\[ \newcommand{\Arg}{\mathsf{Arg}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Im}{\mathsf{Im}} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\ball}{\mathsf{B}} \newcommand{\wind}{\mathsf{wind}} \newcommand{\Log}{\mathsf{Log}} \]

Week 5 Worksheet

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Contour Integration

  1. Determine the value of \[ \int\limits_\gamma x - y + ix^2 \intd z \] where $z = x+iy$ and $\gamma$ is each of the following.
    1. The straight line joining $0$ to $1+i$.
    2. The imaginary axis from 0 to 1.
    3. The line parallel to the real axis from $i$ to $1+i$.
  2. Calculate from the definition the following contour integrals.
    1. $\displaystyle\int\limits_{\gamma_1} f$ where $f(z) = \dfrac{1}{z-2}$ and $\gamma_1(t) = 2+2e^{it}$ on $[0,2\pi]$
    2. $\displaystyle\int\limits_{\gamma_2} f$ where $f(z) = \dfrac{1}{(z-i)^3}$ and $\gamma_2(t) = i+e^{-it}$ on $[0,\pi/2]$
  3. Let $\gamma$ denote the circular path with centre $1$ and radius $1$, described once anticlockwise and starting at the point $2$. Let $f(z)=|z|^2$. Write down a formula for $\gamma$ and calculate the contour integral of $f$ over $\gamma$.

Antiderivatives

  1. For each of the following functions find an antiderivative and calculate the integral along any smooth path from $0$ to $i$.
    1. $f(z) = z^2 \sin(z)$ on $\C$
    2. $f(z) = z \exp(iz)$ on $\C$
  2. Put $f(z) = |z|^2$.
    1. Calculate the contour integral of $f$ over the contour that goes vertically from $0$ to $i$ then horizontally from $i$ to $1+i$.
    2. Calculate the contour integral of $f$ over the contour that goes horizontally from $0$ to $1$ then vertically from $1$ to $1+i$.
    3. What does this tell you about possibility of the existence of an antiderivative for $f(z)=|z|^2$?
  3. The \define{reverse} of a path $\gamma : [a,b] \to \C$ is the path $\tilde{\gamma} : [a,b] \to \C$ defined by $\tilde{\gamma}(t) = \gamma(a + b - t)$. Let $\gamma$ denote the semi-circular path along a circle with centre $0$ and radius $3$ that starts at $3$ passes through $3i$ and ends at $-3$.
    1. Write down a formula for $\gamma$.
    2. Let $f(z)=1/z^2$ on the domain $\C \setminus \{0\}$. Calculate the contour integral of $f$ over $\gamma$.
    3. Write down a formula for $\tilde{\gamma}$ and calculate the contour integral of $f$ over $\tilde{\gamma}$.
    4. Verify in this case that $\displaystyle\int\limits_{\tilde{\gamma}} f= - \displaystyle\int\limits_\gamma f$.
  4. Let $f, g : D \to \C$ be holomorphic. Let $\gamma$ be a smooth path in $D$ starting at $z_0$ and ending at $z_1$. Prove that \[ \int\limits_\gamma f g' = f(z_1) g(z_1)- f(z_0) g(z_0) - \int\limits_\gamma f' g. \] which is the complex analogue of the integration by parts formula.

The Principal Logarithm

  1. We introduced the principal logarithm via the inversion $\exp(\Log(z)) = z$ for all $z \ne 0$. Here is the other side of that composition.
    1. What is the range $\{ \Log(z) : z \ne 0 \}$ of the principal logarithm?
    2. Verify that $\Log(\exp(z)) = z$ for all $z$ in the range of the principal logarithm.
    1. Give an example of complex numbers $a,b$ for which $\Log(ab) \ne \Log(a) + \Log(b)$.
    2. Prove that for any two complex numbers $a$ and $b$ there is $n \in \Z$ such that \[ \Log(ab) = \Log(a) + \Log(b) + 2 \pi i n \] holds.
  3. Calculate $\Log(1+i)$ and $\Log(-1-i)$.
  4. Given complex numbers $a,b$ with $a \ne 0$ we define $a^b = \exp(b \, \Log(a))$.
    1. Calculate $i^i$.
    2. Verify that $a^{b+c} = a^b a^c$ for all $a,b,c$ with $a \ne 0$.
    3. Verify that $(a^{1/2})^2 = a$ for all $a \ne 0$.
    4. Give an example to prove that $(ab)^{1/2} \ne a^{1/2} b^{1/2}$ in general.