8.5: Exercises

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Exercise \(\PageIndex{1}\)

Find the values of \((i)^i\).

Answer: We can write \(i\) in polar coordinates as \(\exp(i\pi/2).\) Hence, \[\begin{align} (i)^i &= \exp\Big\{i \ln\big[\exp(i\pi/2)\big]\Big\} \\ &= \exp\left\{i \left[\frac{i\pi}{2} + 2 \pi i n\right]\right\}, \quad n \in \mathbb{Z} \\ &= \exp\left[- 2\pi\left(n+\frac{1}{4}\right) \right], \quad n \in \mathbb{Z}.\end{align}\]

Exercise \(\PageIndex{2}\)

Prove that \(\ln(z)\) has branch points at \(z = 0\) and \(z = \infty\).

Answer

Let \(z = r\exp(i\theta)\), where \(r > 0\). The values of the logarithm are \[\ln(z) = \ln(r) + i (\theta + 2\pi n), \;\;\;n \in \mathrm{Z}.\] For each \(n\), note that the first term is the real part and the second term is the imaginary part of a complex number \(w_n\). The logarithm in the first term can be taken to be the real logarithm.

For \(z \rightarrow 0\), we have \(r \rightarrow 0\) and hence \(\ln(r)\rightarrow -\infty\). This implies that \(w_n\) lies infinitely far to the left of the origin on the complex plane. Therefore, \(w_n \rightarrow \infty\) (referring to the complex infinity) regardless of the value of \(n\). Likewise, for \(z \rightarrow \infty\), we have \(r \rightarrow \infty\) and hence \(\ln(r)\rightarrow +\infty\). This implies that \(w_n\) lies infinitely far to the right of the origin on the complex plane, so \(w_n \rightarrow \infty\) regardless of the value of \(n\). Therefore, \(0\) and \(\infty\) are both branch points of the complex logarithm.

Exercise \(\PageIndex{3}\)

For each of the following multi-valued functions, find all the possible function values, at the specified \(z\):

\(z^{1/3}\) at \(z = 1\).
\(z^{3/5}\) at \(z = i\).
\(\ln(z+i)\) at \(z = 1\).
\(\cos^{-1}(z)\) at \(z = i\)

Exercise \(\PageIndex{4}\)

For the square root operation \(z^{1/2}\), choose a branch cut. Then show that both the branch functions \(f_\pm(z)\) are analytic over all of \(\mathbb{C}\) excluding the branch cut.

Exercise \(\PageIndex{5}\)

Consider \(f(z) = \ln(z+a) - \ln(z-a)\). For simplicity, let \(a\) be a positive real number. As discussed in Section 8.4, we can write this as \[f(z) = \ln\left|\frac{z+a}{z-a}\right| + i(\theta_+ - \theta_-), \qquad \theta_\pm \equiv \mathrm{arg}(z\pm a).\] Suppose we represent the arguments as \(\theta_+ \in (-\pi,\pi)\) and \(\theta_- \in (-\pi,\pi)\). Explain why this implies a branch cut consisting of a straight line joining \(a\) with \(-a\). Using this representation, calculate the change in \(f(z)\) over an infinitesimal loop encircling \(z = a\) or \(z = -a\). Calculate also the change in \(f(z)\) over a loop of radius \(R \gg a\) encircling the origin (and thus enclosing both branch points).