7.4: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
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.
- f(z) = z
- f(z) = z^2
- f(z) = |z|
- f(z) = |z|^2
- f(z) = \exp(z)
- f(z) = \cos(z)
- 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
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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}.