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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Vanishing loop integral for continuous but not holomorphic functions on convex domains?

If $D$ is a convex domain, $\gamma$ is a rectifiable closed path and $f(z): D \to \mathbb{C}$ is continuous, then $$\int_\gamma f(z) \mathrm{d}z = 0?$$ I don't think this is correct, since Cauchy's ...
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73 views

partial derivative of a complex function.

Let $f(z)=\sin\left(z+\mathrm{e}^{3z}\right)$. Find $\frac{\partial f}{\partial \bar{z}}(z)$. I tried to start with the well known result $$\frac{\partial^2}{\partial ...
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2answers
28 views

Number of zeroes of an analytic function inside unit disc

Let $f(z)$ be analytic in $\{z: \ |z|\leq 1\}$ such that $|f(z)-z|<|z|$ on $\{z: \ |z|=1\}$. Find the number of zeroes of $f(z)$ in $\{z: \ |z|<1\}$. My first thought was to use Rouche's ...
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3answers
202 views

Cauchys integral formula on function with pole of order 2

I want to compute the integral $$\int_{|z - 1| = 1/2} \frac{e^{iz}}{(z^2-1)^2} \mathrm{d}z$$ using Cauchy's integral formula (residual theorem is not allowed). Examining the integrand one gets $$ ...
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54 views

Condition that Fourier inversion formula holds

Recently I am reading Stein's Complex Analysis but I haven't read the book Fourier Analysis before. So I do not have any knowledge about Fourier Transformation. In Chapter 4.3 (p.121), it tells that ...
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36 views

Complex analysis question regarding Cauchy's integral formula and holomorphic functions

Let $U \subseteq \Bbb{C} $ be an open convex subset of $\Bbb{C}$ We also assume that $\partial U$ is smooth. We fix a point $z_0 \in U.$ Using Cauchy's integral formula, show that $$\lvert ...
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15 views

Partial converse to $\mathcal{F}$ normal implies $\mathcal{F}$' normal.

Let $\mathcal{F}$ be a family of analytic functions in the open unit disk $\mathbb{D}$. Let $\mathcal{F}$'= {$f' : f \in \mathcal{F}$}. Suppose $\mathcal{F}'$ is normal and that $\sup \{ |f(0)| : f ...
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33 views

The Polar-Coordinate Form of Cauchy-Riemann

Write $f(z) = u(r, \theta) + iv(r, \theta)$; suppose that the first-order partials of $u, v$ with respect to $r, \theta$ are continuously differentiable in some neighborhood of $z$ and satisfy ...
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2answers
68 views

Prove $|f'(z)|\leq{\frac{1}{2 \operatorname{Im}z}}$

Could you help me with proving: Let $f$ be an analytic function defined in on upper half plane(UHP). Suppose that $|f(z)|<1$ for all $z$ in UHP. Prove that for every $z$ in UHP $$ ...
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85 views

Evaluate the integral $\int_0^\infty x^{t-1}e^{-\beta x}dx$

I want to evaluate the following integral $$\int_0^\infty x^{t-1}e^{-\beta x}dx$$ where $\beta$ is a complex number. Now, if $\beta$ was real, we could just set $y = \beta x$ and we will reduce to ...
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70 views

Explain what goes wrong with the following argument.

To compute $$I =\int \frac{1}{1+x^4} \, dx $$ Put $x = iy$ and we get $I = iI$ from which follows $I = 0$. Compute the integral applying the Cauchy formula to a sectoral contour of radius $R$, one ...
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1answer
27 views

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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1answer
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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24 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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1answer
50 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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17 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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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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33 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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44 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
19 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 ...
4
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1answer
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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21 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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62 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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1answer
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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1answer
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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35 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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0answers
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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84 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
112 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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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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61 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
24 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 ...
3
votes
3answers
54 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
22 views

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 ...
3
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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 ...