Tagged Questions
4
votes
4answers
96 views
what is the relation of smooth compact supported funtions and real analytic function?
What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
2
votes
1answer
38 views
Analyticity implying Constant
Question: $f(z)$ is analytic in $C$ and $Im(f(z))\leq 0$. I want to show that $f(z)$ is a constant.
Approach: I know that if $f$ is analytic on a closed curve then the line integral along that curve ...
1
vote
0answers
40 views
Using Montel's Theorem to show locally uniform convergence of analytic functions
If $f_n :U \to \mathbb{C}$ is a sequence of analytic functions, where $U$ is open and connected. And there exists a point $z_0 \in U$ such that for all $k \geq 0$ the sequence $f_n^{(k)}(z)$ ...
2
votes
1answer
46 views
Showing a sequence of analytic functions converges locally uniformly
Let $f_n :U \to \mathbb{C}$ be a sequence of analytic functions on an open and connected set $U$. Suppose that the sequence is locally bounded and that for the set $$D:= \{z \in U : f_n(z) \, \, ...
0
votes
0answers
34 views
Finding Analytic Functions that Satisfy Certain Conditions
How would one go about finding:
All analytic f such that $|f''(x)|>|e^{7z}|$?
All analytic functions such that $f(z) = z + f(z^2)$.
All linear functions that map from the punctured unit disk to ...
2
votes
1answer
70 views
$f, g$ are analytic at $z_0$. Then $f/g$ is analytic at $z_0$ iff $g(z_0) ≠ 0.$
How to show
$f, g$ are analytic at $z_0$. Then $f/g$ is analytic at $z_0$ iff $g(z_0) ≠ 0.$
using $f, g$ are differentiable at $z_0$. Then $f/g$ is differentiable at $z_0$ iff $g(z_0) \neq 0.$
...
3
votes
1answer
43 views
When does a function differentiable in a domain become constant
In the following result
$f$ is differentiable on a domain $D$ & i) $\Re f$ is constant or ii) $\Im f$ is constant or iii) $\arg f$ is constant or iv) $|f|$ is constant on $D\implies f$ is ...
1
vote
2answers
52 views
Complex analytic function on a line
a) Let $D$ be a domain whose boundary $C$ contains a straight-line segment $L$. Let $f(z)$ be analytic in $D$ and continous on $L$. Assume also that $\Im(f) = v(x,y)$ vanishes on $L$. Prove that ...
1
vote
2answers
38 views
Check for analyticity of a complex function
Prove that $f(z)=|z|^4$ is differentiable but not analytic at $z=0$
My Attempt : $|z|=\sqrt{x^2+y^2} so |z|^4=(x^2+y^2)^2$
Now, we see that at $z=0$ all the four partial derivatives are equal to ...
2
votes
2answers
46 views
Analytic functions of a real variable which do not extend to an open complex neighborhood
Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
1
vote
1answer
33 views
Continuous dependence of zeros on a parameter
Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals.
Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation
$$f_\lambda(x)=0\,.$$
Assume its ...
2
votes
1answer
57 views
If $ f:\Delta \mapsto \Delta $ be an analytic and bijective mapping.
My qyesation is whether it imply $ f (z)=kz $ for $ k \in\mathbb {C} $ such that $| k|=1 $ .where $\Delta :=\{z\in \mathbb { C}: |z|<1\} $
2
votes
1answer
73 views
Prove that $f: (a,b)→ℂ$ cannot have infinitely many zeros in $(a,b)$
I have the following nonzero analytic function: $f:ℂ→ℂ$. We will consider only the restriction $f: (a,b)→ℂ$, $a,b∈ℝ$ and $a<b$.
My question is: Prove that $f: (a,b)→ℂ$ cannot have infinitely many ...
1
vote
1answer
48 views
Can Cauchy theorem be applied to $\log{(z)}e^{ixz}$?
I'm reading about asymptotic analysis on the integral $I(x)=\int_0^1{\ln{t}e^{ixt}}dt$. The book tells me that I can use Cauchy theorem to deform the contour into a rectangular contour:0->iT, ...
4
votes
2answers
38 views
Is a function being analytic considered as a local property?
Sorry for being pedantic... I was just wondering if analyticity of a complex function considered as a local property? Apparently differentiability is considered as a local property. But analyticity ...
1
vote
1answer
83 views
Assume that the set of values where $f^{(k)}≠0$ is finite
Let $f:ℝ→ℝ$ be a real analytic function. Let $f^{(k)}$ be the $k$th derivative of $f$. Assume that the set of values where $f^{(k)}≠0$ is finite, then what we can say about the function $f$.
2
votes
1answer
54 views
Is the inverse of a real analytic function still analytic?
If $f:D\to D'$, with $D, D'$ open subsets of $\mathbb{C}$, is a complex analytic invertible function with non-zero derviative, it's easy to see that $f^{-1}:D'\to D$ is analytic too.
Indeed complex ...
0
votes
1answer
29 views
Zeros of the analytic limit of complex rational function
For $n\in\mathbb{N}$ let $r_n,\ s_n$ be two polynomials of $O(n)$ degrees with real positive coefficients and set $f_n=r_n/s_n$.
Suppose there exists $c>0$ such that
$\bullet$ if $z\in\mathbb{C}$ ...
0
votes
1answer
38 views
Zeros of the analytic limit of complex polynomials
For $n\in\mathbb{N}$ let $p_n$ be a polynomial of degree $n$.
Suppose there exists $c>0$ such that
$\bullet$ if $z\in\mathbb{C}$ is a zero of a $p_n$, then $|z^2+c|\leq c$ (note that in particular ...
2
votes
1answer
52 views
Zeros set of analytic functions over complex plane with several variables
I know that the zeros of analytic function (with one variable) over complex plane are isolated. However, I am not aware about the structure of the zeros set of analytic functions over complex plane ...
3
votes
1answer
45 views
Bound for analytic function from unit disk into punctured unit disk
Suppose $f$ is analytic in the unit disk $D$ and satisfies $0<|f(z)|<1$. Show that $|f(z)|\leq|f(0)|^{\frac{1-|z|}{1+|z|}}$ for all $z\in D$.
I tried to work with $\log|f|$. It seems that ...
1
vote
1answer
56 views
Behavior at infinity.
Classify the behavior at $\infty$ for $$f(z)=\frac{\sin z}{z^2},\,g(z)=\frac{1}{\sin z},\,h(z)=\exp\left(\tan\frac{1}{z}\right).$$
So I just considered $f(1/z),g(1/z),h(1/z)$ at $z=0$. For $f$ I ...
1
vote
1answer
74 views
Radius of convergence of Maclaurin series for $\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$
What is the radius of convergence of the Taylor series about $z=0$ for $h(z)=\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$?
Here's a plot
...
1
vote
2answers
59 views
Removable singularity and laurent series
How to show $z=\pm\pi$ is a removable singularity for $\frac1{\sin z}+\frac{2z}{z^2-\pi^2}$?
I tried to compute the Laurent series, specifically the coefficients for $1/z,1/z^2,...$ since if we can ...
0
votes
1answer
62 views
Showing a bounded analytic function on strip is identically zero
Let $f$ be analytic and bounded on $\{x+iy\in\mathbb{C}:|y|<\frac{\pi}{2}\}$. Suppose $f(\ln n)=0$ for all $n\in\mathbb{N}$. Show that $f$ is identically 0.
I tried to perform some transformations ...
0
votes
0answers
38 views
Prove that $f(s)=εf(2-s)$
Let $f$ be an analytic function defined by
$$f(s)=N^{-s/2}(2π)^{s}Γ⁻¹(s)∑_{n=1}^{∞}(a_{n}/n^{s})(F_{n}(s-1)-εF_{n}(1-s))$$
where $$F_{n}(t)=Γ(t+1,2πn/√N).(√N/2πn))^{t+1},ε=±1$$ and ...
2
votes
1answer
56 views
Evaluate $\int_C\frac{dw}{e^w-1}$ over some loop C contained in the annulus $0<|z|<2\pi$.
Evaluate $\int_C\frac{dw}{e^w-1}$ (counterclockwise) over some loop C contained in the annulus $0<|z|<2\pi$.
Considering the coefficient of $1/z$ in the Laurent series for $\frac{1}{e^z-1}$ by ...
1
vote
1answer
57 views
Bounded imaginary part implies removable singularity at 0
Let $f$ be a holomorphic function on the punctured unit disk. If the imaginary part of $f$ is bounded, is it true that $f$ has a removable singularity at 0?
I see that $|e^{-if}|=e^{Im\;f}$ so ...
0
votes
1answer
46 views
Does replacing analytic by continuity in open mapping theorem holds the result? and is it true for closed sets?
1) My first question depends on the open mapping theorem
Which state that a non constant analytic function maps open sets onto open sets
does the result holds if we use continuous function instead of ...
2
votes
1answer
81 views
Can someone please explain Morera's Theorem and the Schwarz Reflection Principle?
I have been reading Complex Analysis, Third Edition by Joseph Bak and Donald J. Newman and am stuck with chapter 7 which deals with the mentioned theorem. I have read through the chapter at least ...
1
vote
1answer
42 views
Analytic continuation of function given as integral
I have a function $I(D)$ defined by the following integral representation
$$
I(D)=\int_0^\infty\mathrm{d}\alpha\,(1+2\alpha)^{-D/2}
$$
which is clearly only sensible for $D>2$. The result of the ...
3
votes
2answers
73 views
If $f$ is analytic in a disk $|z|<R$ then so is $g(z)=\overline{f(\bar z)}$ in the disk
How to prove that if $f$ is analytic in a disk $|z|<R$ then $g(z)=\overline{f(\overline z)}$ is also analytic in the disk and also $f=g$ iff $f$ is real valued in $(-R,R)$
4
votes
1answer
99 views
If $f$ is analytic where $f$ is represented as $f=g.h$ where $g$ is analytic . From here can we conclude that $h$ is analytic?
If $f$ is analytic, where $f$ is represented as $f=g \cdot h,$ where $g$ is analytic. From here can we conclude that $h$ is analytic?
-1
votes
1answer
79 views
Complex has became so hard after the min\max modulus principle. Need some proofs and examples. [closed]
1) $f(z)$ being non constant and analytic in a domain $D$
if $f(z)$ continuous on $\overline{D}$ and $|f(z)|$ is constant on the boundary
I need to prove that $f(z)$ must have a zero inside $D$!
2) ...
3
votes
1answer
106 views
If$f(z)$ is analytic , then what about $f'(z)?? $
If$f(z)$ is analytic , then what about $f'(z)$?
can we conclude that $f^{(k)}(z)$ is analytic for any k$\in $$ \mathbb{N} $
2
votes
1answer
152 views
Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.
I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
3
votes
1answer
79 views
Basic question about analyticity vs. differentiability in complex analysis.
In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula,"
3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
0
votes
3answers
43 views
some proofs in complex related to F.T of algebra and cauchy's inequality maybe
these two questions i didn't even find the way to solve
so please if you can help
1)suppose f(z) is entire with |f(z)| <= |exp(z)| for every z
i want to prove that f(z) = k exp(z) for some |k| ...
2
votes
1answer
62 views
Analyticity of $\frac{Log(z+4)}{z^2+i}$
This problem is from Churchill and Brown. How do I prove that
$f(z)=\frac{Log(z+4)}{z^2+i}$ is analytics everywhere except $\pm\frac{(1-i)}{\sqrt{2}}$ and on the portion $x \le -4$ of the real axis.
...
3
votes
2answers
43 views
Determining whether a family of power series is normal
How should I check whether a given family of power series forms a normal family? I am trying to apply Montel's theorem that says that a family of holomorphic functions is normal iff it is uniformly ...
0
votes
1answer
72 views
in complex analysis need some examples about uniform convergence and analyticity
I am new in solving anything in complex
and I am stuck on two examples :
1) I read that : The limit of the sequence of non-analytic functions converging uniformly inside a simple closed curve can be ...
0
votes
0answers
50 views
Internal point transformed in an external one?
Let $f \colon \Omega \to \mathbb{C} $ be an analytic function over a connected open subset $\Omega$ of $\mathbb{C}$ and let $\gamma$ a rectifiable closed curve in $\Omega$. If $a$ is a point which is ...
0
votes
1answer
36 views
Show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$
This is a homework problem.
For $|z| \le R$ and $|a_j| < R$ for $j=1,\ldots, k$, not all zero, show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$.
...
2
votes
0answers
89 views
Zeros of analytic function and limit points at boundary
Let $S$ be the open ball of center $0$ and radius $1$ with $0$ removed in the complex plane. Is the function $f(z)=\sin(1/z)$ a valid example of analytic function defined in an open subspace whose ...
2
votes
0answers
65 views
Prescribing zeroes, poles, principal parts and finitely many terms with positive exponents in Laurent series
I was given a problem to prove a theorem by Mittag-Leffler about prescribing the items in the title, using Weierstrass's theorem about prescribed zeroes and Mittag-Leffler's theorem about prescribed ...
11
votes
1answer
287 views
Images of compact subsets in the plane
Let $K$ be an infinite compact subset of $\mathbb{C}$. Is it true that there exists a sequence $(f_n)_{n>0}$ of functions holomorphic in some neighborhood of $K$, such that the images $f_n(K)$ are ...
4
votes
2answers
76 views
Analytic off the real axis
If $f:\mathbb C \longrightarrow \mathbb C$ is continuous and $f$ is analytic off the real axis, then show that $f$ is entire.
0
votes
0answers
24 views
Linearly dependent analytic functions [duplicate]
Possible Duplicate:
Problem on exponential of entire function
If f(z) and g(z) are entire functions such that $$
e^{f(z)},e^{g(z)}, {1} $$ are linearly dependent.
That is there is ...
1
vote
0answers
112 views
Schwarz reflection principle and bounded derivatives
Suppose $f$ is a holomorphic function on $\Omega^+$ (an open subset of the upper complex plane) that extends continuously to $I$ (a subset of $\mathbb{R}$). Let $\Omega^-$ be the reflection of ...
3
votes
1answer
85 views
Domain of bijectivity of function $f:\mathbb{C}\rightarrow\mathbb{C}$
There is a type of problems in my course in Complex analysis that I don't fully understand them.
Given function $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(z)=z^2$. You must specify the analytic and ...
