Complex Analysis I (BMAT3), Odd Sem 2022-2023

Consider a holomorphic function $f$ on a region $\Omega$. Let $C$ be a circle inside $\Omega$ whose interior is also contained in $\Omega$. Here is another way to show that $$\oint_{C} f(z)\,\mathrm{d}z=0.$$

1. Consider any regular polygon $P_n$ of $n$ sides inscribed inside the circle. Argue that $$\int_{P_n}f(z)\,\mathrm{d}z=0.$$
2. Show that $$\lim_{n\to \infty} \int_{P_n}f(z)\,\mathrm{d}z= \oint_{C} f(z)\,\mathrm{d}z.$$

Would the above continue to hold if $C$ was instead a twice differentiable curve inside $\Omega$ whose interior was convex and also contained in $\Omega$? Argue.

The angle between two non-zero complex numbers $z$ and $w$ (taken in that order) is simply the oriented angle, in $(-\pi, \pi]$, that is formed between the two vectors in $\mathbb{R}^2$ corresponding to the points $z$ and $w$. This oriented angle, say $\alpha$, is uniquely determined by the two quantities $$\frac{(z, w)}{|z||w|} \text{ and } \frac{(z, -iw)}{|z||w|}$$ which are simply the cosine and sine of $\alpha$, respectively. Here the notation $(\cdot, \cdot)$ corresponds to the usual inner product in $\mathbb{R}^2$, which in terms of complex numbers takes the form $(z, w)= \Re(z\,\bar{w})$.
Let $\Omega$ be an open set. A holomorphic function $f$ defined near $z_0\in \Omega$ is said to preserve angles at $z_0$ if for any two smooth curves $\gamma$ and $\eta$ intersecting at $z_0$, the angle formed between the curves $f \circ \gamma$ and $f \circ \eta$ at $f(z_0)$. In particular, we assume that the tangents to the curves $\gamma, \eta, f\circ \gamma$ and $f\circ\eta$ at the point $z_0$ and $f(z_0)$ are all non-zero.

1. Prove that if $f:\Omega\to \mathbb{C}$ is holomorphic, and $f'(z_0)\ne = 0$, then $f$ preserves angles at $z_0$.
2. Conversely, prove the following: suppose $f:\Omega\to \mathbb{C}$ is a complex valued function, that is real continuously differentiable at $z_0\in \Omega$, and $J_{f}(z_0)\ne 0$. If $f$ preserves angles at $z_0$, then $f$ is holomorphic at $z_0$ with $f'(z_0)\ne 0$.

Prove $$\int_{0}^{\infty} \sin(x^2)\, \mathrm{d}x = \int_{0}^{\infty} \cos(x^2)\, \mathrm{d}x = \frac{\sqrt{2\pi}}{4}.$$ The integral $\int_{0}^{\infty}$ is interpreted as $\lim_{R\to 0} \int_{0}^{R}$.
Hint: Integrate $e^{-z^2}$ from $0$ to $R$, then along the circular arc from $R$ to $Re^{\frac{i\pi}{4}}$ and then along the straight line from $Re^{\frac{i\pi}{4}}$ to $0$.

Prove that $$\int_{0}^{2\pi} \frac{\mathrm{d}\theta}{(a + \cos\, \theta)^2} = \frac{2 \pi a}{(a^2 - 1)^{\frac{3}{2}}}, \text{ whenever } a > 1.$$

Suppose $f$ is holomorphic in a punctured disc $D_r(z_0)\setminus \{z_0\}$. Suppose also that $$|f(z)| \le A |z-z_0|^{-1+ \epsilon}$$ for some $\epsilon >0$, and all $z$ near $z_0$. Show that the singularity of $f$ at $z_0$ is removable.

Let $t>0$ be given and fixed, and define $F(z)$ by $$F(z) = \prod_{n = 1}^{\infty} \left(1 - e^{-2\pi n t} e^{2\pi i z} \right).$$

1. Show that the product defines an entire function of $z$.
2. Show that $|F(z)|\le A e^{a|z|^2}$, hence $F$ is of order $2$.
3. $F$ vanishes exactly when $z = -int + m$ for $n, m$ integers. Then, if $z_n$ is an enumeration of these zeros we have $$\sum \frac{1}{|z_n|^2} = \infty \text{ but } \sum \frac{1}{|z|^{2+ \epsilon}} < \infty.$$

Suppose that $Q_n, n = 1, 2, \dots$ are given polynomials with constant terms equal to $0$. Suppose also thet we are given a sequence of complex numbers $(a_n)$ without limit points. Prove that there exists a meromorphic function $f$ whose only poles are at $(a_n)$, and so that for each $n$, the difference $$f(z) - Q_n\left(\frac{1}{z - a_n}\right)$$ is holomorphic near $a_n$. In other words, $f$ has prescribed poles and principal parts at each of these poles.

Let $f$ be a complex valued $C^1$ function (that is both the real and imaginary parts are $C^1$ functions) defined in the neighbourhood of a point $z_0$. There are several notions closely related to conformality at $z_0$. We say that $f$ is isogonal at $z_0$ if whenever $\gamma$ and $\eta$ are two smooth curves with $\gamma(0) = \eta(0) = z_0$, that make an angle $\theta$ there ($|\theta|<\pi$), then $f\circ \gamma$ and $f\circ \eta$ make an angle $\theta'$ at $t = 0$ with $|\theta'| = |\theta|$ for all $\theta$. Also $f$ is said to be isotropic if it magnifies lengths by some factor for all directions emanating from $z_0$, that is, if the limit $$\lim_{r \to 0} \frac{| f(z_0 + re^{i\theta}) - f(z_0) |}{r}$$ exists, is non-zero, and independent of $\theta$.
Show that $f$ is isogonal at $z_0$ if and only if it is isotropic at $z_0$; moreover, $f$ is isogonal at $z_0$ if and only if either $f'(z_0)$ exists and is non-zero, or the same holds for $f$ replaced by $\bar{f}$.

If $f$ is holomorphic in the deleted neighbourhood $\{0 < |z-z_0| < r \}$ and has a pole of order $k$ at $z_0$, then we can write $$f(z) = \frac{a_{-k}}{(z-z_0)^k} + \dots + \frac{a_{-1}}{(z-z_0)} + g(z)$$ where $g$ is holomorphic in the disc $\{ |z-z_0|< r \}$. There is a generalization of this expansion that holds even if $z_0$ is an essential singularity.
Let $f$ be holomorphic in a region containing the annulus $\{ z: r_1 \le |z-z_0| \le r_2 \}$ where $0< r_1 < r_2$. Then, $$f(z) = \sum_{n = -\infty}^{\infty}a_n (z - z_0)^n$$ where the series converges absolutely in the interior of the annulus. To prove this, it suffices to write $$f(z) = \frac{1}{2 \pi i} \int_{C_{r_2}} \frac{f(\zeta)}{\zeta - z}\,\mathrm{d}\zeta - \frac{1}{2 \pi i} \int_{C_{r_1}} \frac{f(\zeta)}{\zeta - z}\,\mathrm{d}\zeta$$ when $r_1 < |z-z_0| < r_2$, and argue as in the proof of the existence of power series for holomorphic functions done in class. Here $C_{r_1}$ and $C_{r_2}$ are the circles bounding the annulus.

Show that if $f:D(0, R)\to \mathbb{C}$ is holomorphic, with $|f(z)| \le M$ for some $M > 0$, then $$\left|\frac{f(z) - f(0)}{M^2 - \overline{f(0)}f(z)} \right| \le \frac{|z|}{M R}.$$
Hint: Use Schwarz Lemma!

A complex number $w \in \mathbb{D}$ is a fixed point for the map $f:\mathbb{D} \to \mathbb{D}$ if $f(w) = w$.

1. Prove that if $f:\mathbb{D} \to \mathbb{D}$ is holomorphic and has two distinct fixed points, then $f$ is the identity.
2. Must every holomorphic function $f:\mathbb{D} \to \mathbb{D}$ have a fixed point? (Hint: consider the upper half plane)

The pseudo-hyperbolic distance between two points $z, w \in \mathbb{D}$ is defined by $$\rho(z, w) = \left| \frac{z - w}{1- \bar{w}z} \right|.$$

1. Prove that if $f:\mathbb{D} \to \mathbb{D}$ is holomorphic then $$\rho(f(z), f(w)) \le \rho(z, w), \text{ for all } z, w \in \mathbb{D}.$$
   Hint: Consider the automorphism $\psi_{\alpha}(z) = (z - \alpha) / (1 - \bar{\alpha}z)$ and apply the Schwarz lemma to $\psi_{f(w)}\circ f \circ \psi_{w}^{-1}$.
2. Prove that $$\frac{|f'(z)|}{1 - |f(z)|^2} \le \frac{1}{1 - |z|^2} \text{ for all } z \in \mathbb{D}.$$ This result is called Schwarz-Pick Lemma.