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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2answers
52 views

A question related to Montel's Theorem

Given $c>0$, there exist $\varepsilon > 0,$ such that, whenever $\{a_n\} \subset \mathbb C$ and $\sum_{n=1}^{\infty}\lvert a_n\rvert \le c\,$ implies that $$ \sup_{\frac{1}{2}\leq x\leq1}\left|1 ...
1
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2answers
34 views

What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$?

let $f:D=\{z\in \mathbb C:|z|<1\} \to \overline D$ with $f(0)=0$ be a holomorphic function. What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$? My try:By cauchy integral formula : ...
1
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1answer
26 views

Analyticity of log f(z)

In a solution to a problem, I read that, if $f(z)$ is entire, $f(z)\neq0$ and the domain of definition of $f(z)$ is simply connected, then it is possible to choose a branch of log $f(z)$ that is ...
1
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0answers
17 views

Application of Bernstein's theorem

There is a theorem due to Bernstein related to analytic functions : If $f : ]0,1[ \to \mathbb{R}$ is an absolutely monotonous function (that is a $\mathcal{C}^\infty$ function such that for ...
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0answers
15 views

Question about inverse of CDF being a real analytic function

Let F: [0,a] -> [0,1] be a continuous, strictly increasing CDF. Assume also F admists a continuous, positive pdf f. Now define the inverse function h(x) as F(h(x))=x. Is h real analytic? If not, what ...
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3answers
73 views

Analyticity of $\dfrac{1}{z}$ vs. $\dfrac{1}{z^2}$

I am learning complex analysis on my own. I am familiar with the theorems, and I am able to compute by hand and get correct results. But there is something that escapes me. What is the criteria for ...
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1answer
13 views

A Question from complex variable [closed]

Show that an analytic function with constant modulus is itself a constant
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2answers
55 views

Analytic Functions, Cauchys Integral Formula

Let $f: \mathbb D \to \mathbb D$ be analytic or holomorphic with $f(0)=\frac{1}{2}$ and $f(\frac{1}{2}) = 0$ where $\mathbb {D} = \{ z: |z| \leq 1\}$. Then find $|f^{'}(0)|$ and ...
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1answer
42 views

Find the residue of the function $g(z)=f(z^2)$ at a given point.

Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$. I am looking for a solution to this problem. My thoughts: I know in order to find the residue ...
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1answer
43 views

Find the number of roots of a polynomial using Rouche's Theorem

Use Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that ...
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2answers
50 views

Why does the ring of entire functions have no zero divisors?

Why does the ring of entire functions have no zero divisors, while the ring of infinitely differentiable functions on the real line does?
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1answer
42 views

RoucheTheorem and applications

Now, I want to apply Rouche's theorem in this question, so please i want some hints. I think $f$ maybe one to one since the kernel maybe the zero element.
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0answers
12 views

About smooth function whose reciprocal is also smooth

I know that for exponential functions $e^{at}$, or functions like $1/t^m$, t>0, both they and their reciprocals are smooth. Could you please give me more classes of smooth functions, or Analytic class ...
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1answer
46 views

How to find some $C^\infty$ functions that do not satisfy the uniqueness theorem for analytic functions

The uniqueness theorem for analytic functions states that suppose two series $\sum_{n=0}^\infty s_nx^n$ and$\sum_{n=0}^\infty t_nx^n$ converges in the interval $(-R,R)$. If the set of $x$ that ...
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0answers
37 views

An entire function is zero

Suppose $f(z)$ is an entire function that have zero on positive integers. Does it follow $f$ is identically zero? This seems like an application of Liouville theorem. But I cant come with a function ...
4
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2answers
37 views

$f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?

$f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$? I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$,but this ...
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1answer
73 views

Prove that there is no function $f$ that is analytic. [duplicate]

Prove that there is no function $f$ that is analytic in $\mathbb{C}\setminus\{0\}$ and satisfies $$|f(z)|\geq\frac{1}{\sqrt{|z|}},\quad \operatorname{for all}\quad z\in\mathbb{C}\setminus\{0\}$$ I am ...
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2answers
46 views

How to show real analyticity without extending to complex plane

Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$ For example, $$f(x)=(1+x^2)^{-1}.$$ Using complex analysis, we can easily show $f$ is real analytic. Is there an easy, general method ...
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1answer
25 views

Solving a logarithmic equation with variables on each side

Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically. ...
5
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2answers
66 views

Proving that a doubly-periodic entire function $f$ is constant.

Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: ...
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0answers
15 views

Prove that an entire function of exponential type is of order at most $1$.

By Entire functions theory, the order (at infinity) of an entire function $f(z)$ is defined using the limit superior as: $$\rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln\Vert f \Vert_{\infty, B_r} ...
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2answers
41 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
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1answer
39 views

An entire function is a polynomial iff the Taylor expansion around $0$ converges uniformly

Let $g:\mathbb{C} \to \mathbb{C}$ an entire function. Prove that the Taylor expansion around $0$ converges uniformly in all $\mathbb{C}$ if and only if $g$ is a polynomial. 1/2 PROOF I think I ...
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1answer
79 views

Are the integrals of the following function path independent in the following domain?

Are the integrals of the function: $$f(z)=\frac{1}{z+1}+\frac{1}{(z+1)^2}+e^{\frac{1}{z}}$$ path independent in the following domain: $$D= \{Re z >0\}\setminus\{1\}$$ My thoughts on the ...
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1answer
32 views

Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
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1answer
27 views

Bound on the sup norm for derivatives of a particular $C^\infty$ function

I'm reading textbook "A Primer of Real Analytic Functions" and on page 86 the following "obvious" claim is made: Let $|| \cdot ||$ be the sup norm on $[0, 2 \pi]$ and define function $f$ to be ...
0
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1answer
23 views

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1.

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1. Then, (a) the set {z : |h(z)| = 5} is unbounded by the Maximum Principle; (b) the set {z : ...
2
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1answer
32 views

Prove $f$ analytic on $D(z_0;R)\setminus\{z_0\}$ implies $\exists M, f(D(z_0;r)\setminus\{z_0\})\supset\{z\in\mathbb{C}:|z|>M\}$

Suppose $f$ is analytic on $D(z_0;R)\setminus\{z_0\}$, and $z_0$ is a pole of $f$. Prove that for any $r\in(0,R)$, there is $M\in(0,\infty)$ such that ...
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1answer
38 views

Analytic continuation of function differentiable on real line to complex plane

If $f(z)=g(z)$ on $(0, \infty)$ and f(z) is holomorphic on an open set $U \subset \mathbf{C}$ with $(0, \infty) \subset U$, but we do not have any information about where $g(z)$ is holomorphic, can we ...
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1answer
31 views

derivatives of non-analytic smooth functions

I would like to know how to calculate the derivative of a non-analytic smooth function? Suppose $f:\mathbb R\rightarrow \mathbb R$ is in $\mathcal C^\infty\backslash \mathcal C^\omega$ and in ...
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1answer
34 views

Analytic continuation of function continuous on boundary

Suppose one has a function $f$ in the disc algebra ie: $f$ is continuous on $|z|\leq1$ and holomorphic in $|z|<1$. I wondered, can $f$ always be extended to a holomorphic function on some region ...
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2answers
71 views

Let f be an analytic function in the disk … Prove that

Let $f$ be a function analytic in the disk $D={z:|z-1-i| \leq 4}$ that satisfies $|f(z)| \leq 1$ for all $z$ in $D$. Prove that $|f^{(3)} (1-i)| \leq \frac{3}{4}$. Hint: apply Cauchy's Integral ...
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1answer
45 views

Residue theorem application [demonstration]

I really don't know how to solve this problem! Consider $F$, an analytic fuction, so that, $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial So, I tried to ...
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1answer
47 views

Holomorphic function on the open unit disc

Question: Let f be a holomorphic function on the unit disc $\{|z|<1\}$, which of the following is/are necessarily true? If for each positive integer n we have $f(1/n)=1/n^2$ then $f(z)=z^2$ on ...
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1answer
68 views

Number of zeros of a periodic function

Let's consider a periodic real function of a real variable $f(x)$. If the function is analytical and it is not the zero function, can one infer that the number of zeros in one period $[x,x+P)$ is ...
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40 views

Finding region where $f(z)=z^2\bar{z}$ is analytic.

How can I find a region where $f(z)=z^2\bar{z}$ is analytic ? I first let $z=x+iy$ ,then use Cauchy-Riemann equation and obtain $$u(x,y)=x^3+xy^2$$ $$v(x,y)=y^3+yx^2$$ $$u_x(x,y) = 3x^2 + y^2$$ ...
3
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2answers
66 views

$\frac{1}{z^2}$ is holomorphic

I have to show that $z\mapsto\frac1{z^2}$ is holomorpic on $\mathbb C\setminus\{0\}$ and compute its $n$-th derivative I know that $\frac{1}{z^2}=\sum\limits_{n\ge0}(-1)^n(n+1)(z-1)^n$, so it ...
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2answers
31 views

Composition of real-analytic functions is real-analytic

Suppose $f,g: \mathbb{R} \to \mathbb{R}$ are real analytic, i.e, locally given by convergent power series. Then $g \circ f$ is real-analytic as well. How do I prove this? I guess the "standard" proof ...
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2answers
33 views

Properties of holomorphic functions (demonstration)

I don't know how to do this demonstration: "If f is an holomorphic function, and M $\in \mathbb{R}^+$, such that for $z \in \mathbb{C}$, $|f(z)| \leq M(1+ |z|^n)$, then f is a $n$ or less degree ...
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1answer
32 views

Prove that an entire complex-valued $f$ is constant.

If a complex-valued function $f = u + iv$ is entire with $uv = 3$ for all $z \in \mathbb C$, then $f$ is constant. $f$ is not constant $\rightarrow f^2 = (u^2 - v^2) + 2iuv$ is not constant. Since ...
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What are some general strategies to build measure preserving real-analytic diffeomorphisms?

One could prove the following theorem in the smooth setting: Theorem Let $(M,m)$ be a $d$ dimensional $C^\infty$ manifold with smooth volume $m$. Let $\{F_i\}_{i=1}^k$ and $\{G_i\}_{i=1}^k$ be ...
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4answers
52 views

Complex Analysis. Analytic functions

How could I solve this problem?: "Supose an open set A $\subset$ $\mathbb C$ , so that $A^*= \lbrace z \in \mathbb C : \bar{z} \in A \rbrace$. If f is an analytic function in A, demonstrate that ...
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0answers
23 views

Degree of zero of a family of real analytic functions on a common interval

Given a family of real analytic functions on a common interval on the real line expanded about a common zero of the family of analytic functions, what can be said about the multiplicity of this zero ...
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0answers
21 views

If the result of differentiating a function converges can we claim that there are no singularities in the function?

I was trying to understand an answer to another question of mine Showing Weierstrass Elliptic Function is meromorphic in which the answerer has used "You can differentiate the function term by ...
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0answers
43 views

Show that the function $f(z)$ is analytic

Question : If $\phi$ and $\psi$ ae function of $ x $ and $y$ satisfying laplace's equation . Show that $f(z) = s + it$ is analytic , where $$ s = \frac{\partial \phi}{\partial y} - \frac{\partial ...
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0answers
22 views

Complex Function With No Singularities

Suppose it is given that a function f is meromorphic (no singularities except poles) and now if in any region it is given that f has no poles also, then can I assume that f is analytic/holomorphic in ...
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37 views

Branch cut for arcsin(z)

I am referring to this particular example found here: http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/arcsin.pdf On page one, I have difficulty understanding the region where $Arcsin(z)$ is ...
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3answers
48 views

An Elliptic function can not be holomorphic/analytic?

I was reading about elliptic functions on the wiki and it said that a doubly periodic meromorphic function in contention of being an elliptic function can not be analytic/holomorphic as it would then ...
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0answers
37 views

Can a meromorphic function which does not have any poles in a domain be shown to be bounded in that domain?

Basically such a function would not have any singularities in that domain and is completely analytic on it. Then, is the function bounded on that domain?
2
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0answers
68 views

Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...