Problem 1
Show that the function \(w=1 / z\) maps the straight line \(y=a\) in the \(z\) -plane onto a circle in the \(w\) -plane with radius \(1 /(2|a|)\) and center \((0,1 /(2 a))\)
Problem 3
Show that when \(z\) is represented by polar coordinates, the derivative of a function \(f(z)\) can be written as $$\frac{d f}{d z}=e^{-i \theta}\left(\frac{\partial U}{\partial r}+i \frac{\partial V}{\partial r}\right),$$ where \(U\) and \(V\) are the real and imaginary parts of \(f(z)\) written in polar coordinates. What are the C-R conditions in polar coordinates? Hint: Start with the C-R conditions in Cartesian coordinates and apply the chain rule to them using \(r=\sqrt{x^{2}+y^{2}}\) and \(\theta=\tan ^{-1}(y / x)=\cos ^{-1}\left(x / \sqrt{x^{2}+y^{2}}\right)\).
Problem 4
Show that \(\frac{d}{d z}(\ln z)=\frac{1}{z} .\) Hint: Find \(u(x, y)\) and \(v(x, y)\) for \(\ln z\) and differentiate them.
Problem 6
Show that (a) the sum and the product of two entire functions are entire, and (b) the ratio of two entire functions is analytic everywhere except at the zeros of the denominator.
Problem 9
Show that (a) the flux through an element of area da of the lateral surface of a cylinder (with arbitrary cross section) is \(d \phi=d z(|\mathbf{E}| d s)\) where \(d s\) is an arc length along the equipotential surface. (b) Prove that \(|\mathbf{E}|=|d w / d z|=\partial v / \partial s\) where \(v\) is the imaginary part of the complex potential, and \(s\) is the parameter describing the length along the equipotential curves. (c) Combine (a) and (b) to get flux per unit \(z\) -length \(=\frac{\phi}{z_{2}-z_{1}}=v\left(P_{2}\right)-v\left(P_{1}\right)\) for any two points \(P_{1}\) and \(P_{2}\) on the cross-sectional curve of the lateral surface. Conclude that the total flux per unit \(z\) -length through a cylinder (with arbitrary cross section) is \([v]\), the total change in \(v\) as one goes around the curve. (d) Using Gauss's law, show that the capacitance per unit length for the capacitor consisting of the two conductors with potentials \(u_{1}\) and \(u_{2}\) is $$c \equiv \frac{\text { charge per unit length }}{\text { potential difference }}=\frac{[v] / 4 \pi}{\left|u_{2}-u_{1}\right|} \text { . }$$
Problem 11
In this problem, you will find the capacitance per unit length of two cylindrical conductors of radii \(R_{1}\) and \(R_{2}\) the distance between whose centers is \(D\) by looking for two line charge densities \(+\lambda\) and \(-\lambda\) such that the two cylinders are two of the equipotential surfaces. From Problem \(10.10\), we have $$R_{i}=\frac{a}{\sinh \left(u_{i} / 2 \lambda\right)}, \quad y_{i}=a \operatorname{coth}\left(u_{i} / 2 \lambda\right), \quad i=1,2,$$ where \(y_{1}\) and \(y_{2}\) are the locations of the centers of the two conductors on the \(y\) -axis (which we assume to connect the two centers). (a) Show that \(D=\left|y_{1}-y_{2}\right|=\left|R_{1} \cosh \frac{u_{1}}{2 \lambda}-R_{2} \cosh \frac{u_{2}}{2 \lambda}\right|\). (b) Square both sides and use \(\cosh (a-b)=\cosh a \cosh b-\sinh a \sinh b\) and the expressions for the \(R\) 's and the \(y\) 's given above to obtain $$\cosh \left(\frac{u_{1}-u_{2}}{2 \lambda}\right)=\left|\frac{R_{1}^{2}+R_{2}^{2}-D^{2}}{2 R_{1} R_{2}}\right|$$ (c) Now find the capacitance per unit length. Consider the special case of two concentric cylinders. (d) Find the capacitance per unit length of a cylinder and a plane, by letting one of the radii, say \(R_{1}\), go to infinity while \(h \equiv R_{1}-D\) remains fixed.
Problem 13
Find all the zeros of \(\sinh z\) and \(\cosh z\).
Problem 14
Verify the following hyperbolic identities. (a) \(\quad \cosh ^{2} z-\sinh ^{2} z=1\). (b) \(\quad \cosh \left(z_{1}+z_{2}\right)=\cosh z_{1} \cosh z_{2}+\sinh z_{1} \sinh z_{2}\). (c) \(\sinh \left(z_{1}+z_{2}\right)=\sin z_{1} \cosh z_{2}+\cosh z_{1} \sinh z_{2}\). (d) \(\cosh 2 z=\cosh ^{2} z+\sinh ^{2} z, \quad \sinh 2 z=2 \sinh z \cosh z\). (e) \(\tanh \left(z_{1}+z_{2}\right)=\frac{\tanh z_{1}+\tanh z_{2}}{1+\tanh z_{1} \tanh z_{2}}\).
Problem 15
Show that (a) \(\tanh \left(\frac{z}{2}\right)=\frac{\sinh x+i \sin y}{\cosh x+\cos y}\), (b) \(\operatorname{coth}\left(\frac{z}{2}\right)=\frac{\sinh x-i \sin y}{\cosh x-\cos y}\).
Problem 16
Find all values of \(z\) such that (a) \(e^{z}=-3\), (b) \(e^{z}=1+i \sqrt{3}\), (c) \(e^{2 z-1}=1\).
