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.