8.5: Exercises
- Page ID
- 34564
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
Find the values of \((i)^i\).
- Answer
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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).