Articles of complex analysis

Laurent Series and Taylor Expansion of $ 1 / (e^z – 1) $

Could someone please assist me with the second part of the second paragraph, from “By expanding $f_1$…”? I am not convinced that my expansion for $f_1$ is right – I used the standard binomial, but with $$(z^2 + 4 \pi^2 r^2)^{-1} = z^{-2}(1 + ({2\pi r \over z})^2)^{-1}$$ then normal binomial on this, but only […]

Laurent series of $1/(z^2-z)$

I’m re-learning this stuff. It is a little confusing to me how to obtain the Laurent expansion of rational functions. I have to determine the Laurent series of $$ \frac{1}{z^2-z}, $$ centered at $z=-1$ that converges at $z=1/2$. The “centered” part is clear to me, but what does it mean that converges at $z=1/2$? The […]

Cauchy Integral Theorem and the complex logarithm function

I am given the following integral: $\int_C {e^{(-1+i)\log(z)}}$ with $C:|z|=1$ $\operatorname{Log} (z): 0\le \operatorname{arg} (z)\le 2\pi$ Is it possible to resolve this integral using Cauchy theorem? The function is not analytic in a line inside C so my guess would be that it is not possible. Thanks for the help

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of $z$, $\overline{z}$, and then plug in $z = 1/w$. However, I am having difficulty doing this. I completed […]

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z \right| $$ tends to zero as $R \to \infty$. By using the ML-inequality one has $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z […]

Are there periodic functions satisfying a quadratic differential equation?

Question: Are there periodic functions satisfying a quadratic differential equation, as opposed to just linear or cubic? Bonus question: Are there periodic functions satisfying differential equations which are polynomials of any degree $n$? Background: I know that on the real line, any periodic function can be decomposed into a (possibly infinite) sum of sines and […]

Schwarz-Christoffel mapping onto infinite L-shaped region

I’m trying to map the upper half plane onto the infinite L-shaped region $$ \Omega = \{z = x+iy; \ x > 0, \ y > 0, \ \min(x,y) < 1 \} $$ My first try is a Schwarz-Christoffel function $$ F(w) = \int_0^w (\zeta+1)^{-1}\zeta^{-1/2}(\zeta-1)^{-1} \ d\zeta $$ This guy looks promising, because the integrand […]

Why is Cauchy-Riemann equation not sufficient for differentiablity

This question already has an answer here: Is following contradictory? Can you give an example? 2 answers

The total number of poles of an elliptic function in $P_0$ is always $\geq$ 2

I’m trying to follow this proof from Stein & Shakarchi “Complex Analysis”. The statement of the theorem is in the title of the question. The Proof is as follows: Suppose that $f$ (an elliptic function) has no poles on the boundary $\partial P_0$ where $P_0 =\{ z \in \mathbf{B} : z = a + b\tau […]

Extensions of the Hermite Bielher and Hermite-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a zero of the other polynomial, in the sense that they can be ordered from least to greatest. Two interlacing polynomials automatically […]