Articles of complex analysis

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. Specifically, if S contains small sub-squares, they will be mapped into sub-shapes of C that are […]

integral to infinity + imaginary constant

A proof I’m reading tries to evaluate the integral (where $i$ is the regular imaginary unit) $$\int_{-\infty}^{\infty} e^{-(x-\alpha i)^2}\mathrm{d}x$$ by doing a substitution $u=x-\alpha i$. Normally, one would also have to change the bounds of integration. $$\int_{-\infty+\alpha i}^{\infty+\alpha i} e^{-x^2}\mathrm{d}x$$ But this proof leaves the bounds as +/- infinity. $$\int_{-\infty}^{\infty} e^{-x^2}\mathrm{d}x$$ Why is this valid?

Analytic continuation of a power series 2

Another Qual question here, For the function $$\sum_{n=0}^\infty z^{2^n}$$, Prove the following: i) $f$ converges to a function analytic in the open unit disk $D$, ii) $f(z) =z+f(z^2)$ and iii) $f(z)$ can not be analytically continued past any point on the unit circle. I can even see (ii) very easily, but I can not see […]

zero set of an analytic functio of several complex variables

In one variable complex theory, we have the result that zeroes of a non-zero analytic function are isolated. In several variable theory, this result does not hold. I read it somewhere that this fact can be proved using Hurwitz theorem. If anyone can help me with this.

Radii of convergence for complex series

I need to find the radii of convergence for these series: $1. \sum_{n=1}^\infty (2+(-1)^n)^n z^{2n}$ $2. \sum_{n=1}^\infty (n+a^n)z^n, a \in C $ $3. \sum_{n=1}^\infty 2^n z^{n!}$ Starting with the first power series, I attempted this by claiming that $$a_n =(2+(-1)^k)^k$$ if $n = 2k, k\ge 1$, and $0$ otherwise. Then, $$|a_n|^{1/n} = ((2+(-1)^k)^k)^\frac{1}{2k}= \sqrt{(2+(-1)^k}$$ Therefore, […]

Cauchy's Integral parametric conjugate

By considering the conjugate of its parametric form, evaluate $$\frac{1}{2\pi i}\int_{\gamma(0;1)}\frac{\overline{f(z)}}{z-a}dz$$ when $|a|<1$ and $|a|>1$, where $f$ is holomorphic in in the disk $(0;R), R>1$. Typically when doing these kinds of integration and parametrization, $|z|=n$ is given, but it’s different in this case (or is it not?). Can someone help me out?

Sufficient conditions for an entire functions to be constant

Let $f$ be an entire function. Which of the following are correct? $f$ is constant if range of $f$ is contained in a straight line $f$ is constant if it has uncountable number of zeros $f$ is constant if $f$ is bounded on $\{z:\operatorname{Re}(z)\leq 0\}$ $f$ is constant if $\operatorname{Re}(f)$ is bounded 2. is correct […]

Change of variable (translation) in complex integral

If I have a real integral, e.g. $\int f(x+2) \ dx$, I can substitute $y = x+2$, so $dy = dx$. But if my function is complex, am I still allowed to do this? In which cases I cannot apply a translation?

Approaching a contour integral with singularities on each axis

How do I solve an integral like this using complex methods? $$ \int_{0}^{\infty} \frac{\ln(x)}{\left(x^2 + 2\right)\left(x^2 + 1\right)}dx.$$ I tried using two semi circles in the upper half plane but the singularities on the real axis are troublesome for me and I’m not sure how to approach the problem

Generalization of the argument principle

This exercise is from big Rudin: Let $f \in H(U)$ and $D(a,r)\subset U$ be a disk s.t. $f$ has no zero on the boundary of the disk. Let $\gamma$ be a curve parametrizing the boundary of the disk and Compute: $$I =\frac{1}{2\pi i}\int_\gamma \frac{f’}{f}z^pdz \space \text{ where $p\in \mathbb{N}$}$$ Then replace $z^p$ with an arbitrary […]