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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Complex polynomial injective on the unit disk?

Prove or disprove, for all real $\alpha$ and natural $n\geq 2$ $f(z)=z^{n}+ne^{i\alpha } \cdot z$ is injective on the unit disk. I'm not sure how to approach this, I've seen other 2 problems ...
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27 views

Complex analysis question: holomorphic functions on unit disc

A question from an examination on complex analysis: If $f$ and $g$ are holomorphic on $\Omega \subset \overline{B(0,1)}$, and if: 1) $\vert f(z) \vert = \vert g(z) \vert$ for $\vert z \vert = 1$ 2) ...
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1answer
20 views

Minimum requirement for equality of holomorphic functions

Let $f,g\colon \mathbb{C}\to \mathbb{C}$ be holomorphic and let $A = \{x\in \mathbb{R} :f(x) = g(x)\}$. The minimum requirement for the equality $f=g$ is $A$ is uncountable. $A$ has positive ...
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49 views

Multivariate Residue Theorem

Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...
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23 views

sequence of entire functions

For a sequence {f_n} of entire functions converging to f uniformly on compact subsets of C, suppose, for all n≧1, f_n has n zeroes. Then, 1. f must have infinitely many zeroes. 2. f need not have any ...
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22 views

Complex Sequence Convergence

Claim : If complex sequence $z_n$ converges then $|z_n|$ converges Proof: Let $z_n =x_n +i y_n$ where $x_n$ and $ y_n$ are real sequences. If $z_n $ converges to $(L_1+i L_2)$ $ \forall \epsilon ...
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27 views

Conformal Mapping onto the Unit Disc

Given the open vertical strip $G=\{x+iy~|~0<x<1,~-\infty<y<\infty\}$, what is the explicit conformal injective map characterizing $w=f(z):G\to\mathbb{D}$? It is noted that if there ...
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48 views

Complex analysis exercise (Mittag-Leffler related)

I'm trying to make an exercise in a complex analysis textbook, but I'm stuck, so I hope someone can help me out. The exercise is assigned in a chapter about the Mittag-Leffler theorem. 1) If $f$ is ...
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28 views

suppose that the line $\Gamma=\{t+it:t\in \Bbb R\}$ is mapped to itself, If $f(\sqrt2 )=3$, then what is $f(\sqrt 2i)$?

Let $f(z)$ be analytic on $\mathbb C$ and suppose that the line $\Gamma=\{t+it:t\in \Bbb R\}$ is mapped to itself, that is, $f(z)\in\Gamma$ for all $z \in \Gamma$. If $f(\sqrt2 )=3$, then what is ...
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16 views

Elliptical Bijective Conformal Mapping

Given the conformal mapping $w=T_1(z):R\to\mathbb{D}$ and $v=T_2(w):\mathbb{D}\to E$, where $R=\{x+iy~|~0<x<1,\epsilon_1<y<\epsilon_2\}$ is the open rectangular domain, $\mathbb{D}$ is the ...
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22 views

Prove that $\{{z \choose k}\}$ this sequence is bounded iff $Re(z) \geq -1$

If we define ${z \choose n}= \frac {z(z-1) \cdots (z-n+1)}{n!}$ then prove that $\{{z \choose k}\}$ this sequence is bounded iff $Re(z) \geq -1$ and converges iff $Re(z) > -1$. I have proceeded in ...
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32 views

Prove that if $f$ has an essential singularity at $z_0$ and $g$ has a pole at $z_0$, then $f(z)g(z)$ has an essential singularity at $z_0$.

Assume $f(z)$ and $g(z)$ are holomorphic in a punctured neighborhood of $z_0 \in \Bbb C$. Prove that if $f$ has an essential singularity at $z_0$ and $g$ has a pole at $z_0$, then $f(z)g(z)$ has an ...
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38 views

Find the number of zeros, counting multiplicity, of $z^8 -z^3 +10$ inside the first quadrant $\{z\in \Bbb C: \Bbb R e(z)>0,\Bbb Im(z)>0 \}$.

Find the number of zeros, counting multiplicity, of $z^8 -z^3 +10$ inside the first quadrant $\{z\in \Bbb C: \Bbb R e(z)>0,\Bbb Im(z)>0 \}$. I'm assuming to use Rouche's theorem or argument ...
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1answer
21 views

Finding a sub-neighbourhood of a neighbourhood such that it's open, connected and simply connected.

Say there exists a neighbourhood $U\subseteq\mathbb{C}$ of a point $z\in\mathbb{C}$. Does there always exist another neighbourhood $V\subseteq U$ of $z$ such that $V$ is open, connected and simply ...
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40 views

An entire function satisfying $\left|f\left(ne^{i\theta}\right)\right|\le e^{n\cos \theta}$

Let , $f$ be an entire function and $$\left|f\left(ne^{i\theta}\right)\right|\le e^{n\cos \theta}$$for all $n\ge 1$ and $\theta \in [0,2\pi]$. Show that $f(z)=Ce^z$ for some constant $C$ with ...
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1answer
18 views

Holomorphic function and uniform convergence

I am given the following problem: Let $f\colon G\to \mathbb{C}$ be a complex function where $G$ is open and not empty. Assume that $(f_n)$ converges locally uniformly to $f$ and $f_n$ is complex ...
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27 views

Complex Conjugate of Integral

I want to know that the equality $$ \overline{\int_{\mathbb R} f(x)dx} = \int_{\mathbb R} \overline{f(x)}dx $$ holds, if the both integral converges. Here $f:\mathbb R \ni x \mapsto f(x)\in \mathbb C ...
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18 views

Chordal Distance (Stereographic Projection)

I was working out Gamelin's Complex Analysis and read through the part where he finds an expression for the chordal distance on the Riemann Sphere corresponding to the stereographic projection w.r.t. ...
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61 views

Conformal map of the unit disk onto itself which is not 1 to 1

Apart from the well known biholomorphic maps from $D=\{|z|<1\}$ onto itself of the form $f(z)=e^{i \theta}\frac{z-a}{1-\overline{a}z}$ ($|a|<1$, $\theta$ real), are there any holomorphic maps ...
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30 views

If $|f(z)|\le C_1e^{C_2|z|}$ then prove that $f(z)=e^{az+b}$.

If an entire function $f(z)$ has no zeros and satisfies $|f(z)|\le C_1e^{C_2|z|}$ then prove that $f(z)=e^{az+b}$. I am trying to apply Liouville's theorem on $f$. Since $f$ is entire and it has ...
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40 views

Show that $f\equiv 0$ in $|z|<1$.

Let $f$ be analytic in $|z|<1$ and $f\left(\frac{1}{n^2}\right)=\frac{1}{n}$ , for all $n>2$. Show that $f\equiv 0$ in $|z|<1$. Since $f$ is analytic so Taylor's series expansion of $f$ ...
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34 views

Let $h(z) = g(f(z))$. If two of the three functions $f$, $g$, and $h$ are holomorphic and non-constant, must the third also be holomorphic?

If $h$ and $g$ are holomorphic it seems like the answer is no. Let $f(z) = f(re^{i\theta}) = \sqrt re^{i\theta/2}$ for $\theta \in [0,2\pi)$, and let $g(z)=z^2$. Then $f$ is discontinuous on the ...
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1answer
27 views

Constructing the Koenigs function about a repelling fixed point

My question is rather simple and I hope someone has some sort of an answer. I am looking for a simple yes or no answer, and a reference if anyone has one. We have a holomorphic function $f$ defined ...
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37 views

Calculating the value of a complex integral

I have some problem to find the value of the integral $$\int _{\vert z \vert =2} \frac{z}{e^z-z}dz$$. The integrand is not an analytic function.
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40 views

Complex analysis cycle

I am reading "Complex Variables" written by R. B. Ash & W. P. Novinger, and in the 3rd chapter I've got stuck. I have questions concerning the following definition. Let $\gamma_1, \gamma_2, ...
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44 views

What concept would be independent of path and how do I state it?

Below are lists of theorems I have studied: Theorem1. Let $x,z_0\in S^1\times \mathbb{C}$ and $f,g:S^1\rightarrow \mathbb{C}\setminus\{z_0\}$ be coninuous functions and $\alpha$ is a loop in $S^1$ ...
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81 views

Tough integral with exp

Can anybody integrate this: $$ \int_0^K e^{i(A\sqrt{\mathstrut k^2+m^2} - Bk)} dk,$$ where $K$, $A$, $B$ and $m$ are real constants? Sorry folks, I didn't realise anybody would be interested in the ...
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2answers
108 views

Integrating $ \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\infty} e^{-x^2} \cos (2x) \, dx}}$

I need help calculating the following integrals. For the top integral we can use the jacobin, right? But how do I calculate the bottom one?: $$ \frac{{ \int_{0}^{\infty} e^{-x^2}\, ...
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2answers
24 views

Is it possible to prove that the gradients of the real and imaginary parts of a complex analytic functions have the same length?

Suppose I have a complex analytic function $$f(x,y)=(x+iy)^n$$ where both $x$ and $y$ are real and $n$ is an integer. Is it possible to prove that the gradient of the real part of $f$ and the ...
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1answer
34 views

Residue theorem for line segment

I am working through this problem:- Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer. I follow it all, except for part (3) - I think this ...
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49 views

Prove that such analytic function exists

I'm going through the Complex variables by Murray R. Spiegel (Schaum's outline series) and I stack on a problem 69 from chapter 4: Let $P$ and $Q$ be continuous and have continuos first partial ...
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60 views

Why does this example of global residue theorem not work?

This question is related to and inspired by a previous question What is the residue obtained from this integral? , but note that the appearing functions are slightly different. Consider the following ...
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2answers
80 views

Prove that $|f(z)|\leq \left|\frac{4z^2-1}{4-z^2}\right|$

Here is the question I was working on: Let $f$ be holomorphic in the open disk $\mathbb{D}$ and suppose $|f(z)|\leq 1$ for all $z\in \mathbb{D}$. If $f(\frac{1}{2})=f(-\frac{1}{2})=0$, prove that ...
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1answer
42 views

What is the residue obtained from this integral?

Consider the following integral in two complex variables $z_1$ and $z_2$: $$\frac{1}{(2\pi i)^2}\oint_{{|z_1|=\epsilon}\atop{|z_2|=\epsilon}}dz_1 dz_2\frac{1}{z_1 ...
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1answer
23 views

Make a complex polynomial a covering map

Let $p:\mathbb{C}\to \mathbb{C}$ be a complex polynomial. Let $C:=\{p(z):p'(z)=0\}$ and $V:=\mathbb{C}\setminus C$. I want to show that $p:p^{-1}(V)\to V$ is a covering map. By inverse function ...
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2answers
38 views

Nonnegative analytic function as a square

It is know that if $f:\mathbb{C}\to \mathbb{C}^*$ is a continuous function, then for every $n>0$ there exists a continuous function $g:\mathbb{C}\to \mathbb{C}^*$ such that $f=g^n$. Is it true ...
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3answers
53 views

Residue of $\text{sech}^2(z)$

I am trying to find the residue of $\text{sech}^2(z)$ at $z=\pi/2 i$. The function has a second order pole at $\pi/2 i$. I find the residue to be zero. However, the integral $\int ...
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1answer
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What is a 2-norm of a multivariable complex function?

I was wondering, is there a way to specify the $2$-norm of a multivariable complex function? For example if we have a complex function: $$f = f(x_1, x_2,\cdots, x_n) = Re\{f\} + j Im\{f\}, \ x_i \in ...
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2answers
29 views

Image of a function under unit disk.

What can we say about the image of the following function under open unit disk: $$f(z)=\frac{1}{(1-z)(1-a z)},\quad 0<a\leq1.$$ I think the complement of the image domain is a convex set. But I ...
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12 views

Integral of complex valued function independent of homotopic rectifiable curves

Contour Integral of continuous complex valued function is independent of homotopic rectifiable curves as proved on page 90 of Conway second edition. Is there any other proof for this. In case ...
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70 views

Average order of $\mathrm{rad}(n)$

Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
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73 views

$n$th roots of entire functions

I am stuck on this complex analysis problem. Let $f$ be an entire function and $n$ a positive integer. Show that there exists an entire function $g$ such that $f=g^n$ if and only if the order ...
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32 views

Prove that all the negative term of Laurent series is zero

If the Laurent series has isolated singularity $z_0 = 0$. Prove that all the negative term of Laurent series is 0. Can someone show me how to prove this problem. Thank you.
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26 views

branch cut of $z^\frac{1}{2}$

How does removing a ray at angle $\alpha$ from the domain of the function $$f(Z)=Z^\frac{1}{2}$$ makes it a analytic and single valued function.
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19 views

$\text{ A set} F \subset C(G, \Omega) \text{ is normal iff its closure is compact}$

I want to show that : $$\text{ A set} F \subset C(G, \Omega) \text{ is normal iff its closure is compact}$$ By definition a set $F$ is normal if each sequence in $F$ has a sub-sequence which ...
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51 views

Find the iverse of the followning bounded operator?

The following definition and Theorem are given in the book "A short course on operator semigroup" by the author "K-J Engel and R Nagel". Sectoral operator: A closed linear operator $(A,D(A))$ in ...
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3answers
64 views

Improper Integration of A Non-even Non-odd Function From $0$ to $\infty$

I am trying to calculate the integral: $$\int_{0}^\infty \frac{x^2dx}{1+x^7}$$ I used to face this type of integration with even integrand, but the function here is not even nor odd! Is there a trick ...
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53 views

Show that there is exactly one $z$ [duplicate]

Show that there is exactly one $z$ in the right-half-plane such that: $$z + e^{-z} = 2$$ I know somehow we have to use Rouche's theorem to show that there is exactly one root in the right half plane. ...
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1answer
32 views

Find number in complex form.

Inspired from a recent question on SE, how to write: $2^{36}$ in complex form? To the power of $e$? I thought since: $2 = 2e^{2k\pi i}$ that: $2^{36} = 2^{36} 3^{72k\pi i}$ But the answer is: ...
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1answer
76 views

Number of Zeros in the Right Half Plane

I am having a bit of trouble with the following exercise: Determine the number of solutions to the equation $$z-2 -e^{-z} = 0$$ in the right half plane $P = \{z \in \mathbb{C} : \Re(z)> 0\}$. I ...