The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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9
votes
1answer
273 views

Is there a $k$ such that $a_n=\frac{n^k!}{(n^k!!)^2}$ converges?

Lately I have been playing around with the sequence $$a_n(k) := \frac{n^k!}{(n^k!!)^2}.$$ For $k=1$, it does not look much like it converges. I don't know $k=2$ it converges, but it doesn't really ...
5
votes
1answer
126 views

What is the analytic continuation of a multifactorial?

The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials? I am particularly interested in the double factorial. All Google has ...
0
votes
1answer
842 views

Finding the accumulation point

I need to determine whether the following set has accumulation points: $0 \le \arg z<\pi/2 (z\ne 0)$ Would the accumulation point be z=0, as the set does not include 0? If not, does it not have ...
2
votes
1answer
71 views

Holomorphic function - inequalities

$f$ is holomorphic on some neighborhood of $\lbrace z\in\mathbb{C}: \frac{3\pi}{2}\leq |z|\leq\frac{5\pi}{2}\rbrace$. On both $\lbrace z\in\mathbb{C}:|z|=\frac{3\pi}{2}\rbrace$ and $\lbrace ...
3
votes
3answers
215 views

Holomorphic functions on unit disc

Let $f,g$ be holomorphic on $\mathbb{D}:=\lbrace z\in\mathbb{C}:|z|<1\rbrace$, $f\neq0,g\neq0$, such that $$\frac{f^{\prime}}{f}(\frac{1}{n})=\frac{g^{\prime}}{g}(\frac{1}{n}) $$ for all natural ...
4
votes
2answers
101 views

For which $a$ does the equation $f(z) = f(az) $ has a non constant solution $f$

For which $a \in \mathbb{C} -\ \{0,1\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$. My attempt: First, we can see that any ...
0
votes
1answer
161 views

$f,g$ be two analytic function on unit disc st that $f\cdot g=0$.Does it imply$ f=0$or$ g=0$

let $f,g$ be two analytic function defined in a unit disc of the comlex plane such that $f\cdot g=0$. Does it imply either $f=0$ or $g=0$? If it is not true the give an counter example or any hints.
2
votes
3answers
177 views

Is $f(z)=\exp (-\frac{1}{z^4})$ holomorphic?

Let $f(z)=\exp (-\frac{1}{z^4})$ for $z\neq 0$ and $f(0)=0$. Is it obvious that $$\lim_{z\to z_{0}}\frac{f(z)-f(0)}{z-0}=\lim_{z\to z_{0}}\frac{\exp (-\frac{1}{z^4})}{z}=0$$ And if this ...
1
vote
0answers
61 views

Proving that $|\sin z|> |1/z+i|$, $z$ is a complex number

I already found a proof using the argument principle for a more general version. Prove the function $f(z)= \sin z +\frac{1}{z-a}$ has infinitely many zeros in the strip $|\mathrm{Im}z| < \epsilon.$ ...
2
votes
0answers
56 views

for any $a$ belongs to $\mathbb{C}$ $f(z_i)=a$ holds for $z_1,z_2,..;z_n$ [duplicate]

Possible Duplicate: Proof that a certain entire function is a polynomial Let $f$ be an entire function such that for every $a \in \mathbb{C}$ there are $z_1,z_2,.....,z_n \in \mathbb{C}$ ...
0
votes
1answer
320 views

Differences between Cauchy integral theorem and fundamental theorem for integral calculus over a cycle

There is a theorem (a complex analogous to the fundamental theorem of calculus) that states that if $f$ is a continuous function having a primitive in a region of the complex plane containing a ...
1
vote
2answers
139 views

the definition of Riemann zeta function

$\alpha^z=e^{z\log\alpha}$ is multi-valued. Now I am confused with the definition of Riemann zeta function: $$\zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}, s=\sigma+it$$ because $$n^s=e^{s(\log ...
1
vote
1answer
69 views

derivative quotient of holomorphic function as a contour integral

I have forgotten much of my complex analysis, so I would appreciate some help with the following. Suppose $f$ is holomorphic and $\Gamma$ is a circle of radius $r$ about $0$. Why is ...
3
votes
1answer
165 views

Show that $\frac{z-1}{\mathrm{Log(z)}}$ is holomorphic off $(-\infty,0]$

Let $f(z)=\frac{z-1}{Log(z)}$ for $z\neq 1$ and $f(1)=1$. Show that $f$ is holomorphic on $\mathbb{C}\setminus(-\infty,0]$. I know it looks like an easy problem, but I got stuck and need some ...
0
votes
2answers
555 views

How to prove that composition of two conformal functions is conformal

Let $\hat{\mathbb{C}}$ =$\mathbb{C}\cup\{\infty\}$. A theorem from my lecture notes says that a function $f: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is conformal iff f is a linear fractional ...
4
votes
3answers
167 views

Show that the familiar logistic map $x_{n+1} = sx_n(1 - x_n)$, can be recoded into the form $x_{n+1} = x_n^2 + c$.

What change of variables would trtansform the logistic equation into the Mandelbrot equation $z_{n+1}=z_n^2+c$?
5
votes
2answers
205 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.
12
votes
2answers
397 views

A definite integral with hyperbolic cosines

I want to show that $$ \int_{0}^{\infty} \frac{\cosh (ax) \cosh (bx)}{\cosh (\pi x)} \ dx = \frac{\cos ( \frac{a}{2} ) \cos ( \frac{b}{2})} {\cos (a) + \cos (b)} \ , \ |a|+|b| < \pi.$$ I thought ...
5
votes
2answers
325 views

Are there any continuous functions from the real line onto the complex plane?

Is there any measurable continuous differentiable analytic surjective function $f:\mathbb{R}\to\mathbb{C}$?
0
votes
1answer
248 views

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = ...
1
vote
1answer
55 views

Calculating a limit with constraints

Given the function $f(x)$, $$ f(x,y,z,w) = \frac{x+iy}{\sqrt{|w+z|}} \text{.} $$ How do I calculate the limit $$ \lim\limits_{w\rightarrow -z} f $$ under the constraint that the points $(x,y,z) ...
1
vote
2answers
167 views

A real valued function on the complex plane taking a complex number to its real part is an Open map?

A real valued function on the complex plane taking a complex number to its real part is an Open map, just a hint please, first of all is it a continuous map? I did one problem in complex analysis ...
1
vote
0answers
77 views

Finding a bound on $f'$

$f$ is an entire function such that $|f(z+w)| \leq |f(z)| + |f(w)|$ for any $z, w $ in $\mathbb{C}$. I need to show that $f(z) = az +b$ for some complex numbers $a$ and $b$. So, it suffices to show ...
16
votes
3answers
2k views

Why isn't several complex variables as fundamental as multivariable calculus?

One typically studies analysis in $\mathbb{R}^n$ after studying analysis in $\mathbb{R}$. Why can't the same be said $\mathbb{C}$?
0
votes
2answers
271 views

Convergence radius of power series for different centers

This is a solution to an exercise in complex analysis: I don't quite understand the argument in the red box. In a deleted neighborhood of $z_0$, say $B(z_0,r)\setminus\{z_0\}\subset\{z\in{\Bbb ...
12
votes
2answers
325 views

Is $\frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?

Let $z$ be a complex number. Is the alternating infinite series $ f(z) = \frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ an entire function ? Does it even converge ...
2
votes
1answer
114 views

Is the union of two locally closed sets in real line, locally closed?

Formal Definition: A locally closed set in a toplogical space, is an intersection of an open and a closed set. My Definition: A locally open set in a toplogical space, is a union of an open and a ...
2
votes
1answer
1k views

Analytic functions branch

I am having trouble understanding branch cuts. It seems right when I understand one thing another issue arises. The questions asks: Find a branch of each of the following multiple valued functions ...
0
votes
3answers
163 views

does there exist an entire function with this property?

We need to show that there is no polynomial $P$ with complex coefficients such that $P(n)=(-1)^n\forall n\in \mathbb{Z}$. Does there exist an entire function with this property? Hints only, please.
3
votes
3answers
757 views

Prove that the entire function $f$ is a polynomial if it maps every unbounded sequence to an unbounded sequence.

While studying for an exam in complex analysis, I came across this problem. Unfortunately I was not able to solve it. Any help would be greatly appreciated. Let $f$ be an entire function mapping ...
14
votes
4answers
313 views

Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.

While studying for an exam in complex analysis, I came across this problem. Unfortunately I was not able to solve it. Any help would be greatly appreciated. Let $U ⊂ \mathbb{C}$ be a domain and $f_1, ...
0
votes
1answer
193 views

The inverse of Cauchy's Integral Theorem

Cauchy's 1st integral formula : let $f(z)$ be analytic in simply connected domain $D$ containing a simple closed contour $C$ . If $z_0$ is inside $C$ then $$ f(z_0)=\frac{1}{2\pi i} \int_C\frac ...
4
votes
2answers
349 views

integral of the complex function $1/\cos(1/z)$

I am looking for $\underset{|z|=1}{\oint}\frac{1}{\cos\left(\frac{1}{z}\right)}dz$ I was able to do the following: ...
0
votes
3answers
117 views

to show $f$ is a polynomial in $z$

given that $f$ is an analytic function with real part $u$ is a polynomial in the variable of $x,y$, $z=x+iy$, we need to show $f$ is a polynomial in $z$, I am kind of puzzled to see the problem, first ...
-2
votes
1answer
84 views

Integer solutions of a complex equation

The topic of this question is about integer solutions of a complex equation. Let $m,r≥1$ two integers and $α,β∈(0,1)$. Let $f$ be an analytic function but without known or closed formula. I know that ...
2
votes
1answer
249 views

Questions regarding the complex logarithm and complex integration

I'm currently preparing for an exam in complex analysis and I don't quite feel comfortable with some exercises, mostly those including a complex logarithm and some "unusual" paths to integrate along. ...
5
votes
2answers
176 views

how to show that $\lim_{z \to 0}z^z$ does not exist?

What makes $0^0$ indeterminate. Here is a video by numberphile that claims that $z^z$ does not exist as $z \to 0$ where $z \in \mathbb C $. I tried tried $\lim_{x \to 0}(x+ix)^{(x+ix)}$ and replaced ...
2
votes
2answers
102 views

Power series convergence in boundary, regular point?

Given a power series $\sum_{k=0}^\infty a_k z^k$ with radius of convergence $0<R<\infty$ . Given a point in the boundary of the circle $z, |z|=R$, is there a relationship between $z$ being a ...
3
votes
2answers
437 views

Convergence power series in boundary

Say I have a power series $\sum_{k=0}^\infty a_k z^k$ with radius of convergence $0<R<\infty$. What can be said topologically about the set $\{z\in\Bbb C\mid |z|=R\,\mbox{ and }\sum_{k=0}^\infty ...
1
vote
2answers
2k views

Branches of analytic functions

Find a branch of $\log(z^2+1)$ that is analytic at $z=0$ and takes the value of $2\pi i$ there. Also, determine a branch of $\log(z^2+2z+3)$ that is analytic at $z=-1$. If I plug in $z=0$ and ...
0
votes
1answer
98 views

Analytic in the domain

Show that the function $Log(-z) + i\pi$ is a branch of $logz$ analytic in the domain $D_0$ consisting of all points in the plane except those on the nonnegative real axis. I know that ...
0
votes
1answer
117 views

Fréchet mean of the spherical shape space

The Fréchet mean of a general subspace is defined as $$F(x)=\int_M dist(x,y)^2d\mu(y),$$ where $\mu$ is the probability measure on a general metric space $(M,dist)$. I think the sample mean of ...
6
votes
1answer
332 views

Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
0
votes
1answer
49 views

roots with strictly negative real part

Consider the equation $$\lambda+1-\alpha e^{-\lambda\tau}=0\;,$$ where $\lambda\in\mathbb{C}$, $\alpha\in\mathbb{R}$, and $\tau>0$. I need to establish conditions on the parameters $\alpha$ and ...
3
votes
1answer
254 views

removable singularity

f(z) is analytic on the punctured disc $D(0,1) - {0}$ and the real part of f is positive. Prove that f has a removable singularity at $0$.
1
vote
1answer
97 views

Complex Integral of $ f(z)=z\cdot \exp(z^2)$

Any hint for calculating the integral of $ f(z)=z\cdot \exp(z^2)$? on $r=[i, i+2]$, on $t=\{x+ix^2:0\leq x\leq 1\}$ thx!
13
votes
1answer
403 views

Proof of Cauchy's Beta Integral $\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}$

The Cauchy's Beta Integral is given by $$\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}=\frac{\pi 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ I would like to know how it is proved.
2
votes
1answer
78 views

$|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$

$f$ is analytic function on open unit disk, and $|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$, just a hint please.
2
votes
3answers
268 views

Why the zeta function?

Why is the zeta function, $\zeta(s)$ used to obtain information about the primes, namely giving explict formula for different prime counting functions, when there are many other functions that encode ...
1
vote
0answers
192 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 ...