Articles of complex analysis

Calculating $\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz$

Find the value of $\displaystyle\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz$ The first idea I had was to apply Cauchy’s integral formula. But can this be done? $e^{1/z^2}$ is not holomorphic on the interior of $|z|=2$. Still, if we would apply this we would obtain: $\displaystyle\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz=-2\pi ie$. Another way I think, is to apply the residue theorem. For $1-z$ has a […]

Bounded imaginary part implies removable singularity at 0

Let $f$ be a holomorphic function on the punctured unit disk. If the imaginary part of $f$ is bounded, is it true that $f$ has a removable singularity at 0? I see that $|e^{-if}|=e^{Im\;f}$ so $e^{-if}$ is a bounded holomorphic function on the punctured unit disk and it follows that $e^{-if}$ has a removable singularity […]

Solve $z^6=(z-1)^6$.

In the answer of Surb here : How to solve for the complex number $z$? I don’t understand the subtlety. To me it’s natural to do $$z^6=(z-1)^6\iff \left(\frac{z}{z-1}\right)^6\iff \frac{z}{z-1}=e^{ki\pi/3}$$ for $k=0,…,5$. I can see that the case $k=0$, but I don’t understand why… We do as usual: $$v^6=1\iff v=e^{ki\pi /3}$$ and now replace $v$ by […]

difficulty understanding branch of the logarithm

Here is one past qual question, Prove that the function $\log(z+ \sqrt{z^2-1})$ can be defined to be analytic on the domain $\mathbb{C} \setminus (-\infty,1]$ (Hint: start by defining an appropriate branch of $ \sqrt{z^2-1}$ on $\mathbb{C}\setminus (-\infty,1]$ ) It just seems typical language problem. I do not see point of being rigorous here. But I […]

Derive branch cuts for $\log(\sqrt{1-z^2} + iz)$ as $(-\infty,-1)$ and $(1,\infty)$?

Attempt: First, we examine $\sqrt{1-z^2}$. Note that it can be written $\sqrt{1-z}\sqrt{1+z}$, so the appropriate branch cuts are $(-\infty,-1)$ and $(1,\infty)$ for the inner square root term. Next, we look at $\log(w)$ and note that we can define the cut for $\log(w)$ as $(-\infty,0)$. But now what? I tried setting $w= \sqrt{1-z^2} + iz$, solving […]

Evaluating an integral across contours: $\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$

This is for an assignment, describing the procedure is most beneficial for me, rather that solely computing the result. I want to evaluate the following integral: $$\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$$ Along the line segments from $-4$ to $-4-4i$ to $4-4i$ to $4$, e.g: So I want to evaluate three integrals of the above […]

Determining holomorphicity

I need to determine where the following function is differentiable and holomorphic in $\mathbb C$: $$f(z)=(z-3)^i$$ I have the derivative as $df/dz= i(z-3)^{-1+i}$. The answer in my book says f is differentiable and holomorphic on $\mathbb C$ where $y\neq0$ and $x>3$. I don’t see where this comes from. wolframalpha plotted the derivative and I can […]

Motivation for Mobius Transformation

Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the motivation to study Mobius Transformation? Why should one look at the map defined in the above way?

Lagrange inversion theorem application

Can someone give me an example of where the Lagrange inversion theorem is applied in such a way it inverts a formal series? For example, say I have $$\sum_{i>-1} a_it^i = u.$$ Can someone show me the step by step process by which $$\sum_{i>-1}b_iu^i = t$$ is obtained. I can seem to find any links […]

On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\nabla u)=(\gamma u_x)_x+(\gamma u_y)_y=0,\\ u(x+2\pi,y)=\mu u(x,y),\\ u(x,1)=0,\\ u(x,0)=0 \mbox{ or } u_y(x,0)=0. \end{cases}$$ Let’s denote the set of possible values of the eigenvalue parameter $\mu$, for which the system has a solution $\mu_\gamma$. Does there exist a conductivity $\alpha(y)$ on […]