# Tagged Questions

The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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### calculating curvilinear integral by residue theorem

Calculate the following integral by transposing to a curve integral and then using the residue theorem: $\displaystyle \int_{0}^{2\pi}{\frac{e^{int}}{C-e^{it}}dt}, \qquad |C|\ne1, n\in \mathbb N$. ...
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### Non-constant holomorphic and bounded functions $f:\Omega_j\rightarrow\mathbb{C}$

Are there holomorphic, non-constant and bounded functions $$f:\Omega_j\rightarrow\mathbb{C}$$ with $\Omega_1=\mathbb{C}\setminus\{0\}$ $\Omega_2=\mathbb{C}\setminus[0,\infty)$? Since $\Omega_2$ ...
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### Is possible that an entire function satisfies this condition?

Let $f$ an entire function whose only zeros are all the negative integers. Is possible that $f$ satisfies $|f(z)|\leq C_1 e^{C_2 |z|^p},$ for some real constants $C_1, C_2,$ and $p<1$? Any help?
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### Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
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### Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
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### Laurent series with $e^z$

I'm trying to find the Laurent series Expansion for $$f(x) = \frac{e^z-(z-1)}{z-1}$$ on the annulus $0<|z|<\infty$. I'm aware that I am supposed to use substitution of known series. I am ...
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### Cauchy-Riemann equations on $f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}$

Let $$f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}.$$ I need to show that the Cauchy-Riemann equations hold for $f$ in $0$ but $f$ is not (complex) differentiable in $0$....
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### extension of Cauchy's Integral formula

This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6. Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
Prove that: $$\lim _{z\to i} Arg(z) = \frac{\pi}{2}$$ I tryed to prove it by definition but I did not succeed. Any suggestion? Thanks for helpers!