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, ...

learn more… | top users | synonyms (2)

4
votes
1answer
25 views

Singularities in Complex Analysis

Determine the singular points of the following functions, the nature of these singular points and compute the residues in these points. $$(a)\:\dfrac{\cos z}{z^3},\qquad (b)\:\dfrac z{\sin ...
1
vote
0answers
31 views

Prove a function attains each complex value [on hold]

Let $p$ and $q$ be non constant polynomials. Prove that the function $f(z) =p(z) e^{q(z)} $ attains every complex value.
0
votes
1answer
22 views

Intuition for the limit of complex functions

We have intuition and somehow geometrical point of view about the limit in the Real functions.I mean we can think of the limit on the graph of the Real function and imagine how close we can get on the ...
1
vote
1answer
27 views

Find an harmonic function in $\mathbb R^n$ which is a polynomial of degree 4 and is 1 at the origin

Find an harmonic function in $R^n$ which It is a polynomial of degree 4 and is =1 at the origin. It is a polynomial of degree 5 and its partial derivatives are both =0 at the origin. Important ...
0
votes
1answer
27 views

Using De Moivre's theorem with relation to the argument of a complex number

Given that $Z^4 = 64(\cos\pi+ i\sin\pi)= 64(-1+0i) = -64$ I understand that the argument [$arg(Z^4)$] is $\pi$, now if instead given the form $Z^4 =64(-1+0i)$ and I desired to find the argument ...
3
votes
0answers
20 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
-2
votes
1answer
45 views
2
votes
2answers
61 views

A function is real-differentiable iff it has a complex-differentiable extension

Is this conjecture true? A function $f:\Bbb R\to\Bbb R$ is real differentiable at $a$ if and only if there exists a complex-differentiable function $g:A\to\Bbb C$ for some neighborhood of $a\in ...
2
votes
1answer
45 views

What's the difference between the different types of poles, zeroes and singularities in complex analysis?

I am trying to get an understanding on the difference between the different types of poles, zeroes and singularities in complex analysis and how to identify them. When is it a removable singularity, ...
6
votes
1answer
45 views

A generalization of Bell numbers to arbitrary complex arguments

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because ...
-3
votes
0answers
20 views

Finite Blaschke product [on hold]

I want to do my research in Finite Blaschke products. Also, I want to enter graduate school in the US. I'm a final year mathematics student in Sri Lanka.
0
votes
0answers
12 views

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
2
votes
1answer
53 views

Prove or disprove the existence of polynomials.

Prove or disprove that there exist non-constant polynomials $p$ and $q$ such that $p(z)e^{p(z)}+q(z)e^{q(z)}=1$ for all $z\in \mathbb{C}$.
0
votes
2answers
58 views

Show that $e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}$ for all $y \in \mathbb{R}$ with $|\mu_1| \leq 1$

How to show the expansions \begin{gather*} e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}\\ e^{iy} = 1 + iy - \frac{1}{2}y^2 + \frac{\mu_2 |y|^3}{3!} \end{gather*} where $y \in \mathbb{R}$ and $|\mu_1| \leq 1$ ...
1
vote
1answer
21 views

proof verification analytic function

Exists an analytic function $g(z)$ such that $g^{'}(z)= \frac{1}{z^{2}-1}$ in the annulus $1<|z|<2$. My answer is yes i calculate the integral of $g$ which is $\frac{1}{2}(\log(z+1) + ...
1
vote
1answer
34 views

Question on radius of convergence

Can anyone help me with the following problem: I have a solid geometric picture of what is going on in my head, but I can't seem to turn that into a proof.
0
votes
1answer
61 views

not following two steps in proof that $\int_{0}^{\infty} cos(x^2) = \frac{\sqrt{2 \pi}}{4}$

Hi: I'm reading some notes I found on complex analysis on the internet. In the example, they prove that $$\int_{0}^{\infty} \cos(x^2) = \int_{0}^{\infty} \sin(x^2) = \frac{\sqrt{2\pi}}{4}.$$ I ...
2
votes
2answers
68 views

Do there exist nonconstant holomorphic functions $f$ and $g$ on the open unit disk such that $e^{f(z)}+e^{g(z)}=1$?

This is a question from an old qualifying exam that I was trying to solve for practice: Prove or disprove that there exist nonconstant holomorphic functions $f$ and $g$ on the open unit disk ...
1
vote
1answer
20 views

Computing the radius of convergence of a given series

Could anyone help me with the following problem? I'm getting $1$ as the answer, but I found the solution (without justification) to this problem online and it says the answer is $1/2$. Determine ...
0
votes
1answer
17 views

Showing uniform continuity of function giving radius of convergence

Let $f$ be an analytic function on an open disk $D$ and let $R(z)$ denote the radius of convergence of the power series of $f$ about a point $z$. Is there an easy way to show that $|R(z_1) - R(z_2)| ...
0
votes
1answer
22 views

zeros of Sequence of Analytic Function [on hold]

Let $f$ be a non-constant entire function. Suppose that there is a sequence of polynomials ${P_n(z)}_\{n=1,2...\}$ such that i) $P_n(z)$ converges uniformly to $f$ on every bounded set in ...
0
votes
1answer
23 views

Complex numbers and quadratic equation [on hold]

Have to convert it to rectangular form $$z^4-(1+j)z^2+j=0$$ Please help me do it fast.... Thanks
2
votes
0answers
33 views

Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...
1
vote
1answer
37 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
0
votes
0answers
39 views

Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$ \int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0 $$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
-2
votes
0answers
35 views

Complex Analysis Fundamental Theorem of Calculus [on hold]

$(a)$ Calculate $\int_Cz^2\,\mathrm dz$ when $C$ is the straight line joining $0$ to $i$. $(a)$ Calculate $\int_Cz^2\,\mathrm dz$ when $C$ is any piecewise smooth simple arc joining $0$ to ...
2
votes
1answer
38 views

A question about complex power series.

My book says the following: Let $\sum_n a_nz^n$ be a convergent complex power series with radius of convergence $r$. Then there exist $C,A\in\mathbb{R}$ such that $|a_n|<CA^n$, where ...
2
votes
1answer
29 views

Is it possible to realize a general compact Riemann surface in $\mathbb CP^2$?

Let $X$ be a compact Riemann surface with smooth boundary $\partial X$. Is it always possible to realize $X$ as a complex submanifold of $\mathbb CP^2$? In other words, is it true that there exists a ...
2
votes
0answers
23 views

Analytic cohomology on Zariski site vs analytic cohomology on analytic site

If I have an affine algebraic complex manifold (in fact it is Stein), what is known relating the cohomology of analytic sheaves using only Zariski opens vs the cohomology of analytic sheaves using the ...
1
vote
1answer
28 views

Radius of convergence of series given radius of convergence of another series

I'm hoping someone might be able to verify my solution to the following problem: Suppose that the series $\sum c_n z^n$ has radius of convergence $R$. Find the radius of convergence of the $\sum ...
1
vote
1answer
23 views

holomorphic functions 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 $f(D)$?
0
votes
1answer
21 views

Cauchy-Riemann and Analytic Functions

Using the Cauchy-Riemann conditions, tell if $f(z) = z^{*}$ is analytic I have tried this: $Z = x + iy$ $f(x + iy) = Z^{*} = x - iy$ $U(x,y) = x$ $V(x,y) = -y$ $U_x = 1$ Deriving respect to $x$ ...
1
vote
1answer
29 views

An example of a complex power series. [on hold]

I am looking for a complex power series which is convergent for some $z\in\Bbb{C}$ but not absolutely convergent. In other words, $a_0+a_1z+a_2z+\dots$ is convergent but ...
2
votes
2answers
87 views

Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
5
votes
2answers
121 views

Find a conformal map from semi-disc onto unit disc

This comes straight from Conway's Complex Analysis, VII.4, exercise 4. Find an analytic function $f$ which maps $G:=$ {${z: |z| < 1, Re(z) > 0}$} onto $B(0; 1)$ in a one-one fashion. ...
3
votes
1answer
38 views

Conformal mapping of the domain bounded by a line segment and a circular arc

I am trying to construct a conformal map from the region $R$ which is the set of points in the complex plane bounded by the segment connecting $i$ and $1$ and the part of the unit circle in the first ...
2
votes
0answers
45 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
4
votes
0answers
57 views

Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want ...
1
vote
2answers
34 views

Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
-3
votes
0answers
25 views

Please justify this statement: “[A] holomorphic function is (n+1)-to-1 near a zero of its derivative of order n”.

Another member of the community posted this in one of their answers to a question a few years back and I can't seem to understand why this is true. Help?
2
votes
2answers
31 views

Usage of the term $\arg(z)$

Consider the complex number $z = -1 - i$. Is it mathematically correct to say that $\arg(z) = 5\pi/4$? Sure, $5\pi/4$ is not the principle argument of $z$, but it is an element of the set $\arg(z)$. ...
1
vote
0answers
33 views

At most one connected component of unbounded portion of entire function.

Suppose $f$ is an entire complex analytic function and $R$ a positive real number. Define the set $E:= \{z\in\mathbb{C};|f(z)| < R\}$ to be the set of $z$ whose image is bounded by $R$. I want to ...
2
votes
1answer
97 views

Why does convolution not maintain asymptotic equality of coefficients?

Assume I have four (generating) functions $f$, $f'$, $g$ and $g'$. If that is interesting, we can assume that they all share the same radius of convergence $\rho > 0$. In addition, we know that ...
2
votes
1answer
29 views

Quotient of 2 holomorphic functions which may be holomorphic

Let f and g be two holomorphic functions on a domain $\Omega$. Suppose that $\frac{f}{g}$ is always finite (while g can be zero at some points). Is it true that then $\frac{f}{g}$ is holomorphic? ...
2
votes
1answer
109 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
0
votes
0answers
26 views

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}$

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}; f(z)=\frac{1}{(z-i)^2}$ and $f(z)=e^{(z-1)^-1}$ Good evening, I have been trying to solve the above exercises. However, I'm not sure if my procedure ...
3
votes
1answer
31 views

Conformal mapping between regions symmetric across the real line

In Conway's Functions of One Complex Variable, the section on the Riemann Mapping Theorem has the following exercise: Let $G$ be a simply connected region which is not the whole plane, and suppose ...
1
vote
1answer
23 views

Sketching regions is complex plane

When sektching the region $\left|\frac{2z-1}{z+i}\right|$$\geq$1 on the argrand diagram, how should we go about identifying the region, should we take $\left|2z-1\right|\geq\left|z+i\right|$ or ...
0
votes
1answer
21 views

Describe the family of analytic functions with the following properties:

Find the family of all functions $f$ analytic in $\mathbb{D}$ (the open unit disk) and continuous on $\overline{\mathbb{D}}$ such that $|f(z)|=e^{\text{Re}(z)}$ for all $z\in\mathbb{D}$. My intuition ...
0
votes
0answers
34 views

Entire function with $L^2$ modulus is identically zero [duplicate]

I want to show that if $f$ is entire and $\int_{\mathbb{R}^2}\left| \:f\: \right|^2 < \infty$, then $f \equiv 0$. I was thinking of assuming $f$ is not identically zero; then, since a bounded ...