# Questions tagged [meromorphic-functions]

Meromorphic functions are complex-valued functions which are holomorphic everywhere on an open domain except a set of isolated points which are poles. Consider also using the (complex-analysis) tag.

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### Prove $\sum_{n=0}^{\infty} \frac{(-1)^n}{z+n}$ is meromorphic

The question is prove that the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{z+n}$ determines a meromorphic function. So the way that I prove these kinds of questions is fix $R>0$ and prove that it is ...
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### Is there exists a function f which is meromorphic on C and satisfies |f(z)|≥|z| at all those points where f is holomorphic? [closed]

Is there exists a function $f$ which is meromorphic on $\mathbb{C}$ and satisfies $|f(z)|\geq |z|$ at all those points where $f$ is holomorphic? Is such a function is unique and entire?
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### If $a_i\in\mathbb{R}$, $\omega^2+\omega+1=0$, and $\sum_{i=1}^n\frac{1}{a_i+\omega^k} =2\omega^{2k}$ for $k=1,2$, find $\sum_{i=1}^n\frac{1}{a_i+1}$.

In this question, $\omega$ is the complex cube root of $1$ and $a_i \in \mathbb R$. If $$\sum_{i=1}^n \frac{1}{a_i + \omega} =2\omega ^2$$ and $$\sum_{i=1}^n \frac{1}{a_i + \omega ^2} =2\omega\,,$$ ...
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### Suppose that for $\theta \in R$ we have $f(z)=g(z,\theta)$. Now fix $z=z_0$, is $g(z_0,\theta)=f(z_0)$ for all $\theta$ in its domain?
Let $f(z)$ and $g(z,\theta)$ be complex functions with $f$ analytic and $g$ meromorphic. Suppose that for $\theta \in R$ we have $f(z)=g(z,\theta)$. Now fix $z=z_0$, is $g(z_0,\theta)=f(z_0)$ for ...