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 $$\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta}, \quad \frac{\partial v}{\partial r} = - \frac{1}{r}\, \frac{\partial u}{\partial \theta}.$$