0
votes
1answer
17 views

Question about constructing complex entire function

How to construct an entire function for infinitely many prescribed values? i.e. I hope to find an entire function $f$ given $f(z_k) = w_k$ ($w_k$ might not be zero) for a sequence of $\{z_k\}$ with ...
1
vote
2answers
41 views

Complex function defined by contour integral along a smoothly varying path

Let $D$ be a domain in the complex plane. Consider the function $F: D\to \mathbb{C}$, defined by $$ F\left( z \right) = \int_{\mathscr{C}\left( z \right)} {f\left( {z,t} \right)dt} . $$ Suppose that ...
3
votes
1answer
36 views

Poles of Fourier transform

Let $f\in L_2(\mathbb R_+)$ and consider its Fourier transform $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ Is it true that analytic continuation of $F(\zeta)$ has at most finitely many poles in a ...
0
votes
1answer
33 views

Upper bound for modulus of a function

Let $f(t,x)$ be a bounded and continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_x$. Moreover, assume that for each fixed $t$, ...
0
votes
1answer
35 views

Does analytic at a point implies differentiable at that point?

A function $f:A \subset \Bbb C \to \Bbb C$ is said to be analytic if it is locally given by a convergent power series i.e. every $z_o \in A$ has a neighbourhood contained in $A$ such that there exists ...
1
vote
0answers
29 views

Zeros of the derivatives of a finite Blaschke product.

Let $B$ be an $n$ degree finite Blaschke product. By considering the level curves of $B$, one can show that $B'$ has $n-1$ critical points in the disk (counting multiplicity). Is anything known ...
1
vote
0answers
23 views

separate vs joint real analyticity

Let $$f(x,y) := xy\exp\left(-\frac{1}{x^2+y^2}\right),$$ if $(x,y)\neq (0,0)$ and $f(0,0):=0$. I read the claim that $f$ is (a) separately real analytic on $\mathbb{R}\times\mathbb{R}$ (i.e. for ...
2
votes
0answers
35 views

Non-Trivial Self-Inverse Analytic Function In The Complex Plane

Let $f$ be a complex-valued function which is analytic for all values of $z$ in the complex plane and which satisfies $$f(f(z))= z $$ i.e. it is a self-inverse function. The trivial solutions for $f$ ...
0
votes
1answer
31 views

Analytic Function In The Complex Plane Which Always Gives Real Values

Let $f$ be a non-constant, complex-valued function which is defined and analytic for all $z$ in the complex plane. Also, $f$ has the additional property that it is always real. To me, such a function ...
14
votes
3answers
291 views

What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
2
votes
2answers
37 views

Proof that Radius of Convergence Extend to Nearest Singularity

Can someone provide a proof for the fact that the radius of convergence of the power series of an analytic function is the distance to the nearest singularity? I've read the identity theorem, but I ...
1
vote
1answer
30 views

Analytic Extension: Imaginary Stripe

I was always wondering the following: Given a real analytic function there exists a positive radius of convergence for every point. This won't be affected by allowing complex numbers so it extends ...
1
vote
1answer
35 views

Uniformly analytic functions

Consider the following definition: Let $\Omega$ be an open set of $\mathbb{R}_x^n$, $x = (x_1, ..., x_n)$. A $\mathcal{C}^{\infty}$-function $\varphi(x)$ on $\Omega$ is said to be uniformly analytic ...
1
vote
0answers
33 views

Constructing an analytic continuation

I'm hoping someone could verify my answer to the following problem: Consider a function $f$ that is continuous for $Im(z) \geq 0$ and analytic for $Im(z) > 0$. Furthermore, assume that $f$ ...
4
votes
1answer
50 views

Boundary behaviour of holomorphic function on unit disk

Let $\mathbb{D}=\{z \in \mathbb{C} \ | \ |z|<1 \} $ be the open unit disk in the complex plane. I would like to see explicit examples of the following phenomena: a holomorphic function $f$ on ...
1
vote
1answer
40 views

Riemann removable singularity theorem for annuli

Let $\mathbb{D}^*=\{z \in \mathbb{C} \ | \ 0 < |z| < 1 \}$ denote the unit punctured disk in the complex plane. Riemann's theorem about removable singularities in particular implies the ...
0
votes
3answers
62 views

Does there exist an analytic function that satisfies these properties?

Does there exist an analytic function $f:\{z\in\mathbb{C}:0<|z|<1\}\to\mathbb{C}$ such that $\displaystyle\lim_{z\to0}[z^{-3}f^2(z)]=1$? I'm assuming that there is not such a function, so I've ...
2
votes
1answer
36 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
7
votes
1answer
137 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
1
vote
1answer
36 views

Identity theorem for polynomials in several variables

Let us assume that we are given two polynomials $f,g$ with real coefficients in several variables, say $x_1, \ldots, x_n \in \mathbb{R}$. Further, assume that $f_{|X} \equiv g_{|X}$, with $X$ being ...
0
votes
0answers
31 views

Uniqueness of Analytic Continuation

I wasn't very well introduced to Analytic Continuations, but from what I have seen, showing that the analytic continuation is unique is pretty simple. In Real Analysis, from what I can imagine, there ...
0
votes
1answer
37 views

Analytic continuation of a real function

I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique. Now i am wondering if this is ...
0
votes
0answers
40 views

entire function in complex analysis

$$ y = \left\{ \begin{array}{ll} \dfrac{\cos z}{z^2-\left( \dfrac{\pi}{2} \right)^2}, & z \ne \pm \dfrac{\pi}{2}\\ -\dfrac{1}{\pi}, & z = \pm \dfrac{\pi}{2}\\ \end{array} \right. $$ $$ ...
1
vote
1answer
15 views

Extending a holomorphic function to a radial limit function for almost every angle

I've read in several places about the "well known theorem" which states that a holomorphic function on the (open) unit disk $D=\{z\in\mathbb{C}:\ |z|< 1\}$ can be extended to its boundary on almost ...
0
votes
1answer
34 views

Calculating an integral with a branch cut, using some “uniqueness property”

Consider a complex function $$\tilde{f}(z)=z\int_{M}^{\infty}ds' \frac{\rho(s')}{z-s'} \qquad (1)$$ , where $M>0$ and $$\rho(s')=\frac{1}{s'}\sqrt{1-M/s'}.$$ This function is analytic in the ...
1
vote
1answer
28 views

Finding the number of zeros of $f(z) = z^n$ if $|f(z)| < 1 $ for all $z$ with $|z|=1$.

Suppose $f: \overline{\mathbb{D}} \to \mathbb{C}$ is continuous, analytic in $\mathbb{D}$ and satisfies $|f(z)|<1$ for $|z|=1$. Find the number of solutions to the equation $f(z) = z^n$ where $n$ ...
0
votes
0answers
86 views

An infinite compact set which allows no boundedness and analyticity

I need an example of an infinite compact set $K$ in $\mathbb {C}$ such that there does not exist any non-constant function which is both bounded and analytic on $\mathbb{C} - K$. First, any hints ...
1
vote
1answer
42 views

Integral Continuation $\Gamma(z)=\int_{0}^{1} e^{-t} t^{z-1} dt +\int_{1}^{\infty} e^{-t} t^{z-1}dt$

I am trying to obtain an analytical continuation for $\Gamma(z)$ into the region of the complex plane characterized by $\Re(z) \leq 0$ but am stuck. Starting from the integral definition of ...
0
votes
0answers
24 views

Show existence of an analytic which cannot be extended beyond the boundary

$G$ is an open strip $\{z:1<\text{Im } z<2\}$. Prove that there exists an analytic function $f(z)\in H(G)$ that does not extend analytically beyond any boundary point of $G$. Also determine ...
0
votes
1answer
24 views

To show a function is analytic

Let $G\subset\mathbb C$ be open and connected, and function $h$ is analytic on $G$. $\{f_n(z)\}$ is a sequence of analytic functions on $G$ for which $\lim_{n\rightarrow \infty}f_n(z)$ exists for any ...
1
vote
0answers
20 views

Existence and uniqueness of an analytic function

I'm reviewing complex for the exam and just got stuck here. Let $g$ be an analytic function at $z=0$. We want to show there exists a unique analytic function $f$ such that (1) $f(0)=0$ (2) ...
1
vote
1answer
19 views

Finding a Real valued Function to Create Holomorphism

I am asked whether it is possible to find a real function $v$ such that $$x^3+y^3+iv$$ is holomorphic. Should I basically be working backwards from the Cauchy-Riemann equations? That makes logical ...
1
vote
1answer
42 views

Integrating the function Im(z) on a variety of contours.

I've been asked to evaluate $\int_C Im(z) dz$ for a variety of contours, which I've had no issue in doing. For the sake of clarity, these contours included the upper and lower halves of the circle ...
0
votes
0answers
46 views

maximum modulus principle analytic function

I am trying to show that: Let $f$ be analytic on a given closed unit disc $D$ then prove that for every $k\in\mathbb N$ there is $w\in Bd(D)$ such that $|f(z)-w^{-k}|≥1.$ where z is in the unit disc ...
0
votes
0answers
46 views

Tthe inverse of a Mellin transform of a polynomial…

Let $\mathcal{M}$ be the symbol of the Mellin transform as define in http://en.wikipedia.org/wiki/Mellin_transform In a calculus, I finally end up with $$\mathcal{M^{-1}(f)}=\mathcal{P}$$ where ...
1
vote
0answers
48 views

How to explain this result due to Pôlya

How to explain this result due to Pôlya: An entire function is determined uniquely by the inverse images, counting multiplicities of three distinct non omited values. I cannot understand how this ...
0
votes
1answer
63 views

How to prove that $a$ is unique

Assume that $f : ℂ→ℂ$ is a non-constant non polynomial and entire function and there exist $a∈ℂ$ such that the fiber $f⁻¹(a)$ is finite. My question is: How to prove that $a$ is unique.
3
votes
1answer
36 views

Relation between continuity of $f$ and analyticity of $f(z)^8$

If $f(z)$ is continuous on some domain $D$ and $f(z)^8$ (the function to the eighth power, not the eighth derivative) is analytic, then why does this imply that f is analytic on a neighborhood of each ...
0
votes
1answer
196 views

Is f(z)=1/z truly an analytic function

For an analytic function $f(z)$, we have $$\frac{\partial f}{\partial \bar{z}}=0.$$ Consider the function $f(z)=\frac{1}{z}$, which, at first sight, is a bona fide analytic function. However, we can ...
0
votes
1answer
28 views

Analytic Map from $B(0,1)$ to $B(0,1)$

Is the analytic map from $B(0,1)$ to $B(0,1)$ such that $f(0)=1/2$ and $f'(0)=3/4$ unique?
0
votes
0answers
33 views

Maximum modulus principle, is it true?

Suppose f is analytic in an open set containing the open disk D(2+3i, 7) and its boundary circle C(2+3i, 7) such that |f(z) + 7i + 24|<25 for all z in C(2+3i,7). Then f has no zeroes inside D(2+3i, ...
0
votes
0answers
42 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
0
votes
0answers
34 views

Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
0
votes
0answers
45 views

Liouville's theorem for functions not bounded on an isolated set

On Liouville's theorem. Prove that if $f(z)$ is defined, analytic, bounded in the entire complex plane, except for an isolated set of points, then $f$ must be constant. What happens if said set has a ...
0
votes
1answer
142 views

Double exponential Taylor series $\exp(-\exp(k-ex))$

k is real constant $\gt = 1$. Is $a_n$ for $f(x)$ positive, increasing, and $\lt 1$, where $n\lt= e^{k-1}$? $$f(x) = \sum_{n=0}^{\infty} a_n x^n = \exp(-\exp(k-ex))$$ $f(x)$ is the double ...
1
vote
1answer
96 views

Proof of a result related to Liouville's Theorem

I was investigating the following corollary to Liouville's Theorem in Complex Analysis: if $f(z)$ is entire and $\lim_{z\rightarrow \infty}z^{-n}f(z)=0$, then $f(z)$ is a polynomial in $z$ of degree ...
0
votes
0answers
34 views

$f$ is analytic with range as a circle

I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...
1
vote
1answer
48 views

Complex Analysis analytic function

If $f$ is an analytic function on a domain $D$ and $\mathrm{Im} f$ takes on only the value $71$ then for some constant $C \in \Bbb{R}$, is it true that $f = C + 71 i$ on $D$?
0
votes
4answers
89 views

On the subject of holomorphic functions on an open disc, D.

The question I am pondering over is an interesting one: If $f(z) = u + iv$ is holomorphic on an open disc $D$, and the range of $f$ lies in either a straight line or a circle, prove that $f$ is ...
2
votes
2answers
84 views

Prove that the entire function $f$ is linear.

Suppose $f=u+iv$ be an entire function such that $u(x,y)=\phi(x)$ and $v(x,y)=\psi(y)$ for all $x,y\in\mathbb{R}$. Prove that $f(x)=az+b$ for some $a\in\mathbb{C},b\in\mathbb{C}$. My approach was: ...