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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Degree of zero of a family of real analytic functions on a common interval

Given a family of real analytic functions on a common interval on the real line expanded about a common zero of the family of analytic functions, what can be said about the multiplicity of this zero ...
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39 views

Complex Limit $\lim_{n\rightarrow +\infty}\left [ i^i\left ( 2i \right )^{2i} \cdots\left ( ni \right )^{ni}\right ]$

Evaluate the limit: $$\lim_{n\rightarrow +\infty}\left [ i^i\left ( 2i \right )^{2i} \cdots\left ( ni \right )^{ni}\right ]$$ where we take into account the branch of the log. function that the ...
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3answers
85 views

Translation of a polynomial

I am given a complex polynomial $p(z) = a_0 + a_1z + \cdots + a_nz^n$, with $a_j \in \Bbb C$ for all $j$. Then, we fix $z_0 \in \Bbb C$ and define $P(z) = p(z+z_0)$, and I must prove that $P$ is also ...
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75 views

Is there a geometric insight from this exercise?

I am solving some exercises in a book I'm reading and so far all the exercises contained some insight. But then I got to the following exercise: Let $z,a\in \mathbb C$. Show that $$ ...
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31 views

Defining a harmonic function on an annulus with prescribed boundary values

Give a harmonic function $u$ on the annulus $\frac12 \le z \le 2$ such that on the outer boundary circle $|z|=2$ the boundary-value function is $u(2e^{i \theta})= e^{i \theta}$, while on the inner ...
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1answer
51 views

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0.

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0. Note: $f$ isn't defined on 0. I'm having a bit of trouble getting this ...
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3answers
84 views

Necessary condition for analyticity of $f(x+iy)=x^3+ax^2y+bxy^2+cy^3$

My aim is to show that the function $$f(x+iy)=x^3+ax^2y+bxy^2+cy^3$$ is analytic only if $a=3i$ , $b=-3$ , $c=-i$. For these values of $a$, $b$ and $c$, that $f$ is analytic follows from ...
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82 views

ML lemma question

Denote by $\Gamma_R$ the semicircle $\{z \in \mathbb{C} : |z|=R, 0\leq \arg(z) \leq \pi\}$ traversed in the counter clockwise direction. Using the $ML$ Lemma, show that $$|\int \limits_{\Gamma_R} ...
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2answers
67 views

unobvious cauchy integral formula

Use Cauchy's integral formula to compute the following: $$\int \limits_{\Gamma} \frac{\cos(z)+i\sin(z)}{(z^2+36)(z+2)}dz$$ where $\Gamma$ is the circle of centre $0$ and radius $3$ traversed in the ...
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1answer
40 views

If $f$ is a holomorphic function in a rectangle in the first quadrant, and $|f(z)| \leq Re(z)$, prove that $f = 0$ for all the rectangle.

If $f$ is a holomorphic function in a rectangle in the first quadrant, and $|f(z)| \leq Re(z)$, prove that $f = 0$ for all the rectangle. The rectangle is all $z \in \mathbb{C} = x + iy$ s.t. ...
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45 views

Show that $(z, w)$ is linearly dependent iff the imaginary part of $z\bar{w}$ is 0.

Consider $\mathbb{C}$ as $\mathbb{R}$-vector space. If $z,w \in \mathbb{C}$, show that $(z, w)$ is linearly dependent iff the imaginary part of $z\bar{w}$ is 0. I'm just unsure about the question and ...
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1answer
37 views

Complex analysis ~ Binomial theorem

Given the identity $ \binom {2n} {n} = \frac{1}{2\pi i} \int_{C_r} \frac{(1+z)^{2n}}{z^{n+1}}dz,$ with $C_r$ the unit circle, prove that $\forall n \in \mathbb{N}$: $\binom {2n} {n} \leq 4 ...
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142 views

Show that $ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1.3.5.7}{2^5} $

I'm trying to show the following. $$ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1\cdot3\cdot5\cdot7}{2^5} $$ This is a problem regarding contour integration. My complex analysis ...
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103 views

Determining whether a function is uniformly continuous

Determine whether $(4x-3)/(x-2)$ is uniformly continuous on the open interval $(1,2)$. I'm not sure how to start this as I have only answered these questions with closed intervals?
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113 views

Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where $R>1$

Let C be the circle of Radius $R>1$, centered at the origin, in the complex plane. Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where we employ a branch of the integrand defined by a ...
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23 views

Holomorphic functions and open map theorem

I'm working on a problem in matroid theory. Particularly, I'm interested in the realizability problem for a certain class of matroids over the complex filed $\mathbb{C}.$ I reduced my problem to the ...
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1answer
106 views

Radius of convergence of a power serise involving the Fibonacci sequence.

Consider the power series $$\sum_{n=0}^{\infty}a_nz^n.$$ where, $a_0=0$ , $a_1=1$ , $a_n=a_{n-1}+a_{n-2}$. Find the radius of convergence of the power series. MY Attempt : Clearly $\{a_n\}$ is a ...
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52 views

Control theory: Why doesn't the separation principle hold in nonlinear control theory?

It is widely known in control that separation principle is one of the best tool for pole placement and design of stabilizing controller in linear system. Many results also note the inability of ...
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32 views

An application of Rouches Theorem

Let $f$ be an entire function on the complex plane, with Taylor's expansion around zero as $f(z) = \sum_{k=0}^{\infty}c_{k}z^{k}$. Let $N(r)$ be the number of zeroes of $f$ in $D(0, r)$. Show that for ...
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57 views

Complex distributions - what are the appropriate test functions?

In the theoretical physics literature on conformal field theory, one encounters distributional formulas like $$ \frac{1}{\pi}\partial_{\bar z}\frac{1}{z} = \delta(z), $$ where $\partial_{\bar z}$ is ...
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68 views

contour integrals parametrising and solving

Use Cauchy's integral formula to compute the following: $$\int \limits_{\Gamma} \frac{e^{-z}}{z-1}dz$$ where $\Gamma$ is the square with parallel sides to the axes, centre $i$ and side length $5$ ...
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42 views

Integration imaginary and real part with branch cut

I have some problems with this integral $$ I=\int_{0}^{1}z(1-z)log(1-z(1-z)\frac{q^2}{m^2})dz $$ I see $z(1-z)$ get max value at $\frac{1}{4}$ and if $q^2>4m^2$ log function will be negative and ...
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1answer
104 views

If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$

If $f \in \operatorname{Hol}(D),f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$ $D = \{ z \in \mathbb{C} : |z| < 1 \} $ My thoughts so far: Let's say $f(0) = a$. ...
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1answer
43 views

Proof that there is a unique linear fractional transformation that maps three distinct points to three distinct points in the extended complex plane.

The following is a theorem and a proof from Complex Variables by Herb Silverman. The bolded are parts that I don't understand in the proof. Theorem: Given three distinct points, $z_1,z_2, z_3$ in the ...
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1answer
37 views

Show that $e^g=cf$ for some $ c\in \mathbb{C}\setminus\{0\}$

need help proving that g satisfies $e^g=cf$ for some $ c\in \mathbb{C}\setminus\{0\}$ where g is the anti derivative of $ f'\over f$ f holomorphic function. I tried expressing e in terms of power ...
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26 views

Holomorphic functions on a connected and compact domain

Consider the following theorem (see references at the end): If $X$ is a connected and compact complex manifold, then any holomorphic function $f : X \rightarrow \mathbb{C}$ is constant. What about ...
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91 views

Show f(z)/z is bounded

if $\lim\limits_{|z| \to \infty} \frac{f(z)}{z} = 0$ then how do I show that f is bounded. Intuitively, this makes sense to me but I having trouble writing it out formally. I was thing for $|z|>N$, ...
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51 views

If $|f| \leq e^{1/|z|^2}$ for every $z\in G$…

Let $G = \{z| Im(z)>0\}$, and $f \in Hol(G)$. If $|f| \leq e^{1/|z|^2}$ for every $z\in G$ then we have $ f\equiv_{|_G}0$ The teacher gave an hint but it is still a hard question , and I ...
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31 views

Examine the continuity of complex function

There is confusion regarding continuity of the following function. When solving in polar form it comes continuous but when solving in $x$ and $y$ then not continuous. Examine the continuity of ...
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58 views

Conformal mapping question: Joing two rays using the transformation $f(z) = \frac{z-b}{z-a}$.

I am currently trying to solve some problems in order to refresh my knowledge on linear fractional transformations. I am trying to conformally map the complex plain with cuts along the rays ...
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1answer
37 views

Complex Series Convergence of Quotient

For $|z-a|<r$ let $f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n$. Let $g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n$. Assume $g(z)$ is nonzero for $|z-a|<r$. Then $b_0$ is not zero. Define $c_0=a_0/b_0$ and, ...
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1answer
46 views

Jordan Canonical Form of Real Matrices

Let $A$ be an $m \times m$ real matrix, and let \begin{equation} A=C^{-1} J C, \end{equation} \begin{equation} A=\tilde{C}^{-1} J \tilde{C}, \end{equation} be two Jordan decompositions of $A$, where ...
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3answers
111 views

How do I integrate $\int_{0}^{\infty}\frac{\cos(ax)-\cos(bx)}{x^2}\text{d}x$?

How do I integrate $\int_{0}^{\infty}\frac{\cos(ax)-\cos(bx)}{x^2}\text{d}x$, for positive and real $a,b$? I know the contour that I have to use is a semicircle with a small semicircle cut out near ...
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1answer
38 views

Conditions on coefficients of complex power series to ensure it is real

Given a complex valued function $f(z)=\sum_{n=0}^{\infty} a_nz^n$ with radius of convergence $R>0$, and $\rho\in (0,R)$, is there an if and only if giving that $f([0,\rho])\subset \mathbb{R}$? So ...
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2answers
359 views

Why does a branch need to be defined in complex analysis?

$\newcommand{\arg}{\operatorname{arg}}$Say we have the principal branch, $\arg_\tau(z)$. This is defined so that $\arg_\tau(z) \in (-\pi,\pi]$. Why is it necessary to define the limits on the ...
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1answer
25 views

Prove that their product function $h(x,y)=f(x,y) \cdot g(x,y)$ also satisfies Laplace's equation

Let $w=f(x,y)$ and $u=g(x,y)$ be two real functions of the real variables $x$ and $y$ which both satisfy Laplace's equation. Prove that their product function $h(x,y)=f(x,y) \cdot g(x,y)$ also ...
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1answer
56 views

Rodrigues' formula and Legendre’s polynomials

Let $$P_n(z)=\frac{1}{n!\space2^n}\frac{d^n}{dz^n}(z^2-1)^n$$(i) Show that this is a polynomial of order n. I can see that this is Rodrigues' formula, which means that it is a Legendre polynomial. I ...
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61 views

Möbius Transformation

Let C be the circle with center 0 and radius 1. Find a Möbius transformation which trans- forms C onto C and transforms 0 to 1/2. Notes: consider h(z)= az+b/cz+d. then h(0) = 1/2 so 2b = d Then to ...
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2answers
77 views

Integration using Cauchy's Theorem

I am attempting to evaluate the integral $$\int_C\left(z+\frac{1}{z}\right)^{2n}\frac{dz}{z}$$ where C is the unit circle centered at the origin. Using parameterized $z=e^{i\theta}$ and got that ...
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2answers
65 views

Compute Power Series Convergence to a function

Consider the next power series $$ \sum_{n=1}^{\infty} \ln (n) z^n $$ Find the convergence radius and a the function $f$ to which the series converges. I have easily found that $R=1$ is the ...
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1answer
27 views

differentiability of partial derivatives

Prove that if f a function of n variables is continuously differentiable in an open subset U of $R^n$ then the partial derivatives of f are continuously differentiable. I used the definition of f ...
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1answer
52 views

If $E(z)= \sum _{n=0 }^{\infty }\frac {z ^n } {n! } $, how is $E(0) $ defined?

If $E(z)= \sum _{n=0 }^{\infty }\frac {z ^n } {n! } $, how is $E(0) $ defined? The exponential function for complex $z $ is defined in Rudin's principles as the power series $ \sum _{n=0 }^{\infty ...
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30 views

is that function must be constant under the following conditions

I'm talking about complex function $f$ is analytic function on a region $D$ that include the point $z=0$. for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition ...
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88 views

Complex Manifold: Stokes

This is a lemma for: Helffer-Sjöstrand Given the complex plane. Consider a smooth function: $$f_E\in\mathcal{C}^\infty_c(\mathbb{R}^2):\quad\bar{\partial}f_E\restriction_\mathbb{R}=0$$ How to apply ...
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1answer
37 views

Radius of Convergence of Power Series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$

What is the radius of the power series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$? Justify your answer. My steps toward a solution I can express $\tanh$ simpler as: \begin{align*} \tanh z ...
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1answer
29 views

Upper bound on complex integral

If $f(z)=\sum_{n=0}^{\infty}c_nz^n$ and we know $$c_k=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z^{k+1}}dz$$ for $\gamma$ a circle of radius r centred at the origin, traversed once in the positive ...
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0answers
28 views

$\sum_{n\ge0}f_n$ converges locally uniformly on $U$

How can I conclude that $\sum_{n\ge0}f_n$ converges locally uniformly on $U$, with $U=\mathbb C\setminus\mathbb R_-$ $f_n(z)=\frac{(-1)^n}{z+n}, \quad z\in U$ I've already proved that ...
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46 views

Integrate $\frac{f(x)}{x^{n+1}}$ on $S_{\epsilon}^{+}$

Let $f(z)$ be a holomorphic complex function that has $0$ as a root with multiplicity $n$ Denote by $S_{\epsilon}^{+}=\{x+yi| x,y \geq 0 , \sqrt{x^2+y^2} = \epsilon\}$. What is ...
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1answer
26 views

Complex Integration with Power Series

Let $f(z)=\sum_{n=0}^{\infty}c_nz^n$ have radius of convergence $R>0.$ Use the fact that $$\sum\limits_{n=0}^{\infty}\int_\gamma c_n z^ndz=\int_\gamma ...
23
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2answers
501 views

Existence of an holomorphic function

Is there a simple way to prove this fact : For all holomorphic functions $f : \mathbb C \to \mathbb C$, there is an holomorphic function $\psi : \mathbb C \to \mathbb C$ such that $$\psi(z+1) = ...