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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Calculating the complex line integral along a square

Calculate the complex line integral of the holomorphic function g(z)=1/z along the counterclockwise-oriented square of side 2, with sides parallel to the axes, centred at the origin. I parametrized ...
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43 views

Why does shifting the integration line from the real-axis not affect the integral $ \int_{-\infty}^\infty e^{-x^2} dx $? [duplicate]

I know that $$ \int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}, $$ and clearly this integral is invariant under translation along the real axis. But today I learned that $$ \int_{-\infty}^\infty ...
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1answer
20 views

Power series and zeros

When is a power series equal to zero? Example: Take $\sum_{n=0}^\infty a_n(z-z_0)^n$. Is this power series equal to zero only at $z=z_0$ if we assume that we have infinitely many nonzero $a_n$? ...
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16 views

Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
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24 views

Express complex function in the form $u+iv$

One of the parts of the question I'm working on goes something like this: Express $z^i = \exp(i \log_I(z))$ in the form $u+iv$, where $u,v$ are real-valued functions, and the log is defined on the ...
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22 views

maple plot of Belyi function

I would like to understand how to construct Figure 5 of the paper Composition is a generalized symmetry by Alexander Zvonkin: The hypermap/dessin d'enfant of Figure 4 is while the Belyi function ...
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1answer
117 views

Why can the function only have a finite amount of poles in the plane?

There is a proof in my book that I do not understand. I have highlighted it in red. Why can they say that f only has a finite number of points? What makes them able to exclude the countable number in ...
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2answers
47 views

complex limit,. theoretical justification for beeing able to calculate the limit this way

I have a complex limit I have to calculate. I know how to calculate it, but I would like a theoretical justification for why we can calculate it this way. Look at this: I want to calculate: $\lim_{z ...
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28 views

For all $\xi \in \mathbb{C}$ we have $e^{-\pi\xi^2}=\int_{-\infty}^\infty \! e^{-\pi x^2}e^{2\pi ix\xi}\ \mathrm{d}x.$

This is Exercise 2.4 in Stein & Shakarchi's Complex Analysis. Prove that for all $\xi \in \mathbb{C}$ $$e^{-\pi\xi^2}=\int_{-\infty}^\infty \! e^{-\pi x^2}e^{2\pi ix\xi}\ \mathrm{d}x.$$ They prove ...
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23 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
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408 views

the zeros of sin(z), where z is a complex number

How do I find the zeros of sin(z), where z is a complex number? I know that along the real line we have zeros along $k\pi$, where k is an integer. But what about the rest of the plane? The taylor ...
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17 views

Analyze branch cuts and discontinuities of function $f(z)=\sqrt{1-z^2}$

Analyze the function $f(z)=\sqrt{1-z^2}$, where the square root is defined by the principal branch of the log function. Where does it have discontinuities? Here's what I did: We have $I = ...
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1answer
35 views

(complex variables) Find roots of a complex variable such that $1+\omega^m+\omega^{2m}+ \cdots + \omega^{(n-1)m} = 0$

I need some help understanding the intuition behind the following question: Consider the root of $z^n=1$ given by $\omega = \cos\frac{2\pi}{n} + i\sin\frac{2\pi}{n}$. For which integers $m$ is ...
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20 views

Is the exponential map of GL(n,C) holomorphic?

Let $GL(n, \mathbb{C})$ be the complex general linear (Lie) group consisting of all invertible complex $n\times n$ matrices, and $gl(n,\mathbb{C})\cong C^{n^2}$ be its Lie algebra. The exponential map ...
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46 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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25 views

Finding the set of analytic functions whose image is a subset of a given set

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help ...
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37 views

Constant function

Let $z \mapsto f(z)$ and $z \mapsto \overline{f(z)}$ be analytic functions on a plane $\Omega$. Show that $f$ is constant function. I know that $f$ is constant if it's derivate is $0$ on all ...
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33 views

Calculating the real and imaginary parts of a holomorphic function

Calculate the real and imaginary parts of the holomorphic function $f(z)=z^2\cos(z)-e^{z^3-z}$ and verify directly that each of these functions is harmonic. I believe I know how to the question, ...
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21 views

(complex variables) Express $\cos3\phi$, $\cos4\phi$, and $\cos5\phi$ in terms of $\cos\phi$ and $\sin\phi$.

I'm not sure what the intuition is supposed to be behind this question. This is my attempt at $\cos3\phi$. Does this look agreeable? We can use the identity $e^{iz}=\cos z + i \sin z$. Let $z = 3 ...
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25 views

Finding where function is analytic

\begin{align*} f: z \mapsto f(z)&=\frac{az+b}{cz+d} \\ &\text{when $a,b,c,d \in \mathbb{C}$} \end{align*} Where $f$ is analytic Where $f'(z)=0$ Find inverse function for $f$ (if there is) ...
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26 views

The differentiability of the complex valued function $(Rez)(Imz)z\over|z|^2$

$$ f(z) = \left\{ \begin{array}{ll} \Re(z)\Im(z)z\over|z|^2 & \quad z \neq 0 \\ 0 & \quad z = 0 \end{array} \right. $$ I want to prove that this ...
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18 views

Complex number theory - limits

Can anyone please help me with those two limits? I have missed the first two weeks of a new semester, so really not experienced with this. $$a) \lim\limits_{z \to \infty} \frac{z^2-\overline z^2 + ...
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29 views

Roots of polynomial

I came across when reading paper: Given $f'(z)+\alpha zf''(z) + \gamma z ^2f'''(z) $ where $\mu = \tfrac{(\alpha-\gamma)-\sqrt{(\alpha-\gamma)^2-4\gamma}}{2}$,$\quad$ $\nu+\mu=\alpha-\gamma$, ...
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1answer
10 views

Limit Point of Zero Points Implying a Zero Function

I'm reading through a proof on the uniqueness theorem of Laurent series in my lecture notes and came across this sentence: The point $c$ ($\in \mathbb{C})$ is a limit point of zero points of the ...
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Choosing of root in proving $\mathrm{Arccos} z = -i \mathrm{Ln}(z + \sqrt{z^2-1})$

Prove: $$\mathrm{Arccos} z = -i \mathrm{Ln}(z + \sqrt{z^2-1})$$ Where $z \in \mathbb C$ My Problem: When I set $$z = \cos t = \frac{1}{2}(e^{it} + e^{-it})$$ I got $$(e^{it})^2 - 2z \cdot e^{it} + 1 ...
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28 views

Understanding the statement that $\varphi(\emptyset)=0$ implies $\varphi$ is not identically $\infty$

The proposition is from "Real and Complex Analysis" by Rudin.It states: Let $s$ be a nonnegative measurable simple function on $X$ . For $E\in\mathfrak M$ (where $\mathfrak M$ is a $\sigma$-algebra ...
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38 views

Question about a step in the proof of the Cauchy-Schwartz inequality in $\mathbb{C}$

I'm studying the proof of the Cauchy-Schwartz inequality, which states that for complex numbers $z_1,\ldots. z_n,w_1,\ldots, w_n$ we have $$ \Big\vert\sum_{j=1}^nz_jw_j \Big\vert^2\le ...
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15 views

Maximum Value - Analytic function

I am having a hard time figuring out where to start and what results to use to address the following question: Suppose $f(z)$ is analytic in the unit disc $D=\{z:|z|<1\}$ and continuous in the ...
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48 views

Limit of the quotient of two holomorphic functions

I want to calculate the limit as $\displaystyle \lim_{z \rightarrow 0} \frac{\cos(z)-1}{\sin(2z)}$. I know that for real variables, this value is $0$, using L'Hopital. Is there some way to justify ...
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14 views

Topological conjugacy in Hénon map

$\textbf{Definition:}$ $\textit{(Topologically conjugate)}$ Let $f:A\rightarrow A$ and $g:B\rightarrow B$ be two maps. $f$ and %g% are said to be topologically conjugate if there exists a ...
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34 views

How to calculate an integral of the form $\int_{-\infty}^{\infty} e^{-\left(\alpha x + i\beta\right)^2}\, dx $?

How can I calculate the following complex definite integral, where $\alpha > 0$ and $\beta \in \mathbb{R}$? $$ \int_{-\infty}^{\infty} e^{-\left(\alpha x + i\beta\right)^2}\, dx $$
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41 views

Engineer searching for calculus and complex analysis books without limits

I am an engineer and I need to study calculus and complex analysis without too much limits or Riemann sums or proofs. I mean on the differentiation and integration levels and higher (not digging ...
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44 views

Competencia Iberoamericana Interuniversitaria

Let $f$ a rational function with complex coeficients and without mutiple roots in the denominator. Let $u_0,u_1,...,u_n$ ($n \ge 1$) complex roots of $f$ and $w_1,w_2,...,w_n$ roots of $f'$ (each root ...
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Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
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1answer
35 views

Does $\lim_{x \to 0}({z^2\over \overline z})$ exist? $(z\in \mathbb{C})$

I am trying to figure out if $\lim_{x \to 0}({z^2\over \overline z})$ exists or not. This is a way I though to show that this does not exist but I am not entirely sure. Let $a_n={1\over n}$ and ...
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28 views

$f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
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1answer
30 views

Is it true that $Arg (\prod _{n=1 } ^N p _n )= \sum _{n=1 } ^N Arg (p _n )$?

Is it true that $Arg (\prod _{n=1 } ^N p _n )= \sum _{n=1 } ^N Arg (p _n )$? I know that $Arg(z _1 +z _2)=Arg(z _1)+Arg(z _2) $ and I wonder how this result generalizes. (I think I have seen this ...
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Remmert, exercise 5, chapter 7 section 3. Theory of Complex Functions.

Let $f,g$ be holomorphic on a domain $D$, and let $r>0$, $\bar B(0,r)\subseteq D$. Suppose that $a$, $|a|=r$ is such that $g(a)=0,g'(a)\neq 0$, $f(a)\neq 0$ and that $g$ doesn't vanish in $\bar ...
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51 views

Quantities $g_2$, $g_3$, $\Delta$

This question is somewhat related to this one. Let $\lambda$ be the modular lambda function. Greenhill (Elliptic Functions, p. 57) states that we may put $$g_2 = \frac{1 - \lambda + \lambda^2}{12}, ...
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I need some clarification on the term of “measurable on” and “continuous on”

I run across theorems similar to this one: "If $f$ is a complex measurable function on $X$, there is a complex measurable function on $X$ called $\alpha$ such that $\vert\alpha\vert=1$ and ...
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135 views
+50

Prove complex function goes counterclockwise around the unit circle at least once

Suppose $f$ is non-constant and holomorphic in a neighborhood of the closed unit disk, s.t. $|f(z)| = 1$ for all $|z| = 1$. Then show that as $f(e^{i\theta})$ traverses the unit circle and makes at ...
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23 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
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21 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
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36 views

Value of the integral $\int_0^{2\pi}\int_0^{2\pi}\delta(k_2\cdot e^{i\theta}+k_3\cdot e^{j\phi} +z )d\theta d\phi$

I would like to compute the following integral, which arises from some physics problems, where $k_2$, $k_3$ are real, $z$ is in general complex, $$ \int_0^{2\pi}\int_0^{2\pi}\delta(k_2\cdot ...
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1answer
17 views

Is this log identity true?

I'm wondering if the exponent property carries forward to the complex log. In other words, for some complex numbers $z$ and $w$ does $\ln(z^w) = w\ln(z)$?
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24 views

Principal branch of the complex logarithm does not always satisfy the product formula

My book asks to prove: $\text{Ln}[i \cdot (-1+i)]$ does not equal to $\text{Ln}(i) + \text{Ln}(-1+i)$ where $\text{Ln}$ gives the principal log of the complex number. I don't see why this is true ...
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11 views

Integration over subsets of the complex plane.

Original Problem: Let $\Omega\subset \mathbb{C}$ be an open set and let $f:\Omega\to\mathbb{C}$ be holomorphic such that $f\in L^{2}(\Omega)$. Show that if $B(z,r)$, the ball of radius $r$ ...
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30 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
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16 views

If $a,b,c$ are the vertices of a triangle in the complex plane, prove that the area of a triangle is $\frac{1}{2}|b-c|^2|Im\frac{c-a}{c-b}|$

I have trouble with this proof. I can get as far as the fact that we must position the vertex $c$ on the origin and then rotate by a factor of $|b-c|$. But then this gives: \begin{align*} ...
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1answer
32 views

Show that $T$ has a fixed point $z_{0}$ [on hold]

Let $T$ be a Mobius transformation with a periodic orbit $ O=\{ z_{1},\ldots , z_{q} \}$ of period $q\ge2$. i.e the $z_{i}$'s are distinct and $z_{1}\mapsto T(z_{1})=z_{2}\mapsto ...