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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51 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
11 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
36 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
32 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
61 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
31 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
26 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 ...
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1answer
18 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
30 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
25 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
23 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
28 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
66 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 ...
2
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1answer
40 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 ...
2
votes
1answer
65 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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0answers
38 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 ...
0
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2answers
24 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 ...
4
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2answers
31 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
29 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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0answers
12 views

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 ...
3
votes
4answers
49 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 ...
2
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0answers
14 views

A multivariable real analytic funtion related to a family of multi-complex-variable analytic functions

Let $D \subset \mathbb{C}^n$ be a open region. Let $f_k(k=1,2,...)$ are holomorphic on $D$ and $u(z):=\sum_{k=1}^{\infty}|f_k(z)|^2$ is uniformly convergent on every compact subset of $D$. Show that ...
0
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0answers
19 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
42 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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0answers
35 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
40 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
35 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?
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0answers
59 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 ...
0
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1answer
38 views

Does analytic continuation apply only to analytic functions?

I'm a high school senior attempting to do a project on the riemann zeta function. I've looked online, tried reading college textbooks but still don't have a completely clear idea of analytic ...
2
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0answers
55 views

Is the determinant an analytic function?

I came accross a paper stating that the analytical property of determinants of complex matrices allows us to use some theorem for analytic functions. I am not able to confirm this since I am not sure ...
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0answers
18 views

Is definite integral of such function multiplication analytic?

If $f(x)$ is a general function (integrable) and $g(s,x)$ is an analytic function except for on its poles. Then, can some one judge about $$H(s)=\int_{a}^b f(x) g(s,x) dx $$ Is $H(s)$ analytic ...
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1answer
39 views

The radius of convergence of a power series about a point interior to the domain of an analytic function

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real analytic function with domain an open, non-empty set $(a, b) \subseteq \mathbb{R}$, $-\infty \leq a < b \leq \infty$ and let $c \in (a, b)$. ...
2
votes
1answer
61 views

Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$

Let $U\subset \mathbb{C}$ be a nonempty connected open set such that for every $z\in U$, $\overline z\in U$. Let $f$ be analytic on $U$. Suppose $f(x)\in\mathbb R$ for every $x\in U\cap\mathbb R$. ...
2
votes
1answer
93 views

If $f$ is analytic, prove that $\overline{f(\overline{z})}$ is also analytic

Let $f$ be an analytic function in an open set $U \subseteq \mathbb{C}$. Let $V=\{z\in\mathbb C:\overline z\in U\}$. Define $g$ on $V$ by $g(z)=\overline{f(\overline{z})}$. Show that $g$ is analytic ...
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3answers
35 views

Are the coefficients of power series expansion for a real analytic function bounded?

Are the coefficients of power series expansion for a real analytic function bounded? $f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n$ We have a sequence $\{a_n\}, n=0,\cdots,\infty$. Is this sequence bounded? ...
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0answers
49 views

Analytical solution to nonlinear ode

I solved this equation that I attached numerically in matlab by the Newton Raphson method. Now I want to solve it analytically in matlab or even in Maple if it is possible. Would you please help me ...
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1answer
99 views

Computing Complex Integral to Determine Analytic Continuation of $f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt$

My question is the following: Find the analytic continuation of the function $f(z)$ defined by $$ f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt, \ \ \vert \arg(z) \vert < {1 \over ...
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0answers
44 views

Why is $1/z$ analytic at infinity?

I was given this proof: Let $w(z)=1/z$, so $w$ maps origin to inifinity and infinity to origin. Consider $f(z) = z$. It has no singularities in finite $z$-plane. So $f(w) = 1/w$ has a pole at the ...
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0answers
29 views

Analytic branch of root

I'm trying to prove that if $f$ is holomorphic function in an open subset $G$ of the complex plane, and $z_0 \in G$ is a zero of $f$ of order $m$ then there exist a branch of the $m$th root of $f$ - ...
0
votes
1answer
25 views

Finding all the analyitical function in the unit annulus that satisfy a given condition for natural numbers

Let $f$ be an analytic function in the annulus $0 < |z| < 1 $ such that it's singularity in $z=0$ is not essential. I want to find all of such functions $f$ that satisfy for $n = 3, 4,...$: ...
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0answers
21 views

Showing that $n^z$ for $n\in\mathbb{Z}$ and $z\in\mathbb{C}$ is analytic

I just had a quick question. Can I just say that because we know $n^m$ for any integer $m$ is entire, and since we know that $z$ is entire, then the composition, $n^z$ is entire?
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2answers
65 views

A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function

I am looking for an example of a uniformly convergent sequence of real analytic functions which does not converge to a real analytic function. Also I would appreciate any pointers on how to think of ...
4
votes
1answer
84 views

If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts

If all the roots of a polynomial $P(z)$ have negative real parts, prove that all the roots of the derivative $P'(z)$ also have negative real parts. Could anyone provide a proof for this please?
3
votes
0answers
68 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
1
vote
1answer
32 views

Holomorphic function is zero on an analytic set then $df=0$.

Assume we have an homomorphic function $f:U\rightarrow \mathbb{C} $ which is holomorphic on the open set $U$ of $\mathbb{C}^n$. Assume there is $V\subset U$ analytic and that $f$ restricted to $V$ ...
2
votes
0answers
29 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...