z}{z-1-i}\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Circular arc \(r=2,0 \leq \theta \leq \pi / 2\), under \(w=z^{1 / 2}\)

Use the identity \(\sinh z=-i \sin (i z)\) to find the image of the strip \(-\pi / 2 \leq y \leq \pi / 2, x \geq 0\), under the complex mapping \(w=\sinh z\). What is the image of a vertical line segment in the strip?

A complex velocity potential \(G(z)\) is defined on a region \(R\). (a) Find the stream function and verify that the boundary of \(R\) is a streamline. (b) Find the corresponding velocity vector field \(\mathbf{V}(x, y)\). (c) Use a graphing utility to sketch the streamlines of the flow. \(G(z)=z^{4}\)

Construct a linear fractional transformation that maps the given triple \(z_{1}, z_{2}\), and \(z_{3}\) to the triple \(w_{1}, w_{2}\), and \(w_{3}\). \(-1,0,2\) to \(0,1, \infty\)

Find the image of the region defined by \(-\pi / 2 \leq x \leq \pi / 2\), \(y \geq 0\), under the complex mapping \(w=(\sin z)^{1 / 4}\). What is the image of the line segment \([-\pi / 2, \pi / 2]\) on the \(x\)-axis?

A complex velocity potential \(G(z)\) is defined on a region \(R\). (a) Find the stream function and verify that the boundary of \(R\) is a streamline. (b) Find the corresponding velocity vector field \(\mathbf{V}(x, y)\). (c) Use a graphing utility to sketch the streamlines of the flow. \(G(z)=z^{2 / 3}\)

Construct a linear fractional transformation that maps the given triple \(z_{1}, z_{2}\), and \(z_{3}\) to the triple \(w_{1}, w_{2}\), and \(w_{3}\). \(i, 0,-i\) to \(0,1, \infty\)

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Circle \(|z|=1\) under \(w=z+1 / z\)

If \(G(z)=\phi(x, y)+i \psi(x, y)\) is analytic in a region \(R\) and \(\mathbf{V}(x, y)=i G^{\prime}(z)\), then the streamlines of the corresponding flow are described by \(\phi(x, y)=c\).

Find the image of the region \(|z| \leq 1\) in the upper half-plane under the complex mapping \(w=z+1 / z\). What is the image of the line segment \([-1,1]\) on the \(x\)-axis?

Circle \(|z|=1\) under \(w=z+1 / z\)

A complex velocity potential \(G(z)\) is defined on a region \(R\). (a) Find the stream function and verify that the boundary of \(R\) is a streamline. (b) Find the corresponding velocity vector field \(\mathbf{V}(x, y)\). (c) Use a graphing utility to sketch the streamlines of the flow. \(G(z)=\sin z\)

A frame for a membrane is defined by \(u\left(e^{i \theta}\right)=\theta^{2} / \pi^{2}\) for \(-\pi \leq \theta \leq \pi .\) Use the Poisson integral formula for the unit disk to estimate the equilibrium displacement of the membrane at \((-0.5,0),(0,0)\), and \((0.5,0)\).

Construct a linear fractional transformation that maps the given triple \(z_{1}, z_{2}\), and \(z_{3}\) to the triple \(w_{1}, w_{2}\), and \(w_{3}\). \(0,1, \infty\) to \(0, i, 2\)

A region \(R\) in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image region \(R^{\prime}\) in the \(w\) -plane. . First quadrant under \(w=1 / z\)