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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29 views

Distance to infinity in the complex plane

I am trying to understand calculating the distance between a point in the complex plane and infinity by using the project of the point, say $z_0$ onto the Riemann sphere. We can define the distance ...
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1answer
17 views

Implicit change of path of contour integral

I want to compute a contour integral say: $\oint_{C_z} \frac{dz}{S(z)-a}$ or $\oint_{C_z} g(S(z))dz$. The problem is that I don't have an explicit form for S(z); rather I have the implicit equation $z=...
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3answers
44 views

Example of analytic function that maps circle to self intersecting curve

Is there an explicit example that an analytic function maps the unit circle to a self intersecting curve? As unit circle is not homogenous to self intersecting curve, I am considering finding an ...
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1answer
58 views

Operator theory to study a difference equation

I'm not an expert in operator theory (so I'm going to be very informal sorry), but I would like to be given some advice about a problem I have. Let $f$ be a function defined in $C^{\infty}(\mathbb{R})$...
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19 views

Zeros of Eisenstein Series?

I'm wondering if anyone knows how to, or has seen any literature on, analytically handling zeros of Eisenstein series? Or perhaps, asymptotics on the zeros? For example, I'm specifically interested ...
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5answers
4k views

What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
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2answers
83 views

$f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|\leq 2\}$ such that $\iint_D=|f(z)|^2\,dx\,dy\leq 3\pi$. Maximize $|f''(0)|$

Determine the largest possible value of $|f''(0)|$ when $f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|<2\}$ with the property that $\iint_{D}|f(z)|^2\,dx\,dy\leq 3\pi$. I don'...
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1answer
25 views

Any convex Reinhardt domain is logarithmically convex

I have the following question in Shabat p.59: Prove that any convex Reinhardt domain is logarithmically convex. I think I have a good idea about how to show this, but need to be clear on the ...
3
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3answers
61 views

Show that $f$ is identically zero in $\mathbb{C}$

Let $f$ be holomorphic in $\mathbb{C}$. Prove that if $|f(z)| \leq M|z|^{\alpha}$ with $0 <\alpha <1$, then $f$ is identically zero in$\mathbb{C}$ I know that $f$ has a Taylor expansion $f(...
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0answers
12 views

$f(z)$ analytic in the panctured disk $\{|z|<1 \}\setminus\{0\}, \text{Im}(f(z))>0$, then $z=0$ is removable singularity

Let $f(z)$ be analytic in the panctured disk $\{|z|<1 \}\setminus\{0\}$ and let $\text{Im}(f(z))>0$. Prove $z=0$ is removable singularity. I try to show that $f(z)$ is bounded near $z=0$ but ...
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2answers
609 views

Proving $f$ has at least one zero inside unit disk

Let $f$ be a non-constant and analytic on a neighborhood of closure of the unit disk such that $|f(z)|=\text{constant}$ for $|z|=1$. Prove $f$ has at least one zero inside unit disk. I thought of ...
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0answers
33 views

$\wp$ via Jacobi triple product

$$\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c $$ $$\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right)$$ Then ...
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1answer
41 views

Find all solutions of $e^{e^z}=1$ in the complex space.

Find all solutions of $e^{e^z}=1$ in the complex space. Attempt: $e^{e^z}=1$. Assuming $e^z$ is a complex number, I will start off solving $e^z=e^{x+yi}=1$: $e^x(\cos y+i\sin y)=1\Rightarrow \sin y=...
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1answer
60 views

Jordan curve theorem for a polygon

Let P be a Jordan polygon on the complex plane. I would like to prove Jordan curve theorem for P using an elementary method. It seems that we can prove it using the winding number with respect to P. ...
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2answers
40 views

Analytic Functions

Prove or give a counter-example: If $f_j(j=1,2,...,n)$ is analytic on the domain $D$ such that $\sum_{j=1}^n |f_j(z)|^2$ is constant on $D$. Then each $f_j$ is a constant function. Inputs: We know ...
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1answer
14 views

Linear combination of a real-valued function and its inverse is analytic Implies the real-valued function is analytic.

If $u$ is a real-valued function on a disc $\Delta_R$ such that $u^{-1}+iu$ is analytic on $\Delta_R$, then does this imply that $u$ is analytic on $\Delta_R$? I am actually trying to prove some ...
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2answers
78 views

Complex Analysis: Zeros of an analytic function

What approach should I take to solve the attached problem. I was looking along the lines of 'Great Picard Theorem', which states that 'If an analytic function f has an essential singularity at a point ...
3
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2answers
37 views

Calculation of Fourier transform

How to calculate the Fourier transform of $f(x)=x$. I know using the formula $f(\varepsilon)=\int_xe^{-ix\varepsilon}x \, dx$. But I have problem calculating this complex integral.
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0answers
14 views

Proving a set is connected using the definition of Relatively Open set Of a set in $C$

I came across this definition Let $U\subset S\subset C$. We say that $U$ is relatively open in $S$ if for every $z_0 \in U$, there is $r > 0$ such that $$D(z_0 ;r)\cap S\subset U$$: Now the ...
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1answer
33 views

True or false on two statements(about entire function)

$f (z)$ is an entire function. If $f (x)$ is bounded for all real x, then f is a constant function. If | f (z)| → ∞ as |z| → ∞, then f is a polynomial. Can you tell me how to judge these 2 ...
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2answers
28 views

Bound on derivatives of holomorphic function

Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function on a open and bounded subset $U$ of $\mathbb{C}$. Then suppose $\frac 1 C <|\frac{df}{dz}|<C$ for a certain constant $C>0$. Does this ...
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3answers
25 views

Find the real and imaginary parts of an equation

Find the real and imaginary parts of $\frac{1}{3z+2}$ So I have expanded it out to get $\frac{1}{3x+3iy+2}$ Thus giving $Re(\frac{1}{3z+2})=\frac{1}{3x+2}$ and $Im(\frac{1}{3z+2})=\frac{1}{3y}$ ...
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1answer
40 views

Hadamard-like complex variable substitution

\begin{align} \frac\pi a &= \int_{-\infty}^\infty dxdye^{-a(x^2+y^2)}\\ \tag{1}&= \int_{-\infty}^\infty dxdye^{-a(x+iy)(x-iy)} \end{align} So far so good. Now introduce a complex variable $z$ ...
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0answers
38 views

How do I prove that the compact open topology is metrizable?

Reference: Conway - Functions of one complex variable Let $G$ be open in $\mathbb{C}$ and $\{K_n\}$ be a sequence of compact subsets of $G$ such that $\bigcup_n K_n = G$ and $K_{n}\subset Int(K_{n+...
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2answers
56 views

Determine the number of roots in complex plane

Prove that the equation $z^{3}e^{z}=1$ has infinitely many complex solutions.How many of them are real? Use the argument principle,I choose a disk centered at $0$ with radius $R$ and get $\int_{\...
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1answer
27 views

Show that if $g$ and $h$ are holomorphic in $\mathbb{C}$ and $g\circ h$ is a no constant polynomial then $g$ and $h$ are polynomials.

Let two entire functions(holomorphic in all $\mathbb{C}$) $g$ and $h$ so that the composition $g\circ h$ is a no constant polynomial. Show that $g$ and $h$ are polynomials.
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1answer
45 views

Derivative of a analytic function at its fixed point [duplicate]

Let $D$ be a bounded domain, and let $f(z)$ be an analytic function from $D$ to $D$.Show that if $z_{0}$ is fixed point for $f(z)$,then $|f'(z_{0})|\leq 1$ All the conditions above make me think ...
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44 views

A Conformal Mapping problem

Can you give me a proof of this? I know it may use comformal mapping theorem but don't know how to do. Thanks you very much for your help!
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1answer
44 views

Complementary Text to Gunning and Rossi - Analytic functions in several complex variables

I'm currently a second year student who has a background in group theory, ring theory, galois theory, metric spaces and point set topology. I'm currently taking courses in algebraic topology, advanced ...
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2answers
35 views

Jordan curve in $C^2$

Can we find a Jordan curve $\gamma$ in $\mathbf{C}^2$ of class $C^1$ such that the projection to the first coordinate plane divides the plane into infinite components of connectivity.
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2answers
79 views

How do I solve equation $\bar{z} = |z|$ correctly?

I'm having troubles, finding how solution would look like for complex equation of the form $\bar{z} = |z|$. Taking $z = x + iy$, we get the following: $$x - iy = \sqrt{x^2 + y^2},$$ then raising it to ...
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0answers
37 views

A question about the log of a rational function

We have the rational function : $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$ It's not hard to prove that : $$\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}=(-1)^{n}\prod_{k=1}^{n-1}\...
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0answers
17 views

Symmetry in Analytic Continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we have the following function: $$F(x)=\sum_{n=0}^{\infty} e^{-x E_n}$$ Where $E_n$ is a positive monotonically increasing sequence, bounded from below. Is there a general condition on $E_n$ ...
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2answers
38 views

A property about a bounded harmonic function

Suppose $f(x,y)$ is a bounded harmonic function function in the unit disk and $f(0,0)=1$. Show that $$\iint_{D}f\left(x,y\right)\left(1-x^{2}-y^{2}\right)dxdy=\dfrac{\pi}{2}$$ I don't understand why ...
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1answer
55 views

Holomorphic function between $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ and $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$

Does there exist a holomorphic function $h$ that sends the set $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ to the set $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$? I tried proving it but I could not. Thanks ...
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0answers
16 views

Existence of a holomorphic square root

The problem is as follows. Define\begin{equation*}K=\{iy:y\geq0\}\cup\{x:x\geq0\}\cup\{e^{i\theta}:\frac{3\pi}{4}\leq\theta\leq\frac{7\pi}{4}\}\end{equation*} and $G=\mathbb{C}\setminus K$. Define $f(...
4
votes
2answers
46 views

Double periodic entire function

Suppose f is entire and $f(z)=f(z+1)=f(z+\pi)$. Does this imply $f$ is constant? I want to prove that it is constant.I see that it is enough to consider the value of $f(z)$ in between the lines $z=1$ ...
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1answer
33 views

Complex Number Systems involving trig.

I have to integrate from $0$ to $1$ in complex numbers the quantity $e^{-t} \cdot sin(2\pi t)$ I know what sign should look like if that $2\pi$ was not there, but it is, so do i just but it in for ...
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1answer
53 views

Residue of $e^{z+\frac1z}$ at $0$.

I am trying to compute $$\text{Res}\: (e^{z+\frac1z}, 0)$$ and can't get a solvable integral using the definition of a residue. I already know other ways to compute residues of poles of arbitrary ...
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1answer
50 views

Geometric Description Of a Set In The Complex Plane

$$S_1=\left\{z:Im\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ $$S_2=\left\{z:Re\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ Can someone help me with the ...
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2answers
28 views

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$, suggestion: Reduce this inequality to $(|x|-|y|)^2 \geq0$ (z is a complex number. R stands for real part and Im stands for imaginary part) Approach: Let $...
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1answer
46 views

Prove that if $z_1*z_2$=0 then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers

Prove that if $z_1*z_2=0$ then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers by using the following property: $|z_1z_2|=|z_1||z_2|$ Approach: if $z_1*z_2=0$ then $$|...
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votes
0answers
17 views

Analytic continuation of $\sum_{n=0}^{\infty} (E_n)^{-s}$

Suppose $E_n$ is a monotonically increasing sequence. Under what conditions on $E_n$ may the sum $$q(s)=\sum_{n=0}^{\infty} (E_n)^{-s}$$ Be analytically continued from $q(s)$ to $q(-s)$. How would ...
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26 views

A problem about proving some spaces are not conformally equivalent

Consider the unit disk $D$,the complex plane $C$ and extended complex plane $C^{\ast}$.Show that no two of them are conformally equivalence. From Liouville theorem,it's easy to see that disk is not ...
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0answers
28 views

Proof of a necessary and sufficient condition between annuli centered at the origin [duplicate]

What is a simple way to prove that two annuli $A_1 = {z: r_1 < |z| < R_1}$ and $A_2 = {z: r_2 < |z| < R_2}$ are conformally equivalent if and only if $R_1/r_1 = R_2/r_2$, using standard ...
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2answers
26 views

Question about triangle inequality

By factoring $z^4-4z^2+3$ into two quadratic factors an using the triangle inequality, show that if $z$ lies on the circle $|z|=2$ ($z$ is a complex number) then $$\left|\frac{1}{z^4-4z^2+3}\right| \...
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1answer
379 views

Complex Arctan function and its power series

I face a sequence of confusing questions: In complex plane, note that $arctan(z)$ denote the principal branch of inverse complex tanget function ,by requiring $$\frac{-\pi}{2} < \mbox{Re}(\arctan(...
3
votes
2answers
102 views

Find the value of $\sum_{r=0}^{\left\lfloor\frac{n-1}3\right\rfloor}\binom{n}{3r+1}$

Show that $$\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\ldots=\dfrac{1}{3}\left[ 2^{n-2} + 2\cos{\dfrac{(n-2)\pi}{3}}\right]$$ My solution:- $$(1+x)^n=\binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+\...
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1answer
20 views

Inversion of lines and circles using explicit parametrizations

Is there a way to parametrize a line and a circle in the complex plane [by $z = z(t)$], to show that under the inversion function $f(z) = 1/z$, a line is mapped either to a line or a circle, and a ...
0
votes
1answer
22 views

Complement of image of an entire, holomorphic map contains an open disk implies the map is bounded

I need to show the above in order to then use Liouville's Theorem and conclude the map is constant. Thus, I am not sure that the entirety of the map is necessary. $\exists B(x, r) \subseteq f(\mathbb{...