Math 185 Assignment 3

1. Complex exponentiation is tricky

   Let $n \in \N$ with $n \gt 1$. Show that there is no continuous function $f : \C \to \C$ so that for all $z \in \C$, \[f(z)^n = z.\] (Hint: suppose $f$ were such a function, and consider the continuous function $[0, 2\pi] \to \T$ given by $t \mapsto f(\exp(it))$.)

2. Reversals

   Let $\gamma$ be a smooth curve in $\C$, and $f$ a continuous function on $\gamma \subset \C$. (Here we are identifying $\gamma$ with the set of points in the image of any of its parameterizations.) With $\gamma^-$ the reversal of $\gamma$, prove that \[\int_{\gamma^-} f(z)\,dz = -\int_\gamma f(z)\,dz.\]

3. Variety of primitives

   Suppose that $\Omega$ is a domain, and $f : \Omega \to \C$ is continuous. Show that any two primitives of $f$ differ by a constant.

4. Weierstrass's Theorem

   Recall that Weierstrass's theorem states that if $K \subseteq \R$ is compact and $f : K \to \R$ is continuous, then there is a sequence of polynomials (with real coefficients) $(p_n)_n$ which converges uniformly to $f$.

   Is it true that if $K \subseteq\C$ is compact and $f : K \to \C$ is continuous, then there is a sequence of polynomials (with complex coefficients) $(p_n)_n$ which converges uniformly to $f$? (Hint: try integrating polynomials.)

5. Some explicit integrals

   For the purposes of this question, given $a_1, \ldots, a_n \in \C$, let us denote by $[a_1, \ldots, a_n]$ the piecewise smooth curve obtained by concatenating the individual curves $[a_1, a_2]\, . [a_2, a_3]\, . \cdots . [a_{n-1}, a_n]$.

   Compute $\int_\gamma\frac{dz}{z}$, where $\gamma$ is each of the following:

   1. $\sq{1, -1+i, -1-i, 1}$
   2. $\sq{i, -1, -i, 1, i}$
   3. $\sq{1+\frac i2, -1-i, -1+i, 1-\frac i2, -2, 1+\frac i2}$

   You may use $\R$ calculus freely, including its applications to functions such as $\ln$ and $\arctan$ on their real domains. You will save yourself a headache by first devising a formula for $\int_{[a,b]}\frac{dz}z$, provided $0 \notin [a, b]$. Answer using exact values; depending on what approach you take, this may require you to do some bashing of trigonometric functions. If you find yourself trying to compute the sum of several arctangents, one possible route forward is compute both the $\sin$ and $\cos$ of the entire sum, using sum-of-angles formulae and the fact that for $x \in \R$, $\sin(\arctan(x)) = \frac{x}{\sqrt{x^2+1}}$ and $\cos(\arctan(x)) = \frac{1}{\sqrt{x^2+1}}$.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• More integrals

  1. Suppose that $T$ is a triangle and $z_0 \in T \setminus \partial T$. Show that \[\int_{\partial T} \frac{dz}{z-z_0} = 2\pi i.\] Conclude that the orientation on $\partial T$ can be defined as the direction which causes the integral to have this value rather than $-2\pi i$.
  2. Suppose that $\gamma$ is a curve parameterized by $w(t) = z_0 + R\exp(it)$ for $t \in [0, 2\pi]$, some fixed $z_0 \in \C$, and $0 \lt R \lt \abs{z_0}$ (so $\gamma$ is a circle not surrounding $0$). Let $n \in \Z$. Compute \[\int_\gamma z^n\,dz.\]

• The Fundamental Theorem of $\C$alculus

  Suppose that $a \lt b$ are real numbers, and $F : [a, b] \to \C$ is a smooth parameterized curve. Verify that \[\int_a^b F'(t)\,dt = F(b) - F(a).\] (This is the step we glossed over in class on February 6^th. It mostly requires checking how the real and imaginary parts of the derivative of $F$ relate to the derivatives of the real and imaginary parts of $F$.)