91Ó°ÊÓ

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 \(w=1 / z\)

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

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. \(\mathbf{F}(x, y)=\frac{x}{x^{2}+y^{2}} \mathbf{i}+\frac{y}{x^{2}+y^{2}} \mathbf{j}\)

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 \(w=z^{2}\)

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

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 \(w=z^{2}\)

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

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. \(\mathbf{F}(x, y)=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{i}+\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{j}\)

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