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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11 views

Branch of logarithm which is real when z>0

I am familiar with the complex logarithm and its branches, but still this confuses me. I read this in a textbook: "For complex $z\neq 0, log(z)$ denotes that branch of the logarithm which is real ...
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
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0answers
17 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 - ...
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0answers
16 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 ...
2
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1answer
43 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). $f(z)\ne 0$, ...
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0answers
27 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 ...
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2answers
23 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 ...
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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
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0answers
33 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 $E$, ...
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0answers
15 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 ...
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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
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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 ...
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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 ...
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vote
1answer
50 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
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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
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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} ...
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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?
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4answers
70 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 ...
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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!
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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.
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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
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1answer
29 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 ...
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1answer
36 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 ...
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0answers
36 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
26 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 ...
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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 ...
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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 ...
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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 ...
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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?
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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, ...
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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 ...
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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
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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 $$|z_2+z_3|$$
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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 ...
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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
33 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
240 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 ...