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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2
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1answer
19 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. ...
0
votes
2answers
24 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 ...
1
vote
1answer
19 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 ...
-1
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0answers
63 views

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

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 ...
-2
votes
0answers
25 views

Show that $ \int^{\infty}_{0} \frac{\cos x}{\cosh x} dx = \frac {\pi}{2\cosh \pi / 2}$ [on hold]

Show that $ \int^{\infty}_{0} \frac{\cos x}{\cosh x} dx = \frac {\pi}{2\cosh \pi / 2}$ using the contour $ (-R,0) \rightarrow (R,0) \rightarrow (R,\pi) \rightarrow (-R,\pi) \rightarrow (-R,0) $ ...
0
votes
1answer
20 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?
-1
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0answers
30 views

Show that $\int^{\infty}_{0} \frac{x^m}{x^n+1} dx= \frac{\pi}{n \sin(\pi(m+1)/n)} $ [on hold]

Show that $\int^{\infty}_{0} \frac{x^m}{x^n+1} dx= \frac{\pi}{n \sin(\pi(m+1)/n)} $ where $n$ and $m$ are nonnegative integers with $n-m \geq 2$. A suggested contour is using a wedge of angle $2\pi / ...
-2
votes
2answers
47 views

Show that $\int^{2\pi}_{0} cos^m \theta d\theta = \frac{2 \pi}{2^m} \frac{m!}{[\frac{m}{2})!]^2} $ when m is a positive even integer [on hold]

Show that $\int^{2\pi}_{0} cos^m \theta d\theta = \frac{2 \pi}{2^m} \frac{m!}{[(\frac{m}{2})!]^2} $ when m is a positive even integer
0
votes
1answer
30 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 ...
0
votes
0answers
14 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 ...
4
votes
0answers
26 views

large parameter asymptotics for the integral of an alternating sum

Consider the alternating sum $$G(y)=\sum_{n=1}^\infty(-1)^{n+1}\frac{n}{n^2y^2+1}\exp(-n^2y^2t)$$ with parameter $t>0$. I want to find the asymptotics of $\int_0^\infty G(y)dy$ as $t\to\infty$. ...
1
vote
1answer
36 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 ...
0
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0answers
10 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 ...
0
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0answers
17 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 ...
0
votes
0answers
13 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 ...
1
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1answer
23 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 $5$ ...
2
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0answers
21 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 ...
2
votes
1answer
53 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 $f(0) = a$. ...
1
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1answer
19 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 ...
1
vote
1answer
25 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 ...
0
votes
2answers
18 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 ...
1
vote
3answers
48 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 $|z|>N$, ...
0
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0answers
22 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 ...
-3
votes
1answer
21 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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votes
0answers
12 views

Proving the existence of some delta for a convergent complex function [on hold]

How would you go around proving the existence of some $\delta$, for an arbitrary convergent complex function? Given; $\lim_{z \to z_0} g(z) = B$, where $B\ne 0$.
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0answers
27 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 ...
0
votes
1answer
20 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, ...
1
vote
1answer
26 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 ...
1
vote
3answers
82 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 ...
1
vote
1answer
21 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 ...
1
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0answers
14 views

Schwarz Christoffel Formula

Does every function of the form given by the Schwarz Christoffel formula necessarily be a mapping from the circle into a polygon? I'm just curious if this is true because ordinarily it just seems that ...
5
votes
2answers
319 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 ...
2
votes
1answer
19 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 ...
2
votes
1answer
26 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 ...
0
votes
0answers
34 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 ...
1
vote
2answers
53 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 ...
2
votes
1answer
38 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 ...
1
vote
1answer
22 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 ...
0
votes
1answer
49 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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0answers
34 views

Direct proof that if $f$ is holomorphic then $f$ has continue partial derivatives in every point

If $U$ is a open set of $\mathbb C$ and $f:U\to \mathbb C $, then $f$ is holomorphic if there exists $ \lim\limits_{h\to 0} {\dfrac{f(z+h)-f(z)}{h}} $ for every $z=x+iy\in U$. It can be easily proved ...
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0answers
30 views

show that the complex function is non-holomorphic everywhere [on hold]

Can someone please show me how to answer this question: $f(z) = (z^2)\times\overline{z}$
0
votes
1answer
23 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 ...
0
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0answers
41 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 ...
0
votes
1answer
25 views

length of parabolic curve

How do you find the length parabolic curve z(t)=t-3it from t=0 to t=1 using the hyperbolic definition of sine and cosine. I am kind of stumped and if I can have a general set up of the problem that ...
1
vote
1answer
27 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 ...
0
votes
1answer
15 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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
21 views

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

Let $f(x)$ be a continous complex function that has $0$ as a root with multiplicity $n$ Denote by $S_{\epsilon}^{+}=\{x+yi| x,y \geq 0 , \sqrt{x^2+y^2} \leq \epsilon\}$. What is ...
0
votes
1answer
23 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 ...
8
votes
0answers
125 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) = ...