# Tagged Questions

A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers is quite different from the properties ...

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### Example of two analytic functions that differ at countably infinity many point

$\displaystyle f_1(x) = \frac{x^n-1}{x-1}$ and $f_2(x) = x^{n-1} + \cdots + 1$ have the same values except at $x=1$ (where $f_1$ fails to be analytic ). Is there an example of two analytic function ...
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### If $f(z)$ is meromorphic but not entire, is $\exp(f(z))$ meromorphic? Could it even be entire?

First, I can show that $f$ meromorphic is a rational function. Now, I want to consider $g=e^{f(z)}$. I have heard that there is something interesting that goes on with $g$, that there is some room ...
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### Is the rank of a matrix-valued mapping $\mathbb R^n \rightarrow \mathbb R^{n_1 \times n_2}$ constant almost everywhere?

How can I prove the following statement or even better: which source can be quoted for the proof. Consider an $n_1 \times n_2$ matrix $A(z)$ where $z\in \mathbb{R}^n$ is arbitrary and the the entries ...
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### Laurent series $\frac{1}{e^z-1}$ [duplicate]

How can I expand $$f(z)=\frac{1}{e^z-1}$$ into Laurent series? I know that $f$ has singularities in $2k \pi i, \ \ k \in \mathbb{Z}$. Just substituting Taylor series for $e^z$ in the denominator ...
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### Singularities of difference of two functions

We are given $$f(z) = \frac{1}{\sin z} - \frac{1}{e^z-1}$$ $\frac{1}{\sin z}$ has poles in $k \pi, \ \ k \in \mathbb{Z}$ and $\frac{1}{e^z-1}$ has singularities in $2 k \pi i, \ \ k \in \mathbb{Z}$. ...
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### An entire function bounded outside a strip which contains the reals is constant

Let $f$ be an entire function, which takes real values on the real axis and has no zeros. Suppose $f$ is bounded for $|\operatorname{Im} z| > a > 0$ where $a>0$. Is $f$ a constant? I would ...
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### Analytic functions and constants: Proving $f(z)-g(z)$ is constant

Let $f(z)$ and $g(z)$ be analytic on some domain. Show that if $\Re(f(z)) = \Re(g(z))$ then $f(z)-g(z)$ is constant. I haven't a clue on how to start. What is being asked of me & What am I ...
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### Cesaro limit of analytic functions

Let $f_n$ be a uniformly bounded sequence of analytic functions on $\Omega\subset\mathbb C$. If $f_n(z)\to f(z)$ forall $z\in\Omega$, then by the Montel's theorem I know that the convergence is ...
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### Bounded holomorphic function, limsup, boundary

Let $f \in \mathcal{O}(D)$, $\ D$ is a bounded region in $\mathbb{C}$, be such that $$\limsup _{D \ni z \to z_0} |f(z)| < \infty$$ for any $z_0 \in \partial D$. Prove that $f$ is bounded. So we ...
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### Show that an analytic function defined on unit ball with these properties does not exist

Show that there does not exist an analytic function $F : B(0,1) \to \mathbb{C}-\mathbb{Q}$ with $F(0)=i$ and $F(1/2)=-i$. I have already used the open mapping theorem to show that if we assume ...
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### Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$.

Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$. I had no clue, I was trying to use the facts that ...
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### Why is a harmonic conjugate unique up to adding a constant?

If $v$ and $v_0$ are harmonic conjugates of $u$, then $u + iv$ and $u + iv_0$ are analytic functions. Then $i(v - v_0)$ is analytic, but how does this imply $v - v_0$ is a constant function?
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### Example of real analytic function

We were taught real analytic functions in class today. I am playing around trying to construct examples. I see exponential, sine, cosine and logarithmic functions (for $x > 0$). One function I am ...
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### Proove $|f(z)|\leq M|z|$ inside unit disk if $f(0)=0$ and $|f(z)|\leq M$

The problem goes as follows: Assume $f(z)$ is analytic inside the unit disk $D=\{z:|z|<1\}$. Also, $f(0)=0$ and $|f(z)|\leq M$ in $D$. Proove that $|f(z)|\leq M|z|$ in $D$. When does ...
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### $f(x+1)=f(x)+f(\alpha\cdot x)$

I try to find an analytic increasing function $f_\alpha$ ($0\le\alpha\le1$) from $\mathbb R$ (or $\mathbb R^+$) to $\mathbb R$ such that for all $x$ f_\alpha(x+1)=f_\alpha(x)+f_\alpha(\alpha\cdot x)\$...