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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28 views

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D, is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally?

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D (an open connected subset of $\Bbb C$), is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally, for any constants $c_1$ and $c_2$? ...
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66 views

Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?

Does there exists a nonconstant entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line? Clearly,$ f(z)=sin(z)$ is an example of an entire function which is ...
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0answers
34 views

Integral curves of complex function components

How to generally find integral curves $F(x,y,c) = 0$ from $u(x, y), v(x, y)$ when $$ x'(t) = u (x, y),\; y'(t) = v (x, y) $$ where $u, v$ are real and imaginary parts functions of a complex ...
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2answers
40 views

plot graph of function $f(z)=\frac{1+z}{1-z}$

I am not able to plot graph of function $f(z)=\frac{1+z}{1-z}$. can anyone tell me how to do this without using any software?
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2answers
68 views

Proving that the cross ratio is a Möbius transformation

I'm trying to show that given three distinct points $z_1,z_2,z_3\in\mathbb C$, the rational function $$ f(z) = \frac{(z-z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} = \frac{(z_2 - z_3)z + (z_1z_3 - ...
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1answer
21 views

Extending functions on proper subsets of $\mathbb C$ to functions on proper subsets of $S^2$.

There are a number of nice results about extending holomorphic and meromorphic functions from the complex plane $\mathbb C$ to the Riemann sphere $S^2$. See for instance Does entire function extend ...
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2answers
49 views

Complex integral with exponential and tangent

Suppose that $k \in \mathbb{R}.$ Evaluate as a function of $k$ the integral $$I(k) : = \int_{-\pi/2}^{\pi/2} e^{i \ k \ \mathrm{tan}(\phi)} d \phi.$$ Any suggestions on how to approach this problem? ...
3
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1answer
44 views

Why this map is a mobius transformation

Question: Let $D_2=\bar D(2,1)$ and $D_{-2}=\bar D(-2,1)$ be the closed disks of radius $1$ centered at $z=2$ and $z=-2$ in the complex plane, respectively. Set $X= \mathbb C-\{D_2 \cup D_{-2} \}$, ...
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2answers
94 views

How do I compute the indicator function of an entire function?

Let $F(z)$ be an entire function of finite exponential type. The indicator function of $F$ is defined as ...
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0answers
26 views

Is the Riemann–Liouville fractional derivative holomorphic in order?

If my understanding of complex analysis is correct then the arbitrary order generalization of Cauchy's formula for repeated integration $$(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x ...
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62 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
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1answer
63 views

What does $\Bbb R/2\pi$ for a set mean?

I simply cannot figure out what this means. I read this on an article about the scalar product of $2\pi$ periodic functions. it says that < f,g > goes from $\Bbb R/2\pi \to \Bbb C$ (complex) Do ...
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1answer
44 views

A certain Complex line integral

In evaluating the line integral of $\frac{dz}{z-2}$ around the circle $|z-1|=5$ , and also around the square with vertices $3+3i ,3-3i,-3+3i,-3-3i$, I obtain zero in both cases however my textbook ...
3
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1answer
54 views

Find the Laurent series about $z=0$

Let $f(z)=\cfrac{e^{-3z}}{z^2(z-2)^2}$, find the Laurent series about $z=0$. On the region $0<|z|<2$, I get $\cfrac{1}{(z-2)^2}=\displaystyle\sum_{n=1}^{\infty}\cfrac{nz^{n-1}}{2^{n+1}}$, ...
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39 views

set of numbers satisfying a complex exponential equation

Here is the question: Using the principle branch definition of $z^i$ determine the set of all $z\in\mathbb{C}$ for which $(z^i)^2=(z^2)^i$. My ideas: I took the principle branch to be ...
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1answer
49 views

Mapping the upper half plane conformally onto a semi-infinite strip,

Map the upper half y>0 of the z-plane conformally onto the semi-infinite strip u>0, $-\pi<v<\pi$ in the w-plane. I would like some hints for now, please. I'm not sure how to even get started ...
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2answers
96 views

Is the limit $\lim\limits_{x\to\infty} {i}^{-x}$ equal to $0$, or doesn't exist?

Can someone show me if this limit exists: $$\lim_{x\to \infty} {i}^{-x}=0$$ or it doesn't exist? Here, $i$ is the unit imaginary part. Thank you for any help.
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1answer
88 views

Best estimate using Cauchy integral formula: why is a circle the optimal path?

I once encountered this question from Ahlfors' Complex Analysis. An analytic function $f$ has the property that for $|z|<1$, $|f(z)|\leq \frac{1}{1-|z|}$. Find the best estimate of ...
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3answers
52 views

How to remember/rederive the isomorphisms from the half planes to the unit disc

I know that $$z \mapsto \frac{z-i}{z+i}$$ maps the upper half plane to the unit disc, and $$z \mapsto \frac{z-1}{z+1}$$ maps the right half plane. Is there an intuitive way to construct such maps ...
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1answer
66 views

When proving that f(z) is a polynomial, is it enough to consider just one point instead of keeping z arbitrary?

I think so - but I'd rather ask the MSE community too. Say I am given the bound |f(z)| < $|z|^3$, and that f is entire. Show f must be a polynomial. I used Cauchy's Integral Formula for ...
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1answer
38 views

Slopes of curves from complex derivative [closed]

Show that the slopes of the level curves$$u(x,y)=\text{constant} \ \ \text{and} \ \ v(x,y)=\text{constant}$$ are respectively given by $$\cot(\arg(f'(z))) \ \ \text{and} \ \ -\tan(\arg(f'(z)))$$ If ...
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122 views

What does “bounded away from zero” actually mean?

For example, is $f(z) = 1/z$, on the set $0<z<1$ "bounded away from zero"?
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2answers
63 views

Proof or Counterexample:Is every open connected set $D \subset \mathbb C$ is a domain of holomorphy?

Def: An open set $D \subset \mathbb C^n$ is called a domain of Holomorphy if there exists a holomorphic function $f$ on $D$ such that $f$ cannot be extended to a bigger set. Is every non empty open ...
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6answers
272 views

Compute definite integral

Question: Compute $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$ Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational ...
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1answer
56 views

Existence of unique circle passing through interior points of unit disk meeting the boundary orthogonally

I am a self-studies and this is a hw problem from a complex analysis scourse I've been doing. The problem set pertains to the topic Automorphism Groups and has a high concentration of fractional ...
2
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2answers
67 views

How do I determine if a given function is entire?

Consider the three functions $\displaystyle e^{\frac{r}{\ln r}}$, $\displaystyle e^r$, and $\displaystyle e^{r\ln r}$, where $r = |z|$. Note that these are not constant functions. Can someone ...
2
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0answers
26 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
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1answer
67 views

Computing the residue of $\frac{z-2}{z^2} \sin\left(\frac{1}{1-z}\right)$ for $z = 1$.

Consider the function $$f(z) = \frac{z-2}{z^2} \sin\left(\frac{1}{1-z}\right)$$ We have that $0$ is a double pole and $1$ is a single pole (essential singularity) of $f$. It is simple to compute ...
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2answers
36 views

For given $t$ and $x$ and $y$, is there at least one $f$ such that $\cos ft = x, \sin ft =y$?

Suppose that $t$, $x$ and $y$ are given and are all in $\mathbb{R}$. Is there always at least one $f$ such that $\cos ft = x, \sin ft =y$? Edit: OK I forgot to add that given $x$ and $y$ are such ...
2
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1answer
52 views

How to get sine term in Analytical continuation of $\zeta(s)$

I am able to prove the symmetric functional equation that Riemann gives in his paper, using Poisson Summation and properties of $\theta(x)$. The functional equation is given like so, ...
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0answers
29 views

Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
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0answers
39 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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1answer
38 views

$f$ continuous on $\overline{D} \setminus \{1\}$, holomorphic and bounded on $D$. Then $f$ attains its supremum on the boundary

Let $D$ be the unit disc, $f$ continuous on $\overline{D} \setminus \{1\}$, holomorphic and bounded on $D$. The problem is to show that for all $z \in D$, $$|f(z)| \leq \sup\limits_{|\zeta| = 1, ...
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43 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
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23 views

How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the ...
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18 views

Classification of isolated singularity by limit

Let $z_0$ be an isolated singularity of $f$ so there exist a punctured ball $B'$ centered in the singularity where $f$ is holomorphic. Let $f(z)=\sum_n a_n (z-z_0)^n$ be the Laurent series of $f$ in ...
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1answer
48 views

Borel Measures: Lusin

I'm trying to self-learn. Given the complex plane $\mathbb{C}$. Consider a Borel measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\mu\geq0$$ Regard a measurable: ...
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1answer
37 views

Image under conformal mapping

I'm trying to prove that the mapping $$f(z) = \frac{z ( az+1)}{z+a}$$ where $a=\sin(\alpha/4)$ ($0< \alpha < 2\pi$) maps $\mathbb{C}_{\infty} \setminus \overline{B}(0;1)$ conformally onto ...
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1answer
79 views

Show: An entire function $g$ with $\vert g(x) \vert \to \infty$ for $|x| \to \infty$ is a polynomial.

This is part of an exercise sheet in complex analysis. It should by solvable by rather elementary methods like the main theorems of complex analysis. I succeded to show that $g$ has only finitely ...
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16 views

Singular value decomposition about complex matrix

I am now stuck on the following problem. Could anyone help me out? Suppose a rank $1$ $n \times n$ matrix $A$ is form as a follows, $A=a*a'$, were $a$ is a $n\times 1$ vector. Now, suppose we have ...
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1answer
43 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
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12 views

Finding all homographies preserving the upper right quarter of $\mathbb{C}$

i am trying to solve the following problem: find all homographies given by a two by two matrix with coefficients in $\mathbb{C}$ which transform the upper right quarter $ \{ \Re z, \Im z > 0 \} $ ...
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1answer
53 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
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43 views

Conformal mapping

I'm studying Potential theory in the complex plane in Dr.Thomas Ransford book and I get stuck in proving that the mapping $$f\left( z \right) = \frac{1}{2} \left( z-1 + \sqrt{\left( z- e^{i \alpha ...
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1answer
32 views

Why the complex number system is not an ordered field [duplicate]

In high school, we are taught that we do not have $2i < 3i$, i.e., the complex number system is not an ordered field. (Real number, for example, is an ordered field. For example, $2 < 3$). ...
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1answer
35 views

A holomorphic function on a punctured disc has removable singularity iff it can be approximated by polynomials on a circle

Let $r>0$ and $f: D(0,2r)\setminus\{0\} \to \mathbb{C}$ be holomorphic, where $D(0,2r):= \{z \in \mathbb{C} \,:\, |z|<2r\}$. Show that f has removable singularity at $0$ iff ...
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1answer
37 views

Computation of a certain integral

Assume that $\alpha>0, t \in R$. Compute the integral $\int_0^1(-1)^xx^{-\alpha-it}dx.$
3
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1answer
56 views

Complex Inequalities

Let $\mathbb{H}=\{z\in \mathbb{C} | \ \Im(z)>0\}$ and $f:\mathbb{H} \to \mathbb{H}$ analytic. Prove that for every $z_1, z_2 \in \mathbb{H}$, it must happen that $$ ...
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1answer
35 views

Help with proof that that affine plane curves in $\mathbb{C}^2$ are not compact

This is a problem from Kirwan's Complex Algebraic Curves that I'm stuck on. She gives a hint suggesting that for $C = \{(x,y)\in\mathbb{C}^2: P(x,y) = 0\}$, show that at all but finitely many points ...
3
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1answer
35 views

An effective way of finding the order of the zero $z=0$ of $e^{\sin z}-e^{\tan z}$

An effective way of finding the order of the zero $z=0$ of $f(z)=e^{\sin z}-e^{\tan z}$? What I tried is developing both exponentials by their Taylor series around $z=\sin z$ and $z=\cos z$, getting ...