Linked Questions

0 votes
2 answers
619 views

Proving that a "nearly entire" continuous function is entire [duplicate]

If $f$ is a continuous mapping from $\mathbb C$ to $\mathbb C$ such that $f$ is analytic except $[-1,1]$ then $f$ is entire function I want to prove this important theorem. Is there anyone who is ...
Kavita's user avatar
  • 708
6 votes
0 answers
461 views

Show that this function is entire [duplicate]

Possible Duplicate: Which sets are removable for holomorphic functions? Prove analyticity by Morera’s theorem Let $f$ be a continuous complex-valued function on $\mathbb{C}$ such that it is ...
Char's user avatar
  • 61
2 votes
0 answers
67 views

Need help with holomorphic functions on a domain interval removed. [duplicate]

I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$. Is this problem related to ...
Xaviere's user avatar
  • 633
8 votes
5 answers
1k views

Stronger Liouville theorem

"Every bounded function that is holomorphic on $A$ is is constant." For which $A\subseteq\mathbb{C}$ is this true? Are there well-known examples of unbounded sets $A\subseteq\mathbb{C}$ on which ...
Michael Hardy's user avatar
6 votes
2 answers
2k views

Continuous extension of a Bounded Holomorphic Function on $\mathbb{C}\setminus K$

Let $f:\mathbb{C}\setminus K\rightarrow\mathbb{D}$ be a holomorphic map, where $K$ is a compact set with empty interior. My question: Prove or disprove that: $f$ extends continuously on $\mathbb{C}.$...
Abelvikram's user avatar
5 votes
2 answers
1k views

Distinguished map that straightens a curve

Consider a smooth finite curve $\gamma$ without intersections in $\mathbb{R}^2$. Consider the family of smooth maps $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that $T(\gamma)$ is a straight line. ...
Hans-Peter Stricker's user avatar
2 votes
1 answer
2k views

Prove analyticity by Morera's theorem

Let $f$ be continuous on the complex plane and analytic on the complement of the coordinate axes. Show that $f$ is analytic everywhere. Hint: Morera's theorem. I think that I need to show that the ...
Danny's user avatar
  • 877
8 votes
2 answers
528 views

Holomorphic extension to a perfect, nowhere dense set.

Suppose $f$ is holomorphic and uniformly bounded by $M$ on $\mathbb{C}\setminus E$, where $E$ is perfect and nowhere dense? Can $f$ be extended to a holomorphic function on $\mathbb{C}$? Riemann's ...
user39992's user avatar
  • 540
5 votes
1 answer
1k views

Using Morera's theorem to prove analyticity

I have a function $F(x,y)$ which is continuous and analytic on the complement of a certain function $x(y)$. Is it possible to use Morera's theorem to show that it is analytic everywhere? Clearly, this ...
user71551's user avatar
1 vote
1 answer
672 views

A continuous function $f$ is analytic everywhere except along a simple closed contour $C$ in domain $D$, then $f$ is analytic everywhere in $D$.

Claim: Let $D$ be a domain, let $C$ be a simple closed contour in $D$, f is analytic in $D/C$ and continuous in $D$, then $f$ is analytic in $D$. I tried to show the contour integrals of $f$ in $D$ ...
mathlearner's user avatar
3 votes
2 answers
343 views

analytic on a disc with a hole

I am reading a proof of Cauchy's Integral Formula.In the proof,the author let $\phi (z,w)=[f(z)-f(w)]/(z-w)$ if $(z\neq w)$ and $f'(z)$ otherwise and leaves the readers to prove that $g(z)=\phi (z,w)$ ...
Ben's user avatar
  • 3,608
2 votes
1 answer
388 views

Removable singularities for continuous functions

Let $f: D - K \rightarrow \mathbb{C}$ be holomorphic, where $D$ is a planar domain and $K$ is a compact subset of $D$. Suppose that $f$ extends continuously to all of $D$. On which conditions on $K$ ...
Albert's user avatar
  • 9,120
1 vote
2 answers
224 views

Let $f$ be holomorphic on $\Omega - \{z_0\}$. Then $g$ defined by $g(z)=(z-z_0)f(z)$ for $z\in \Omega-\{z_0\}$, $g(z_0)=0$ is holomorphic on $\Omega$.

Edit: Also given $$ \lim_{z\rightarrow z_0} (z-z_0)f(z)=0 $$ It's easy to see that $g$ is holomorphic on $\Omega-\{z_0\}$, so we only need to worry about $g'(z_0)$. By definition, we have \begin{...
William Stagner's user avatar
1 vote
0 answers
302 views

Proving that a complex function is holomorphic, if we know it's holomorphic, except for a real line

Given any real line $G$ in $\mathbb{C} \cong \mathbb{R}^2$, meaning that $G$ is of the shape $\{a + b t | t \in \mathbb{R}\}$ for any $a, b \in \mathbb{C}, b ≠ 0$, I want to proof that: Any complex ...
moran's user avatar
  • 3,037
2 votes
0 answers
179 views

A continous function on $G$, complex analytic on $G\setminus S$ is analytic on $G$.

Let $G$ be a region in the complex plane and $S$ a closed subset of $G$. Assume that a function $f:G\to \mathbb{C}$ is continuous and that it's analytic on $G\setminus S$. Under certain assumptions ...
ploosu2's user avatar
  • 8,470