7.4: Exercises

Exercise \(\PageIndex{1}\)

For each of the following functions \(f(z)\), find the real and imaginary component functions \(u(x,y)\) and \(v(x,y)\), and hence verify whether they satisfy the Cauchy-Riemann equations.

1. \(f(z) = z\)

2. \(f(z) = z^2\)

3. \(f(z) = |z|\)

4. \(f(z) = |z|^2\)

5. \(f(z) = \exp(z)\)

6. \(f(z) = \cos(z)\)

7. \(f(z) = 1/z\)

Exercise \(\PageIndex{2}\)

Suppose a function \(f(z)\) is well-defined and obeys the Cauchy-Riemann equations at a point \(z\), and the partial derivatives in the Cauchy-Riemann equations are continuous at that point. Show that the function is complex differentiable at that point. Hint: consider an arbitary displacement \(\Delta z = \Delta x + i \Delta y\).

Exercise \(\PageIndex{3}\)

Prove that products of analytic functions are analytic: if \(f(z)\) and \(g(z)\) are analytic in \(D \subset \mathbb{C}\), then \(f(z) g(z)\) is analytic in \(D\).

Answer

We will use the Cauchy-Riemann equations. Decompose \(z\), \(f\), and \(g\) into real and imaginary parts as follows: \(z = x + i y\), \(f = u + i v\), and \(g = p + i q\). Since \(f(z)\) and \(g(z)\) are analytic in \(D\), they satisfy \[\begin{align} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y},\;\; -\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}\\ \frac{\partial p}{\partial x} &= \frac{\partial q}{\partial y},\;\; -\frac{\partial p}{\partial y} = \frac{\partial q}{\partial x}.\end{align}\] This holds for all \(z \in D\). Next, expand the product \(f(z)\,g(z)\) into real and imaginary parts: \[\begin{align}\begin{aligned} f(z)\,g(z) = A(x,y) + i B(x,y),\;\;\mathrm{where}\;\; \begin{cases}A = up - v q \\ B = uq + vp. \end{cases}\end{aligned}\end{align}\] Our goal is to prove that \(A\) and \(B\) satisfy the Cauchy-Riemann equations for \(x + i y \in D\), which would then imply that \(fg\) is analytic in \(D\). Using the product rule for derivatives: \[\begin{align} \frac{\partial A}{\partial x} &= \frac{\partial u}{\partial x} p + u \frac{\partial p}{\partial x} - \frac{\partial v}{\partial x} q - v \frac{\partial q}{\partial x} \\ &= \frac{\partial v}{\partial y} p + u \frac{\partial q}{\partial y} + \frac{\partial u}{\partial y} q + v \frac{\partial p}{\partial y} \\ \frac{\partial B}{\partial y} &= \frac{\partial u}{\partial y} q + u \frac{\partial q}{\partial y} + \frac{\partial v}{\partial y} p + v \frac{\partial p}{\partial y}.\end{align}\] By direct comparison, we see that the two expressions are equal. Similarly, \[\begin{align} \frac{\partial A}{\partial y} &= \frac{\partial u}{\partial y} p + u \frac{\partial p}{\partial y} - \frac{\partial v}{\partial y} q - v \frac{\partial q}{\partial y} \\ &= - \frac{\partial v}{\partial x} p - u \frac{\partial q}{\partial x} - \frac{\partial u}{\partial x} q - v \frac{\partial p}{\partial x} \\ \frac{\partial B}{\partial x} &= \frac{\partial u}{\partial x} q + u \frac{\partial q}{\partial x} + \frac{\partial v}{\partial x} p + v \frac{\partial p}{\partial x}.\end{align}\] These two are the negatives of each other. Q.E.D.

Exercise \(\PageIndex{4}\)

Prove that compositions of analytic functions are analytic: if \(f(z)\) is analytic in \(D \subset \mathbb{C}\) and \(g(z)\) is analytic in the range of \(f\), then \(g(f(z))\) is analytic in \(D\).

Exercise \(\PageIndex{5}\)

Prove that reciprocals of analytic functions are analytic away from poles: if \(f(z)\) is analytic in \(D \subset \mathbb{C}\), then \(1/f(z)\) is analytic everywhere in \(D\) except where \(f(z) = 0\).

Exercise \(\PageIndex{6}\)

Show that if \(f(z = x + iy) = u(x,y) + i v(x,y)\) satisfies the Cauchy-Riemann equations, then the real functions \(u\) and \(v\) each obey Laplace’s equation: \[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0.\] (Such functions are called “harmonic functions”.)

Exercise \(\PageIndex{7}\)

We can write the real and imaginary parts of a function in terms of polar coordinates: \(f(z) = u(r,\theta) + i v(r,\theta)\), where \(z = re^{i\theta}\). Show that the Cauchy-Riemann equations can be re-written in polar form as \[\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta}, \quad \frac{\partial v}{\partial r} = - \frac{1}{r}\, \frac{\partial u}{\partial \theta}.\]