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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25 views

How to apply Little Picard Theorem to prove that $a$ is unique

Little Picard Theorem: If a function $f : ℂ→ℂ$ is entire and non-constant, then the set of values that $f(z)$ assumes is either the whole complex plane or the plane minus a single point. Assume that ...
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13 views

Suppose f and g are analytic on a domain G

Suppose f and g are analytic on a domain G. If f and g are non-constant, then for any b in G, there exists a punctured disk D'(b,R) of radius R>0 such that f(z)g(z) is not equal to g(z)-f(z) + 1 for ...
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27 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 ...
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1answer
70 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 ...
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1answer
26 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?
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22 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, ...
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1answer
37 views

If composition with a linear functional is continuous, is the function continuous?

If $G$ is an open subset of $\mathbb{C}$ and $f:G \to X$ is a function such that for each $x^*$ in $X^*$, $x^*\circ f:G\to\mathbb{C}$ is analytic, then f is analytic. Will the statement still hold ...
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20 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 ...
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29 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 ...
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1answer
36 views

The Identity Theorem for real analytic functions

What is the condition for two real analytic functions to be identically equal? We know that there is a nice condition (Identity Theorem) for holomorphic function to check if they are the same. What is ...
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22 views

Showing a function is not analytic

Let $g(x)=e^{-1/x^2}$ for $x\neq0$, and $g(0)=0$. I've shown that $g^{(n)}(0)=0$ and thus that it is infinitely differentiable about 0, but now I must show that it cannot be expressed as a power ...
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44 views

Proofs for $g(x)=e^{-1/x^2}$ when $x\neq0$, and $g(x)=0$ when $x=0$

Sorry for the non-descriptive title - the question is a bit long. I have $g(x)$ as in the title, and we proved previously that $g'(0)=0$ using L'Hôpital's rule. Now I must show by induction that ...
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34 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 ...
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2answers
27 views

Inequality regarding entire function

Let $f$ be entire function. Must there exists $R>0$; such that $|f(z)| \leq |f'(z)|$ for all $|z|>R$ ,OR $|f'(z)| \leq |f(z)|$ for all $|z|>R$ ?
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17 views

Analytic Function, taking value 0 on an open ball

Let $f: D \rightarrow \mathbb{C}$ be analytic on a domain $D$. Suppose that $f(z)=0$ for all $z \in B(z_0, r) \subset D$ for some $z_0 \in D$ and $r>0$. Must $f \equiv 0$ on $D$?
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21 views

flabbiness of hyperfunctions

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...
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109 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 ...
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1answer
75 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 ...
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30 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 ...
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1answer
79 views

Analytic Continuation of Riemann Zeta Function

How do we show that this holds for $\operatorname{Re}(z)>0$ (and not $1$) $$\zeta(z)= \sum_{i=0}^{m-1} n^{-i} + \frac{m^{-z}}2 +\frac{m^{1-z}}{z-1} -z\int_{m}^{\infty} ...
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1answer
41 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 $D$?
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80 views

Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
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76 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 ...
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2answers
66 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: ...
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1answer
42 views

Complex Conjugation question

I had a complex analysis exam yesterday, and one of the questions is bothering me. Suppose $f(z)$ is an entire function. Show that $g(z) = (f(z^*))^*$ is also entire. Here $^*$ indicates complex ...
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74 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
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1answer
42 views

Can we deduce that the zeros of $g$ are also isolated?

Let $f:Ω→ℂ$ be a non-zero holomorphic function and $g:Ω→ℂ$ be a non-zero non-holomorphic function. We know that all the zeros of $f$ are isolated. Assume that $$f(s)=0⇒g(s)=0$$ Can we deduce that the ...
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67 views

On Cauchy-Riemann equations

Given $f:\mathbb C\to \mathbb C$ is a non-constant entire function. Then which of the following is possible? Re $f(z)=$ Im $ f(z)$, Im$\,f(z)<0$, Re$\,f(z)$ is bounded, $f(z)\neq 0,$ for all ...
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22 views

Show that $f$ and $g$ are holomorphic in the set $D=({s=α+iβ∈ℂ: 0<α<1})$

Let us consider two complex functions $g,f$: $$g(α+iβ)=∑_{n=2}^{m}(-1)ⁿ⁻¹((n^{2α-1}-1)/n^{α})n^{iβ}$$ $$f(α+iβ)=(-1)^{m}(((m+1)^{2α-1}-1)/(m+1)^{α})(m+1)^{iβ}$$ My question is: Show that $f$ and ...
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48 views

use example to prove the sum of two nonanalytic functions can be analytic [closed]

Find two functions, each of which is nowhere analytic, but whose sum is an entire function.
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265 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
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24 views

Prove the following property of holomorphic functions.

Let $\rho(x)$ be a holomorphic function on a disk $D \subseteq \mathbb{C}$ with the property that $\rho(x) \notin \mathbb{N^*} = \{1,2,\dots\}$ on $D$. Prove the following: There exists an $R$ ...
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Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
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Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
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118 views

let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant

$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant... Then we are supposed to use this result to ...
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4answers
225 views

Define a branch of $(z^2 − 1)^{1/2}$ which is analytic in the unit disk.

Hint: $z^2 − 1 = (z + 1)(z − 1)$. I'm really struggling with this question. I understand that for this function to be analytic it has to be differentiable in some neighbourhood, but I have no idea ...
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14 views

Milne-thompson for polar form

How can we readily apply MilneThompson method to find the conjugate function if the complex variable is given in polar form? If a complex variable is given as $u(x,y) + i v(x,y)$ then its $f'(Z) ...
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1answer
60 views

Why do u,v components in Cauchy-Riemann conditions are irrotational?

It's very strange to me! When we decompose a complex function to a real part and an immaginary part, we have $f(z) = u(x,y) + j v(x,y)$ following the conditions of analyticity we can derive the ...
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1answer
25 views

Polar partial derivatives continuously differentiable implies holomorphic

I need to show that if $f(re^{i\vartheta}) = U(r,\vartheta) + iV(r, \vartheta)$ and $U,V$ are continuously differentiable and satisfy the Cauchy-Riemann equations, then $f$ is holomorphic. I am ...
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3answers
58 views

Corresponding analytic function?

I have found a general harmonic function of form $a x^3 - 3dx^2 y - 3axy^2 + dy^3$ and it's harmonic conjugate $v = 3ax^2y - 3dxy^2 + ay^3 + dx^3 + K$ where k is constant. I now am asked to find the ...
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1answer
81 views

Analytic continuation of complex square root

This example comes from the book by Elias Wegert: Visual Complex Functions. Consider the function $f(z):=z^{1/2}$ for $z \in \mathbb{C}$ with $|z - 1| < 1.$ For these $z$, the function can be ...
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1answer
65 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
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1answer
50 views

Commutativity of integration and Taylor expansion of the integrand in an integral

I am baffled with a seemingly a straightforward problem. Suppose we are given the following integral: \begin{equation} f(a)\,=\,\int_{0}^{\infty} \frac{x^4}{x^4+a^4} e^{-x}, \end{equation} and we ...
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1answer
112 views

$f $ is analytic and maps the unit disk to itself. Prove that $|f'(0)|\leq1- |f(0)|^2 $

I am having difficulties with the following problem: $\bf Given$: $f $ is analytic and maps from unit disk to itself. $\bf Prove:$ $|f'(0)|\leq1- |f(0)|^2 $. For some reason (unclear to me) it ...
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1answer
34 views

Real analyticity of $\sqrt{x}\coth(\sqrt{x})$ : function that is decreasing with resp. to derivation

I'm trying to show that $\frac{\sqrt{x}}{\tanh(\sqrt{x})}$ is real analytic at $x=0$ (with the principal determination of $\sqrt x$). Apparently (i.e. thanks to graphs with Maple) $n \mapsto ...
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1answer
33 views

When does analytic continuation respect functional equation

This is a subtle point about analytic continuation. Let $\Gamma(s)$ be the analytic continuation of $\gamma(s) := \int_0^\infty e^{-t}t^{s-1}dt$ to $\Bbb C \setminus \Bbb Z_{<0}$, the latter ...
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2answers
70 views

$f$ analytic and $|f|$ a function of $|z|$

Suppose $f$ is analytic inside the unit disc and that $|f(z)|$ depends only on $|z|$. Prove that we can write $f(z) = Cz^N$, for all $z$ in the disc. In the suggested proof, it is stated like it's ...
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2answers
49 views

Show that $\log|\sin(z)|$ is the real part of a holomorphic function

$D$ is a connected, simply connected domain with $\sin(z)$ never zero on D. Show that $\log|\sin(z)|$ is the real part of a holomorphic function. My question is: how to show $\sin(z)$ maps a simply ...
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1answer
87 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...