Week 5 Worksheet

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Contour Integration

1. Determine the value of $$\underset{\gamma }{\int }x-y+i{x}^2\mathsf{d}z$$ 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. ${\displaystyle \underset{{\gamma }_1}{\int }f}$ where $f(z)={\displaystyle \frac{1}{z-2}}$ and ${\gamma }_1(t)=2+2e^{it}$ on $[0,2\pi ]$
   2. ${\displaystyle \underset{{\gamma }_2}{\int }f}$ where $f(z)={\displaystyle \frac{1}{(z-i{)}^3}}$ 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

4. 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 $\mathbb{C}$
   2. $f(z)=z\exp (iz)$ on $\mathbb{C}$
5. 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 $f(z)=|z{|}^2$?
6. The \define{reverse} of a path $\gamma :[a,b]\to \mathbb{C}$ is the path $\tilde{\gamma }:[a,b]\to \mathbb{C}$ 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 $\mathbb{C}\setminus \{0\}$. 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 ${\displaystyle \underset{\tilde{\gamma }}{\int }f=-{\displaystyle \underset{\gamma }{\int }f}}$.
7. Let $f,g:D\to \mathbb{C}$ be holomorphic. Let $\gamma$ be a smooth path in $D$ starting at $z_0$ and ending at $z_1$. Prove that $$\underset{\gamma }{\int }f{g}^{\prime }=f(z_1)g(z_1)-f(z_0)g(z_0)-\underset{\gamma }{\int }{f}^{\prime }g.$$ which is the complex analogue of the integration by parts formula.

The Principal Logarithm

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