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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8 views

To show each $f_i$ is bounded

so for $\Omega$ a connected open subset in $\mathbb{C}$ each $f_I \in \mathcal{H}(\Omega)$ for all $j = 1,2,3...n $ such that $\sum |f_j|^{2}$ is constant on $\Omega$ then I need to show that each ...
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1answer
12 views

to show that $\frac{df}{dz} = \bar{(\frac{d\bar{f}}{d\bar{z}})}$

I need to show to show that $\frac{df}{dz} = \bar{(\frac{d\bar{f}}{d\bar{z}})}$ given that $f : \omega$ ---> $ \mathbb{C} $ and all the partial derivatives are continuous. I tried using $f=u+iv$ and ...
4
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0answers
28 views

Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$

We have the integral : $$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$$ Where $s$ is a complex parameter, and $n$ is a positive integer. The integral ...
0
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0answers
11 views

Deduction from a proposition harder than the proof of that proposition? A question about deducing a property of complex moduli from a previous proof.

So I'm currently reading An Introduction to Complex Analysis by Agarwal R.P. et al, and I'm still in the beginning and one of the problems had me prove: $$ |\alpha +\beta |^2+|\alpha -\beta ...
1
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0answers
16 views

component of the union of two sets

This question is exercise 2 on page 32 of the book Analytic Functions by Stanislaw Saks and Antoni Zygmund: (All sets are in the extended complex plane) If S, disjoint from a certain closed set Q, is ...
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0answers
18 views

Proving that if a sequence converges weakly, then their set of norms is bounded.

We are given that if $\{\Lambda_n\}\subset X^*$ (where $X^*$ is the dual space of some Banach space $X$) converges weakly to some $\Lambda\in X^*$. That is $\Lambda_n x\to\Lambda x$ as ...
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3answers
34 views

How to solve this semi gaussian definite complex integral?

How can I solve this definite complex integral : $L=\int_{-\infty}^{\infty} e^{-{\lambda r^2}-jr \zeta_1+j\zeta_2}dr$ where : $\lambda , \zeta_1, \zeta_2$ are real and constant values and $\,j ...
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2answers
27 views

complex numbers triangle inequality proof question

Prove that $$\left|z_1 + z_2 + z_3\right| \le |z_1| + |z_2| + |z_3|$$ and $$\left|z_1 - z_2\right| \ge |z_1| - |z_2|$$
2
votes
2answers
37 views

Complex analysis.

Here's a problem I'm struggling with, especially the second question. Let $f$ be a holomorphic function in $\mathbb{D}$, say $f(z) = \sum a_n z^n$. Suppose that $f$ satisfies the following ...
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1answer
19 views

Prove that there exists a real number p in the complex equation… [on hold]

Suppose the product of two complex numbers z_1 and z_2 is real and different from zero. Prove that there exists a real number p such that; z_1 = p*(complex conjugate(z_2))
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0answers
47 views

Imaginary part of An Squre Root Integration

I am looking for a particular form of an integral which some simplified version of it has the following form $$ \Im\int_{0}^{\infty} \frac{\sqrt{1+u^4-u^6}}{u^5}du. $$ Could someone gives some idea ...
2
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1answer
28 views

Structure of solutions of $f(z)^n - c = 0$

Let $f(z)$ be a second degree polynomial, $n \in \mathbb{N}$, and a constant $c \in \mathbb{C} \setminus \{0\}$ with $|c| < 1$. We have the equation \begin{equation} f(z)^n - c = 0. \end{equation} ...
0
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1answer
40 views

Roots of polynomial $a z \overline{z} + \overline{b}z + b \overline{z}+c$: is my solution correct

Let $a,c \in \mathbb R$ and $b \in \mathbb C$ with $|b|^2 - ac > 0$. To solve this I distinguished 4 cases. Now I am interested in the case $a =0, c\neq 0$ - Please can you tell me if ...
1
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2answers
26 views

Conformal maps from the unit disc onto itself, given by two sets of three points on the boundary

I want to construct a conformal map from the closed unit disc onto itself that maps given three points on the unit circle to another given set of three points on the unit circle. I know that the word ...
0
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1answer
20 views

Inverse Function Theorem [on hold]

Use the inverse function theorem to show that if $f \colon A \to \mathbb{C}$ is analytic and the derivative of $f$ is never $0$, then $f$ maps open sets into open sets.
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1answer
15 views

$\Pi ({1+z^{2^n}})$ converges uniformly on compact subsets of the unit disk.

Prove that pact sets $\Pi ({1+z^{2^n}})$ converges uniformly on compact subsets of the unit disk. Show that limit is $1/(1+z)$ I cant even start!
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1answer
38 views

Closed and exact differential forms [on hold]

I am trying to solve this question. But how does one define exact and closed differential forms in a domain in $\mathbb R^2$ . Show that if $f$ is holomorphic, then $f (z)dz$ is closed If $ P dx + ...
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0answers
10 views

number of zeroes of $f_n$ is independent of n if n is large enough

Let $\{f_n \}$ be a sequence of functions in a domain $D$ in $\mathbb C$ converging uniformly on compact sets to $f$ . Show that the number of zeroes of $f_n$ is in an open disc whose closure is ...
1
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1answer
36 views

Roots of polynomial $a z \overline{z} + \overline{b}z + b \overline{z}+c$

I am working on finding the roots of $a z \overline{z} + \overline{b}z + b \overline{z}+c$ given $a,c \in \mathbb R$ and $b \in \mathbb C$ with $|b|^2 - ac > 0$. First, I distinguished two cases: ...
1
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1answer
14 views

Simply connected domain (Homotopy) [duplicate]

Can anyone help me showing that : For any simply connected plain domain $D_1, D_2 \subseteq \mathbb{C}$ where $ D_1 \cap D_2 \neq \phi$ is connected, show that $D_1 \cup D_2$ and $D_1 \cap D_2$ are ...
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1answer
22 views

If $f (z_0 ) \not= 0$, then show that there is some z such that $|f (z)| < |f (z_0)|$ [on hold]

Let f be a nonconstant analytic function in a neighbourhood of $z_0$ . If $f (z_0 ) \not= 0$, then show that there is some z such that $|f (z)| < |f (z_0)|$
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0answers
27 views

Need help showing $\oint \frac{f (z)}{z} dz$ = $2πif (0)$ [on hold]

Let $D$ be a domain containing the closed unit disc with origin as centre. I am trying to show that if $f$ is a complex valued analytic function on $D$ whose restriction to $D\setminus\{0\}$ is ...
0
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0answers
19 views

Double checking if contours are correct

Since $$ |z| = 1 $$ is the unit circle centered at (0,0) which is used as a contour for a lot of integration problems, would $$ |z - i| = 1, |z + 3| = 1 $$ simply be translations of the unit ...
0
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1answer
15 views

Definite integral of absolute value complex function

Seems pretty straight forward but absolute values have always given me headaches $$\int_0^1 |1 -t + it|^2$$ Now usually I get roots and split up the intervals for when the function is greater or ...
1
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1answer
37 views

Orthonormal Sets and Compactness

1) Let {$u_n$} $(n=1,2,...)$ be an orthonormal set in Hilbert space $H$. Show that this set is closed and bounded but not compact. 2) Let $Q$ be the set of all $x\in H$ of the form ...
0
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1answer
30 views

Schwarz theorem in complex analysis.

Is there a version of the Schwarz theorem $ \partial_x \partial_y = \partial_y \partial_x $ in the theory of complex functions of several variables and complex analysis ? It would be nice that you ...
0
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0answers
15 views

Exact formulation of Voronin Universality Theorem of Zeta function

The Voronin Universality therem of the Zeta function is stated, in most of the paper I have seen, as follows: Let $0<R<1/4$ and suppose that $f(x)$ is a nonvanishing continuous function on the ...
0
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1answer
23 views

Unique Solution to an Integral Equation

Let $X$ be a bdd open set in $R^n$ and let K(x,y) be a Lebesgue measurable function on the product space $X$ x $X$ such that $$\int_X \int_X |K(x,y)|^2dx dy $$ is finite. Let f(x) be a function in ...
3
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1answer
35 views

Use the limit definition to show that $\frac{d}{dz}(z^2+2)=2z$

I need to show that $$\frac{d}{dz}(z^2+2)=2z$$ I am using the limit definition for complex variables which is $$f^1(z_0)=\lim_{\Delta z\to 0}=\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}$$ to show this. I ...
0
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0answers
19 views

Show that the integral is purely imaginary [duplicate]

Let f be a holomorphic function in a domain containing the closed unit disc Show that$ \int \overline f \frac{∂f}{∂z} dz$ over the unit circle is purely imaginary I tried solving and got the answer ...
0
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1answer
11 views

Intuitive explanation of complex-valued function and a notation

Recently I have introduced to a new concept and a new notation while learning complex analysis. The new concept is complex-valued function and the new notation is $f:\mathbb C\to\mathbb C$. I'm ...
0
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1answer
26 views

Extended Liouville Theorem

Use the extended Liouville Theorem to prove that if $\alpha$ is a zero of the polynomial $p$ of degree $n>0$, then $p$ is divisible by $z-\alpha$. Do not use the Euclidean Algorithm. Well I know ...
2
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1answer
45 views

Solve $\sin(z) = 2$

There are a number of solutions to this problem online that use identities I have not been taught. Here is where I am in relation to my own coursework: $ \sin(z) = 2 $ $ \exp(iz) - \exp(-iz) = 4i $ ...
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0answers
14 views

Goursat's Theorem and Real Differentiability

I've been studying Stein and Shakarchi's proof of Goursat's theorem in complex analysis, which states If $\Omega$ is an open set in $\mathbb{C}$ and $T \subset \Omega$ is a triangle whose interior ...
1
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1answer
28 views

Prove that when $z = p$ is a solution of $az^3+bz^2+cz+d=0$, $z=-p^*$ is also a solution

Given that $z = p$ is a solution of the equation: $$az^3+bz^2+cz+d=0$$ where $a$ and $c$ are real constants while $b$ and $d$ are purely imaginary constants. Show algebraically that $x = -p^*$ ...
0
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0answers
17 views

uniform convergence on a compact set

Show that $\prod_{n=1}^{\infty} (\frac{1+z}{n})(e^{-\frac{z}{n}})$ converges absolutely and uniformly on compact sets. Can one please provide a short method to it, I can do but I'm getting long ...
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0answers
11 views

complex parametizstions of the same curve

Is there a way to prove that two complex parametrizations are for the same smooth curve? The only way I know of is to sketch them but is there another way?
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0answers
8 views

How is arithmetic overflow avoided by using the Dawson function over the erfi function?

I came across the term arithmetic overflow while reading up on the IEEE754 yesterday, and read up its definition and related terms as well. Today, while reading about the error function and its cousin ...
0
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2answers
31 views

Paramtrizing a complex arc

How do you paramatrize the semi circle with radius 1 about the orgin from the initial point i to -i in the counterclockwise direction.
2
votes
1answer
29 views

Reference request: Riemann-Hilbert problems

I am trying to learn more about the applications of complex analysis in solving spectral problems and came across applications of theory built around Riemann-Hilbert problems. So far I have only read ...
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1answer
46 views

How to find roots of polynomial $a z \overline{z} + \overline{b}z + b \overline{z}+c$?

Let $a,c \in \mathbb R$ and $b \in \mathbb C$ with $|b|^2 - ac > 0$. I am stuck trying to find roots of the polynomial $a z \overline{z} + \overline{b}z + b \overline{z}+c$. I know the formula for ...
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1answer
14 views

Existence of Gluing of Riemann surfaces

Consider two copies of holomorphic disks $\{ z \in \mathbb{C} \ | \ |z| \leq 1 \}$. Denote them by $\Delta_1$ and $\Delta_2$. Let $f$ be a diffeomorphism from boundary of first disk to boundary of ...
0
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1answer
28 views

Complex number raised to a complex number equality

If $a$ and $b$ are complex numbers, for which $z = x+iy$ does the formula $(z^a)^b = (z)^{ab}$ hold? My approach is the use the principal arguements $z^c = e^{cLog(z)}$ but this leaves me stuck, I've ...
1
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0answers
25 views

Fundamental questions about Logarithm and finding quadratic roots

Define: $(e^{iz}+e^{-iz})/2= cos z$ where $z \in \Bbb C $, i.e, the cosine function is defined for complex $z$. Now, is it true that for each $w \in \Bbb C $ there is $z \in \Bbb C $ such that $cos z ...
1
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1answer
25 views

Complex integration using parametrization

Let $C$ be the circle $|z-z_0| = r$ traversed counter-clockwise, and let $\alpha$ ne any nonzero real number. Parameterize $C$ by $z=z_0+re^{i\theta}$, with $-\pi < \theta < \pi$, and compute ...
5
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3answers
114 views

Why doesn't $\frac 1 z$ have an antiderivative in $\mathbb{C}\setminus\{0\}$?

Why doesn't $\frac 1 z$ have an antiderivative in $\mathbb{C}\setminus\{0\}$? I understand that the antiderivative could've been $\operatorname{Log}(z)$, but it always has atleast one branch cut. But ...
1
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1answer
23 views

Solving a complex Gaussian integral using a rectangle

Let C be the circumference of the rectangle with the vertices $-a, a, a +ib, -a+ib, a > 0$, traversed counterclockwise. Show that the sum of the integrals of $e^{-z^2}$ along the upper and lower ...
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votes
2answers
29 views

Jordan Curves must be zero? [on hold]

Explain why the following line integral must be zero $\int_C [e^{xy}(xy+1)dx +x^2e^{xy}dy]$ where $C$ is a Jordan curve in the plane.
1
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1answer
28 views

Green's formula on a positively oriented simple closed contour

If C is a positively oriented simple closed contour, use Green's formula to show that the area of the region enclosed by C can be calculated as $$\frac{1}{2i}\int_C \bar{z}dz$$ I approached this ...
0
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1answer
35 views

Modulus of a complex number raised to a real number

How would one go about arguing that if we have a complex number $z$ that isn't $0$, and $a$ as real number that $\lvert z^a\rvert = \lvert z\rvert^a$ without using principal arguments or set-valued ...