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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2
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1answer
22 views

Finding Inverse Fourier Transform of found Fourier Transform

So I have the equation $f(t) = e^{-at^2}$ where $a>0$. First I found the Fourier Transform by solving $F(k) = \int_{-\infty}^{\infty}e^{-at^2}e^{-kt}dt$ from this i got $F(k) = ...
2
votes
0answers
13 views

Question about an extension of an analytic function

Setting: Let $\Omega \subseteq \mathbb{C}$ and suppose that $f$ is analytic on $\Omega' = \Omega - \{a\}$. Suppose that $a \in \Omega$ satisfies the crucial property that $$ \lim_{z \to a} (z - ...
0
votes
0answers
15 views

functions orthogonal to the exponentia Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
1
vote
1answer
38 views

single simple pole, $e^{i\theta } \frac{1-\overline{z_0}z}{z-z_0}$

$D= \big\{z:|z|\leq 1\big\}$, $|z_{0}|\in D$. A function $f(z)$ such that 1).$f(z)$ is analytic on $D\setminus \{z_0\}$; 2).$f$ has a single simple pole at $z_0$ ; 3). $f(z)\ne 0$, ...
0
votes
0answers
26 views

Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and lim |z(f(z)|= 0 as ...
0
votes
2answers
20 views

Uses of Jacobian of a map on $\mathbb{R}^n$.

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix} \frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial ...
1
vote
1answer
14 views

Integreal around a unit circle

I know that when $m \in \mathbb{Z} \backslash \{ 0 \}$, we have $$ \int_0^1 e^{2 \pi i m \beta} \ d \beta = 0. $$ I was wondering if there is a simple formula for the following similar integral, when ...
3
votes
0answers
32 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
0
votes
0answers
14 views

Proving that the function set $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set

I have the the following problem from my Fourier analysis book: Show that $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set in $PC(0,l)$, i.e. class of piecewise ...
0
votes
0answers
9 views

Proving an inequality on $G_n(z)=a_0^n G_0(z)+n a_1^{n-1} G_1(z)+\frac{n (n-1)}{2} a_2^{n-2} G_2(z)+…$

Hypothesis: 1) $a_{n,r}, a_{n,i}\in \mathbb R$ such that $$a_n=a_{n,r}+i a_{n,i}\in \mathbb C$$ and $$|a_{n,r}|\leq k<1,\ \ \ |a_{n,i}|\leq h<1$$ 2) $|z|\leq R, \ \ |G_m(z)|\leq M m!$ ...
2
votes
1answer
23 views

Residue question on contour integral

I'm asked to evaluate the following contour integral $$\int_{\gamma} \frac{1}{(z^4+1)(z-3)}$$ where $\gamma$ is the circle of radius $2$ centered at the origin and travelled once in the ...
6
votes
1answer
25 views

Proof for Complex Analysis Inequality

This is a homework assignment that will be graded; so I'm not specifically asking for an answer. But I could use a hint, as it's been a few days and I'm still not sure if how I've proved it would ...
1
vote
1answer
48 views

Proving $\lim_{R \to \infty} \frac{1}{2 \pi i}\int_{\gamma_{R}} \frac{p(z)}{q(z)}\,dz = \frac{a_0}{b_0}$

Let $m$ and $n$ be integers with $m>n>0$. Let $q(z)$ and $p(z)$ be polynomials of degree $m$ and $n$ $$p(z) =a_0z^n+a_1z^{n-1}+\cdots+a_n \text{ and }q(z)=b_0z^m+b_1z^{m-1}+\cdots+b_m$$ Let ...
4
votes
0answers
62 views

Delicate Integral $I=\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set a=0 we get a similar integral given by $$ ...
3
votes
1answer
21 views

How can the winding number change under a holomorphic map?

This question comes from an old complex analysis qual. First denote $\mathbb{C}^{\times} = \mathbb{C} \backslash \{ 0 \}$, $u = \{ e^{it} : 0 \leq t < 2 \pi \}$, and let $f : \mathbb{C}^{\times} ...
1
vote
3answers
70 views

Are there complex solutions for $z^3=\bar z$

I'm asked to solve $z^3=\bar z$. I got $z=0, 1, -1$. Are there any complex solutions $a+bi$ to this though?
2
votes
4answers
69 views

Values for $(1+i)^{2/3}$

This question might seem easier than I'm making it seem. But how many values are there for $(1+i)^{2/3}$? Do I let $z=(1+i)^{2/3}$ so that $z^3=2i$? I'm asked to write each in polar coordinates and in ...
1
vote
2answers
27 views

An integral involving two variables and the floor function

Let $N$ be some fixed positive integer. I have the following function $$ g(z) = z \int_1^N [t] e^{2 \pi i t z} \ dt. $$ How would one compute $$ \int_0^1 g(z) \ dz ? $$ Thanks!
1
vote
2answers
52 views

$\zeta(2 + it) = \zeta(2-it)$

Let $\zeta(s)$ denote the Riemann zeta-function. Show that $\zeta(2 + it) = \zeta(2-it)$ for all real t. Give some hints how to do this one.Thanks in advance.
0
votes
1answer
18 views

relatively compact family of analytic function

Let $\Phi$ be the family of all analytic functions $f(z) = z + a_2z^2 + a_3z^3 + \cdots$ on the open unit disc such that $|a_n| \leq n$. Show that $\Phi$ is relatively compact. Show me the right ...
2
votes
1answer
28 views

How to obtaining the lattice corresponding to an elliptic curve

Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding ...
0
votes
1answer
35 views

solve the functional equation

Let $\phi : R-> C $ (complex numbers) $\phi(0)=1$ $ \phi(-t) = \overline{\phi(t)} $ ( continuous and bounded) solve the functional equation: $Re \phi(t)= \phi(t) \overline{\phi(t)}$ This is all ...
0
votes
0answers
34 views

Poisson summation formula

Let $f$ be an even real valued function, $\hat f(n)$ the $n^{\text{th}}$ Fourier coefficent of $f$ and $F$ the Fourier transform of $f$. I found the following problem in my lecture notes in which I ...
2
votes
1answer
25 views

Harmonic conjugates on annulus slit

Let $D$ be an annulus slit with $$D= \{a<|z|< b \}$$ excluding $(-b,-a)$. Show that any harmonic function on $D$ has a harmonic conjugate on $D$. The hint says to fix $c$ between $a$ and ...
0
votes
2answers
30 views

Need help proving the power series of a function.

We want to show that $$\log(1-z)=-\sum_{k=1}\dfrac{z^k}{k},\quad z\in D=\{z: |z|<1\}$$ I honestly have no idea where to start so if someone would please help guide me through the problem that would ...
1
vote
2answers
19 views

need help finding a closed form function.

I need help to find a function in closed form "not a power series" for: $\sum_{k\geq2} k(k-1)z^k$. I am not quite sure how to start the problem, i tried using the derivative but that did not work ...
0
votes
1answer
22 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
0
votes
1answer
27 views

Can an entire, non-constant function map $\mathbb{C}$ to a proper subset of $\mathbb{C}$?

Can an entire, non-constant function map the complex plane to a proper subset of the complex plane? And, by what theorem?
1
vote
1answer
31 views

Can an entire, non-constant function map the complex plane to a an open bounded set?

Here is a question from Conway. Let $f$ be entire and non-constant. For any positive real number $c$ show that the closure of $\{z: |f(z)| < c\}$ is the set $\{z: |f(z)| ≤ c\}$. If an entire, ...
1
vote
1answer
33 views

Strong maximum principle

Let $S^{n-1}$ denote sphere in $\mathbb{R}^n$ and let $D$ denote open unit disk in $\mathbb{R}^n$. Let $f$ be homeomorphism of $S^{n-1}$ onto itself. Let $F$ be its harmonic extension given by Poisson ...
0
votes
0answers
19 views

Help with some argument

Let $U$ be an neighborhood of $0\in{\mathbb{C}^2}$. And $K=\{(z_1,0):|z_1|<\rho\}$ be a subset of $U$, $L=\{z:|Rez_1^k|\leq \epsilon |z_1|^k\}$, where $\rho=\rho(\epsilon)$. Then i am supposed to ...
1
vote
0answers
24 views

Show there exist an positive integer $N$ such that $|e^z|>|\sum_{r=N}^{\infty}\frac{z^r}{r!}|$

$z$ an complex number, my question is how to show that there exist an positive integer $N$ such that $|e^z|>|\sum_{r=N}^{\infty}\frac{z^r}{r!}|$. (We know $\sum_{r=0}^{\infty}\frac{z^r}{r!}=e^z$) ...
2
votes
3answers
34 views

Complex Numbers of Unit Modulus

if $z_1$, $z_2$ and $z_3$ are Complex Numbers of Unit Modulus Such That: \begin{equation} |z_1-z_2|^2+|z_1-z_3|^2=4 \tag{1} \end{equation} Find the value of $$|z_2+z_3|$$
0
votes
1answer
14 views

Bernoulli numbers identity with binomial coefficient

The generating function for the Bernoulli numbers $B_k$ is given by $f(z) = \frac{z}{e^z -1}= \sum_{k=0}^{\infty} \frac{B_k}{k!} z^k$. Applying the identity $$1 = \frac{e^z -1}{z} \cdot ...
2
votes
1answer
26 views

About the implicit funtion in a holomorphic situation.

Let $f(x,y)$ be a polonomial with integral coefficients which has a zero $(a,b)\in \mathbb{R}^2$ such that the partial derivative respect to $y$ at this point is nonzero. Then by the implicit function ...
2
votes
1answer
45 views

Does $\mathrm{Im}(f(z))$ bounded above $\implies$ $|f|$ is bounded, for analytic $f$?

If $f$ is analytic on $\Omega$ s.t. $\mathrm{Im}(f(z))$ is bounded from above, then does this imply that $|f|$ is itself bounded? I know that if $\Omega = \mathbb{C}$, then the result follows as a ...
0
votes
1answer
29 views

Show that for $|f(z)| \leq C (|z| + 1)\log(|z| + 1)$, there is an $a$ such that $f(z) = az$

Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and suppose a $C \geq 0$ exists such that \begin{align*} |f(z)| \leq C(|z| + 1) \log(|z| + 1) \end{align*} for all $z \in \mathbb{C}$, where $\log: ...
3
votes
1answer
25 views

Showing that $a$ is a removable singularity if $\mathrm{Im}(f(z))$ is bounded from above

Problem: Suppose $f$ is analytic on the domain $\Omega$ except at the isolated singularity $a \in \Omega$. Show that $a$ is a removable singularity if $\mathrm{Im}(f(z))$ is bounded from ...
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0answers
12 views

Schwarz–Christoffel-like mapping on differentiable simple cubic spline boundary

For a concept of a computer game I have in mind I came to need that. I have a 2D pond, which has a boundary that is a simple differentiable cubic spline. There are ducks floating around, looking at ...
2
votes
2answers
42 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
3
votes
1answer
37 views

Computing $\int_{\gamma} {dz \over (1-z)^3}$

(a) Let $\gamma$ be the circle of radius ${1 \over 2}$ centered at the origin, oriented counter-clockwise. Compute $$ \int_{\gamma} {dz \over (1-z)^3} $$ (b) Same as above, ...
3
votes
1answer
26 views

When does conformal equivalence guarantee the existence of a “conformal homotopy”?

Suppose $f$ is a conformal equivalence between two domains $D_1$ and $D_2$ in $\mathbb{C}$. Does this imply the existence of a map $F_t(z): D_1 \times [0, a] \rightarrow \mathbb{C}$ such that each ...
0
votes
0answers
31 views

Caccioppoli inequality

Assume we have established the following version of Caccioppoli inequality $$\int |\nabla u|^2 \psi^2 dA\leq C \int u^2 |\nabla \psi| ^2 dA$$ for $C^2(\mathbb C)$- smooth functions $u\geq 0$ with ...
1
vote
1answer
25 views

Trying to find conformal map

I'm trying to find a one to one map from $ \{z \in \mathbb{C}: |z-1| < 1 \}$ to the right half plane $Re(z)>0$ My approach: I'm trying to come up with a map that takes $|z-1|<1$ to $0$. ...
0
votes
1answer
23 views

Schwarz Lemma Question $|f(z)| \le \left |\frac{z-1/2}{1-1/2z} \right|$

Let $U$ be the open unit disk $D= \{ z \in \mathbb{C} : |z|<1\}$. Suppose $f: U \to U$ is analytic on $U$ with $f(1/2)=0$. Show that $$|f(z)| \le \left |\frac{z-1/2}{1-1/2z} \right|=\left ...
1
vote
0answers
12 views

Real Hypersurfaces In Complex Manifolds

I have a problem: ================= I don't understand (2.12) and (2.13) :( How to prove that $$PF=\sum_{\min(k,l) \le 1}F_{kl}+G_{11}\left \langle z,z \right \rangle +\left ( G_{10}+G_{01} ...
0
votes
1answer
47 views

Proving $f$ must be a polynomial no more than $n$

Suppose $f: \mathbb{C} \to \mathbb{C}$ is analytic on all $\mathbb{C}$ and that there is a polynomial $p$ of degree $n$ and a point $z_0$ such that $|f(z)| \le |p(z)|$ for all $z$ with $|z| \ge ...
10
votes
1answer
239 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
3
votes
1answer
38 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
0
votes
1answer
18 views

Application of Residues

So in applying the residue theorem to solve improper real integrals, we agree to take our semicircles to be as large or as small as necessary such that all the poles we wish to work with lie inside ...