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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5
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1answer
144 views

Definition simply connected in $\Bbb C$

I recently saw a different definition for simply connected which I had never seen before. A connected subset $\Omega\subset\Bbb C$ is called simply connected if the boundary is the image of a simple ...
2
votes
2answers
138 views

Analytic continuation of Riemann Zeta funtion

I am reading about zeta function from book by Ingham. In that book the following theorem is given. I am unable to understand what does he mean by finite part of plane in the statement.
1
vote
1answer
88 views

Prove that 2 groups $Aut(D),Aut(D')$ are isomorphism

Prove that 2 groups $Aut(D),Aut(D')$ are isomorphism given that there is bijective and holomorphic function $f$ from region $D$ onto region $D'$ (in other words, $f^{-1}$ is also holomorphic) any idea ...
2
votes
1answer
34 views

Evaluate $\int_{D} \frac{dw}{w \cdot (1-w)}$ where $D$ is the rectangle

How can I evaluate the below integral:: $$\int_{D} \frac{dw}{w \cdot (1-w)}$$ using the $\textbf{Cauchy Integral Formula}$, where $D$ is the rectangle with vertices at the points $3 \pm{i}$ and $-1 ...
0
votes
1answer
58 views

Order of poles to make an intergral

I need to check something that I think I got wrong at the first try. If $$ f(z) = \frac{1+z^2}{1-\cos{2\pi z}} $$ Find $$ I = \frac{1}{2\pi i } \oint_C \frac{f'(z)}{f(z)} dz $$ where $ C$ is the ...
0
votes
2answers
32 views

Complex function integral

I have the function $f : D_f \subset\mathbb{C} \rightarrow \mathbb{C} $ defined by $$f(z) = \frac{1}{(z-1)(z^2+2)}, z \subset D_f$$ where $D_f$ is the domain of $f$. How do I calculate ...
1
vote
0answers
118 views

Infinite products (involving complex numbers)

I am learning the Gamma function, based on some lecture notes, and I wish to ask a couple of questions regarding infinite products. Let $z$ be a complex number except $\{0, -1, \ldots \}$. (1) How ...
0
votes
1answer
56 views

Contour integral for $\int _0^\infty \frac{t^2+1}{t^4+1} dt$

I know that the four singularities for $\int _0^\infty \frac{t^2+1}{t^4+1} dt$ are $\pm \frac{\sqrt{2}}{2} \pm i \frac{\sqrt{2}}{2}$. Also, since the function is even, I can calculate $\frac{1}{2} ...
1
vote
2answers
116 views

periodic holomorphic function other than the exponent function

Is there a function f, in the complex plane, such that f(z) is both analytic and periodic and doesn't involve e^z in it? I've tried: g(Real(z)) ( cos(Im(z))+i sin(Im(z)) ) For function of real ...
6
votes
3answers
216 views

If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?

Is the following true if $\theta\in\mathbb{Q}$? $$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$ Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
4
votes
1answer
183 views

Finding the intregral $\int_{0}^{2\pi} (\cos x)^{2n}\,\mathrm dx$ by contour integration

How do I find $$\int_{0}^{2\pi} (\cos x)^{2n}\,\mathrm dx$$ using contour integration ? Should I turn it into an integral around the unit circle? Or should I integrate some other function with real ...
1
vote
1answer
33 views

Calculate the $r$-derivative of the function $f$

Let $f$ be an analytic function defined by $$f(s)=g(s)∑_{n=1}^{∞}a_{n}/n^{s}$$ where $∑_{n=1}^{∞}a_{n}/n^{s}$ is an absolute convergent series for $Re(s)>1$. I have the following question: ...
1
vote
1answer
146 views

Evaluating $\int_0^{2 \pi} \frac {\cos 2 \theta}{1 -2a \cos \theta +a^2}$

In order to evaluate $\int_0^{2 \pi} \frac {\cos 2 \theta}{1 -2a \cos \theta +a^2}$ we can define $$ f(z) := \frac 1 z \cdot \frac { (z^2+z^{-2})/2}{1-2a( \frac {z+z^{-1}} 2) +a^2} $$ I have $0 < ...
0
votes
3answers
420 views

Find an analytic function with real part $\frac{y}{x^{2}+y^{2}}$

How do I find a analytic function such that $\displaystyle \mathfrak{Re}(f) =u(x,y)= \frac{y}{x^{2}+y^{2}}$. I can call the real part $u(x,y)$ and by Cauchy-Riemann I will have $u_{x}=v_{y}$ and ...
3
votes
1answer
70 views

Largest domain on which $z^{i}$ is analytic.

Can anyone help me with this question: What is the largest domain $D$ on which the function $f(z)=z^{i}$ is analytic?
1
vote
2answers
76 views

Differential equation system with complex eigenvalues

I need to solve this equation $$ x'=x+y, y'=-2x+3y$$ I get the matrix rigor and the eigenvectors complex $2-i$ and $2+i$. When I try to apply the eigenvectors associated the solution for $x$ I ...
0
votes
1answer
51 views

Where is $(z^2 +1)\Re(z)$ holomorph?

I have to calculate the points $z$ where $f'(z)$ exists with $f(z) = (z^2+1)\Re(z)$. By checking the Cauchy-Riemann equations I got $z = \pm i$. Is this possible ? Must $f$ not be holomorph on an open ...
3
votes
1answer
73 views

Upper bound on difference quotient with Cauchy Integral Formula

In the proof of a proposition on the exchangeability of complex differentiation and integration with regards to parameter integrals I am stuck at a step involving an upper bound of the difference ...
1
vote
1answer
49 views

Prove g is bounded and holomorphic

Let f be a holomorphic and univalent function on $D\subset\mathbb{C}$ and $a\in D$ be a constant.Define function $\displaystyle{g(z)=\frac{1}{f(z)-f(a)}-\frac{1}{f'(a)(z-a)}}$ then prove that 1.$g$ ...
4
votes
1answer
204 views

Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions

Question statement I would like to solve the following equation in the two variables $x$ and $y$: \begin{gather} 0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] , \end{gather} where $a$ and $b$ are ...
3
votes
1answer
106 views

Rouche's theorem problem

Prove that the polynomial $z^n + nz-1$ has $n$ zeroes inside the circle with centre at $0$ and radius $1+\sqrt{2/(n-1)}$ for $n=3,4,\dotsc$ Please give me some hints as to how to apply Rouche's ...
2
votes
1answer
64 views

Finding an $N$-th root for $\prod_{k=1}^N (z- a_k)$, where $a_1, \dots, a_N \in \mathbb{C}$

Let $a_1, \dots, a_N$ be distinct complex numbers. Let $\Omega$ be the domain obtained from $\mathbb{C}$ by removing each closed line segment connecting $a_k$ to origin, $k =1, \dots, N$. Explicitly ...
2
votes
2answers
149 views

Prove $\frac{1}{2 \pi} \int_0^{2\pi} \frac{1 - r^2}{1 + r^2 - 2r \cos{t}} dt = 1$ using contour integration

The question is to solve the integral using concepts of contour integrals: $$\frac{1}{2 \pi} \int_0^{2\pi} \frac{1 - r^2}{1 + r^2 - 2r \cos{t}} dt = 1$$
2
votes
2answers
175 views

Removable singularities for $L^2$ bounded holomorphic functions

Suppose $V$ is a analytic variety of an open subset $U\subset ℂ^n$. Suppose that $f:U\setminus V\rightarrow C$ is holomorphic and that $f$ is $L^2$-bounded in $U$. Question: Is it true that there ...
0
votes
1answer
77 views

What is the image of $D=\{z:0<\operatorname{Re}z<\pi\}\setminus\{\pi/2\}$ under $f(z)=\tan z$?

What would be the image of the domain $D = \{z:0<\operatorname{Re}z<\pi\} \setminus \{\pi/2\}$ under $f(z) = \tan z$? I havn't met with tan(z) transformation so I don't really know how to ...
4
votes
4answers
317 views

How to find $\int_0^{2\pi} \frac{dt}{1+2\cos(t)}$

The problem is $$\int_0^{2\pi} \frac{dt}{1+2\cos(t)}.$$ I know it is equal to $$\int\limits_{|z|=1}\frac{2dz}{i(1+z)^2}$$ but I don't know how I should calculate the last integral.
6
votes
0answers
75 views

“Natural” interpolation between partial sums of a power series

Suppose $f(z)=\sum_{n=0}^\infty a_n z^n$ has a radius of convergence of $R$. Let the $N$-th partial sum be $f_N (z)=\sum_{n=0}^N a_n z^n$. What smooth (analytic) function interpolates between ...
2
votes
1answer
220 views

A convergence test for improper integrals ($\mu$-Test)

In a textbook I found a convergence test for improper integrals referred to as the $\mu$-Test. I was wondering if it goes by another name (since I can't find it anywhere else) and if anyone could ...
0
votes
1answer
74 views

If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Is $f(x+iy) = \exp(x+iy)$?

$(1)$ If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Then, is $f(x+iy) = \exp(x+iy)$ for every $x$ and every $y$? $(2)$ If $f$ is an entire function such that ...
1
vote
1answer
945 views

Inverse function theorem in complex analysis

I was wondering whether an inverse function theorem in the complex numbers exists? I mean, in the real numbers we have that if the derivative of a function is non zero, then the inverse function is ...
1
vote
1answer
129 views

Existence of meromorphic function implies biholomorphic map onto the sphere.

Let $M$ be a closed simply connected Riemann surface, and let $f: M \to \overline{\mathbb C}$ be a meromorphic map with a simple pole in a point $p \in M$. Is it true that $f$ is injective? That $f$ ...
1
vote
1answer
37 views

Conformal Map Question

I was wondering if someone could give me a hint on how to map the entire plane with cuts from [-1,1] and [-i,i] to the outside of the circle. I'm having trouble figuring out the first map to begin ...
0
votes
0answers
46 views

definition of $e^H$ in MUSIC algorithms

i would like to clarify what does represent $e^H$ in the formula where as i know definition of $e^H$ is hermitian,which means that it's complex conjugate equal original variable with ...
3
votes
2answers
224 views

Application of Rouche's theorem gives two different answers?

So I am supposed to find how many solutions the equation $z^7-5z^4+iz^2-2 = 0$ has in the region $|z|<1$. Here's the dilemma: $|z^7-5z^4+iz^2|= |(-1)(-z^7+5z^4-iz^2)| = |-z^7+5z^4-iz^2| \geq ...
0
votes
1answer
684 views

Derivative of absolute value over the complex numbers

Given the function $f: \mathbb{C} \rightarrow \mathbb{C}, z \mapsto z\bar{z}$. I am supposed to determine where f is differentiable and where it is holomorphic. So I tested the Cauchy Riemann ...
3
votes
2answers
180 views

Finding the number of solutions of a complex valued function $f(z) = z^n$

This past semester I took a graduate course in complex analysis which I completed moderately well in spite of my expectations (that is I honestly think I deserved a lower grade than I received). I had ...
0
votes
1answer
48 views

derivative of parameter integral in $\mathbb C$

Let $f:\mathbb R\rightarrow\mathbb R$ be continuous and let $g(x):=xf(x)$ be absolutely integrable. Then $\widehat f'=-i\widehat g$. I know this would be true if I can differentiate in the integral ...
0
votes
1answer
80 views

How do I calculate the complex integral $\int_c\frac{1}{z^2}\,\mathrm{d}z$?

How do I calculate the complex integral $\displaystyle\int_c\frac{1}{z^2}\,\mathrm{d}z$, where $c$ is a direct line from $z=0$ to $z=1+2i$? Is it possible solving such the integral? I mean ...
7
votes
1answer
106 views

Evaluating $ \sum\limits_{n = 1}^{\infty}{\binom{2n}{n} \frac{1}{5^n}} $

Here is the question a little bigger: $$ \sum_{n = 1}^{\infty}{\binom{2n}{n} \frac{1}{5^n}} $$ This is given as an example (not an exercise), given in my textbook, on how to solve sums involving ...
10
votes
3answers
239 views

Where is the fallacy in the argument using Prime Number Theorem

I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem: Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say ...
3
votes
1answer
43 views

finding rational complex numbers in a disk with least denominators

Suppose that I have a disk of radius $r$ around some complex $\alpha\in\mathbb{C}$: How would one find a complex number $g$ in that disk besides $\alpha$ such that $\mathrm{Re}(g)\in\mathbb{Q}$ ...
3
votes
2answers
74 views

Finding two eigenvalues which add to $1$

$\textbf{Question}$: For $0<t<\pi$, the matrix $$ \left( \begin{array}{cc} \cos t & -\sin t \\ \sin t & \cos t \\ \end{array} \right) $$ has distinct complex eigenvalues $\lambda_1$ ...
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vote
2answers
54 views

Help Showing a function is identically zero derivative

Need help showing that if $f$ is analytic and not identically zero on $A$ then if $f(z_0)=0$, there is an integer $k$ such that $f(z_0) = 0 = \dots = f^{(k-1)}(z_0)$ and $f^{(k)}(z_0) \neq 0$. Any ...
0
votes
1answer
105 views

$f$ analytic on $\mathbb{D}$ and $|f(z)|\le 1$, we need to show

$f$ analytic on $\mathbb{D}$ and $|f(z)|\le 1$, we need to show ${(f(0)-|z|)\over (1-|z f(0)|)}\le |f(z)|\le {(f(0)+|z|)\over (1+|z f(0)|)} $ I am not able to proceed with schwarz lemma/pick lemma, ...
2
votes
1answer
52 views

A doubt about an element of $\text{Aut}(\Bbb{H})$

If $\theta \in \Bbb{R}$, this mapping $\varphi(z)=e^{i\theta}z$ is an automorphism of the disc $\Bbb{D}$.And $F(z)=\dfrac{i-z}{i+z}$ is a conformal map from $\Bbb{H}$ to $\Bbb{D}$, its inverse is ...
1
vote
0answers
156 views

Diagram of Riemann Surface for $\sqrt {z^2-1}$

I'm trying to discuss what $\sqrt {z^2-1}$ looks like. I'm trying to relate it to the square root function $z \mapsto \sqrt z$ but not getting anywhere. Any help would be greatly appreciated. Thanks
1
vote
0answers
38 views

Approximation the function $f(t)=I_0(-rt)e^{-rt}$ with sum of Exponentials.

Consider the function $f(t)=I_0(-rt)e^{-rt}$ where $I_0(t)$ is modified Bessel’s function and $r>0$. I am looking for an approximation for the function with a sum of exponential functions in $t ...
2
votes
1answer
191 views

Analytic continuation on the unit disc

Let $f$ be an analytic function on the open unit disc $D=\{z \in \mathbb{C} \mid |z|<1\}$, suppose it is continuous on the the closure $\bar{D}$, then is $f$ also analytic on $\bar{D}$?
0
votes
2answers
67 views

Why is $\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n) $?

I'm trying to understand the equation: $$\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n).$$ Here $x\in \mathbb{R}, x\geq 0$, and $C = \{s:\operatorname{Re}(s) = ...
1
vote
1answer
36 views

putting a complex structure on a graph

I am studying Riemann Surfaces, and an example that comes up in two of my references, as a preamble to smooth affine plane curves, is the following: Let $D$ be a domain in the complex plane, and let ...