# Tagged Questions

37 views

### Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
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### Proof that a complex function is continuous at $z=0$

Given the function $f\colon \mathbb{C}\to\mathbb{C}$ by $f(z)=\begin{cases} \frac{xy(x+iy)}{x^2+y^2} & \text{if } z\neq 0\\ 0 & \text{if } z=0 \end{cases}$ with $z=x+iy$. How do I ...
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### Branch Points of $f(z)=\log\left( \frac{1+\sqrt{1+z^2}}{2}\right)$

I am doing a problem concerning the branch points of the complex function $$f(z)=\log\left( \frac{1+\sqrt{1+z^2}}{2}\right)$$ The question begins by asking what the branch points of $f(z)$ are, and ...
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### Binomial series property for $|z| \lt 1$

I had a lecture where we proved the following identity: $\forall z \in \mathbb{C}$ with $|z| \lt 1$. $$\frac{1}{(1-z)^{\alpha}}=\sum_{k=0}^\infty {\alpha +k-1\choose k}z^k$$ Now i know this proof is ...
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### Crazy calculation for winding numbers

Find the winding number around $z=-i, z=-1, z=0$ in the following figure. The purpose of this exercise is to complete a complex integral with singularities at the stated points. My attempt is ...
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### Essential singularity HW question

Show that $f(z)=ze^{\frac{1}{z}}e^{\frac{-1}{z^2}}$has an essential singularity at $z=0$. This one should be straightforward, as I should be able to tackle it by use of expanding the power ...
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### Prove function is open and connected

Prove that $f(re^{i\theta}) = \sqrt[3]{r} e^{\frac{i\theta}{3}}$ for $0 < r < \infty$ & $-\pi < \theta < \pi$, has open connected domain and is an analytic function. Can some kind ...
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### need to show a complex function is continuous on C (complex plane)

Prove $$f(z) = \sum\limits_{n=0}^\infty \frac{z^{2n}(-1)^{n}}{(2n)!}$$ is continuous everywhere on $\mathbb{C}$ I want to show, for each $\epsilon > 0$, there is some $\delta > 0$ such that ...
### $f$ is analytic with range as a circle
I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...