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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16 views

Cauchy integral formula

Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! ...
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0answers
15 views

There exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(z)| = \infty$

Suppose $f$ has an Essential Singularity at $Z_0$. Then there exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(z)| = \infty$ Here two cases arise If there exist a nbd of ...
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0answers
12 views

Interior of a closed curve

I'm working through a proof that contains this particular argument which I think is highly non-trivial but no justification is given - the context is complex analysis and the proof is of Lindelof's ...
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1answer
15 views

A question about zeroes and poles of complex functions.

Let $f (z)=\frac {z}{z} $ be a complex function. Is 0 a zero, a pole, or neither of these?
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1answer
25 views

How can I find these partial derivatives?

I'm reading a book which gives this function $f(x,y)=x^2y/(x^2+y^2)$ if $(x,y)\neq (0,0)$ and $f(0,0)=0$ as a $C^1$ function in $\mathbb R^2-\{(0,0)\}$, continuous in $(0,0)$ and it has the partial ...
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1answer
61 views

Differences between real and complex analysis?

To start with, real analysis deals with numbers along the (one dimensional) number line, while complex analysis deals with numbers along two dimensions, real and imaginary, Cartesian style. Could this ...
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2answers
35 views

Type of singularity of $\sin(z)/z^3$ at $0$

I would have thought that this is a pole of order $3$ but on the answers it says it is of order $2$. I don't see why...
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0answers
33 views

Complex function

Can anyone give me a hint to approach this question? I haven't done anything like this before so I'm bit confused about what this question is asking. Thank you very much for all your help.
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2answers
20 views

Finding Laurent series with imaginary numbers

$$f(z)=\frac{2z}{z^2+1}=\frac1{z-i} +\frac1{z+i}$$ Find Laurent series in powers of $z$ in the domain $|z|<1$. So I got to find two Taylor series of the two terms in the function but how do you do ...
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0answers
14 views

How can I get the region of Convergence for zcos1/z? [on hold]

Find laurent Series and the region of convergence for ZCos(1/Z), I can find the series but I can't get the region of convergenc
2
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0answers
18 views

An extension of Kato's Selection Theorem?

One formulation of the well-known Kato Selection Theorem states that, given an analytic family of $n \times n$ complex, symmetric matrices $M(t)$, one can choose an orthonormal basis $\{e_i(t)\}_{i = ...
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1answer
21 views

Analytic paths through converging sequences in the complex space.

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
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1answer
19 views

How can I find Laurent Series and the region of convergence for $z/((z+1)(z+2))$ for$ z= -1$? [on hold]

How can I find Laurent Series and the region of convergence for $z = -1$ of $$ \frac{z}{(z+1)(z+2)} $$
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0answers
17 views

Let f be a analytic map that sends the annulus A(0,1,2) to the unit disk such that $|z|=1,|z|=2$, Furthermore f is not constant. Prove:

Let $f$ be a analytic map that sends the annulus $A(0,1,2)$ to the unit disk such that $|z|=1,|z|=2$ get mapped to the points $|f(z)| = 1$. Furthermore f is not constant. Prove: 1) $f$ has at least ...
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1answer
23 views

Let $f\in H(\mathbb{C})$. Prove that: $\exists_{M\in\mathbb{R}^+} \forall_{z\in\mathbb{C}}\ \ \ \ |f(z)|> M \Rightarrow f$ is constant

Let $f\in H(\mathbb{C})$. Prove that: $\exists_{M\in\mathbb{R}^+} \forall_{z\in\mathbb{C}}\ \ \ \ |f(z)|> M \Rightarrow |f(z)|> M \Rightarrow f$ is constant Completely don't know how to bite ...
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0answers
36 views

Laurent series confusion

I've split it up into partial fractions and got $1/z$ - $2/(z-1)$ + $1/(z-2)$ but I'm unsure sure what to do now. I think I have done part $(i)$. I get $$z^{-1} + \sum_{n=0}^\infty ...
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1answer
48 views

How find the poles/residues of $\int_{-\infty}^\infty \frac{x^2 \, dx}{1 + x^4}$

I'm trying to find the poles/residues of this integral: $$\int_{-\infty}^\infty \frac{x^2 \, dx}{1 + x^4}$$ I've been given this attempt for a solution, but I don't really understand the procedure ...
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1answer
15 views

Question about a certain step in Rudin's General Cauchy Theorem proof

I am having trouble seeing a certain claim that Rudin makes in proving his "Global Cauchy's Theorem": $\textbf{Cauchy's Theorem.}$ Suppose $f$ is holomorphic in $\Omega$, which is an open set in ...
2
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1answer
25 views

Laurent series of $1/({z^3-z})$

Question: Find the Laurent series of the function $$f(z) = \frac{1}{z^3 - z}$$ at the domain $|z-1|>2$. Attempt: So we have $$\frac{1}{z(z-1)(z+1)}$$ and we only have to find a Laurent ...
3
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1answer
39 views

Show that an entire function is a proper if and only if it is a nonconstant polynomial

Show that an entire function (Holomorphic on $ \mathbb C$) is proper if and only if it is a non constant polynomial. Def:A map $f:X\to Y$ is called proper if $f^{-1}(K)$ is compact for every ...
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0answers
41 views

How does $\cos (2z) = e^{2zi}$?

In my notes, they are solving $$\int \limits_{- \infty}^{\infty} \frac{\cos(2x)}{x^2 +1}$$ and they let $$f(z) =\frac{e^{2zi}}{z^2 +1}$$ but how did the numerator become that? I wrote it as $\cos(2z)$ ...
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1answer
51 views

suppose $f(x)$ is an entire function and everywhere $|f'(z)| \leq |z^2+1|$ and further $f(0) = f'(0) = 1$. Determine $f$

Suppose $f(z)$ is an entire function and everywhere $|f'(z)| \leq |z^2+1|$ and further $f(0) = f'(0) = 1$. Determine $f$. I tried using Liouville's theorem but i don't know if $f'(z)$ is an entire ...
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1answer
23 views

About Fourier transform and complex conjugate

why this passage is correct ? \begin{equation*} \mathscr{F}[h(-\tau)] = H^*(f), \end{equation*} when $h(\tau)$ is a real function of real variable $\tau$, and $H^*(f)$ is the complex conjugate of ...
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0answers
16 views

Contour integration, cos(z)sin(z)

Evaluate \begin{equation*} \int_{\Gamma}\cos(z)\sin(z)dz,~\Gamma:\gamma(t):=\pi t+(1-t)i,~0\leq t\leq 1. \end{equation*} I think I should do it using this \begin{equation*} ...
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0answers
36 views

Is $\overline{z} $ independent of $z $?

Ahlfors' Complex Analysis says the following: $x=1/2(z+\overline {z} )$ and $y=-1/2i (z-\overline {z}) $. Hence, for a function $f (x, y) $, we have $\frac {\partial f}{\partial z}=1/2(\frac ...
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0answers
21 views

Show that $f(V_0)\cap f(V_1)\neq\emptyset$

Let $U$ be a connected subset of $\mathbb C$ and $z_0,z_1\in U$ and if $f$ is holomorphic on $U\setminus\{z_0\}$, with essential singularity in $z_0$, prove that for any subsets $V_0,V_1$ of $U$ ...
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0answers
28 views

Algebraic approach to Local Analytic Complex Geometry

I'm attending a second course in Complex Analysis from a geometrical point of view. In the final part of the course we have discussed about germs of complex analytic sets and their algebraic ...
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2answers
34 views

let $f$ be holomorphic on the unit sphere and $|f(z)| = 1$ for $|z| = 1$ and $f(-1) = 1$. Furthermore $f$ has no zero's, determine $f$

let $f$ be holomorphic on the unit sphere and continous on the closure, suppose $|f(z)| = 1$ for $|z| = 1$ and $f(-1) = 1$. furthermore $f$ has no zero's, determine $f$. So far i know with the ...
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1answer
13 views

Interpolation with nonvanishing constraint

Let $x_1,x_2,\ldots,x_n$ be distinct complex numbers. Let $y_1,y_2,\ldots,y_n$ be nonzero complex numbers, and let $K$ be a bounded subset of $\mathbb C$. Does there always exist a polynomial $P$ such ...
0
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1answer
15 views

Differentiation or integration term by term of the Laurent series!

Let $f(z)$ be an analytic function in the annual $r< |z|<R.$ Then $f(z)$ has the Laurent expansion series in this annual. My question is that: Can we derivative (or integrate) term by term from ...
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1answer
22 views

Find value of complex function at a point

Let $f(z)$ be analytic in $ D = \{z \in \Bbb C : |z| < 1\}$, and $f(z) = 1$ when $Im(z) = 0$ and $-\frac{1}{2} \leq Re(z) \leq \frac{1}{2}$. What is the value of $f(\frac{1}{2}+i\frac{1}{2})$? I'm ...
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0answers
35 views

Inequality on complex polynomial

For every $a\geq 0$, let $p_a(z)=1-z+az^3$. What is the maximal value of $a$ such that $$ p_a(|z|)\leq |p_a(z)| $$ for all $z\in \mathbb C$? EDIT: I claim that $a=\frac{4}{27}$ is the maximal value. ...
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2answers
27 views

Taylor expansion of a complex function

Trying to find Taylor series of $$\frac{z^2}{(1+z)^2}$$ I write it in the form $1- \frac{2}{1+z} + \frac{1}{(1+z)^2}$ and I can find Taylor expansion for each factor, is there another method without ...
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2answers
39 views

Taylor series of $1\over z^2$

How to find the Taylor series of $1\over z^2$ near $2$ ( in the power of $z-2$) I have tried to write it in the form: $1\over ((z-2)^2+4z-4)$ But I reached nothing, any help please
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0answers
14 views

Meromorphic complex function [on hold]

I want to find all function $f$ which are meromorphic in $C$ and satisfy $|f(z) - tan(z)| < 2$ for all $z$ which are neither poles of $f$ nor poles of $tan(z)$
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0answers
23 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
-1
votes
1answer
39 views

Show that $f$ is constant in $D(0,1)$. [on hold]

Considere this, $f: D(0,1) \to D(0,1)$ analytic. Suposse that $|f(z^2)|>|f(z)|$, for all $z \in D(0,1)$. Show that $f$ is constant in $D(0,1)$. Any help.
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1answer
27 views

Taylor series in complex analysis [on hold]

I am working on finding the Taylor series of $$\frac1{az+b}$$ in powers of $z.$ How to start with it Any help in details...
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1answer
20 views

Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case.

I'm having difficulty understanding the solution to the following problem. In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a ...
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0answers
19 views

Exactly one supporting line for a $C^1$ Jordan curve [on hold]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a convex Jordan curve (closed, simple, continuous) that has $C^1$ regularity, with $\gamma '(t)\neq 0,\ \forall t\in [a,b]$. Prove that there is exactly one ...
2
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0answers
35 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
2
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1answer
45 views

Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty ...
-1
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1answer
26 views

Antiholomorphic function

Let f be an antiholomorphic function in C. $z_0 \in C - C(0,1). $ Show that $\frac{1}{2 \pi i}\oint \frac {f(z)}{z-z_0} = \begin{cases}f(0) &\text{for } |z_0| < 1\\f(0) - f(\frac{1}{z_0}) ...
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1answer
46 views

Why is $|e^{i \lambda z}| |e^{- \lambda y}|= |e^{- \lambda y}|$ here?

Let $z \in \Gamma (R)$ where this is the upper semi circle centred at the origin with radius $R>1$. Let $z=x+iy$ with $x \in \mathbb{R}$ and $y \geq 0$. So $$|e^{i \lambda z}|=|e^{i \lambda z}| ...
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1answer
37 views

Complex Variables 1 (Montel's thrm

given $0 < c \in R$, there exist $\epsilon > 0 $ such that $a_k \in C$ & $\sum_{k=1}^{\infty}|a_k| \leq c$ implies $\sup_{\frac{1}{2}\leq x\leq1}|1 - \sum_{k=1}^{\infty}a_k x^k| \geq ...
0
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1answer
23 views

Show that if $f(z)=\frac{\operatorname{Log}z}{z-1}$ when $z\neq 1$ and $f(1)=1$, then $f$ is analytic throughout the domain.

$\operatorname{Log}z=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(z-1)^n \; (|z-1|\lt 1).$ Use this fact to show that if $$f(z)=\frac{\operatorname{Log}z}{z-1} \; \text{when} z\neq 1$$ and $f(1)=1$, ...
2
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1answer
36 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
0
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1answer
33 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
-3
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0answers
36 views

Best way to find Residue? [on hold]

I know that this is a strange question to ask on this website but I am dying to know a method that you can always use to find the residue of any complex function. Please help! We learned three types ...
0
votes
0answers
37 views

Laurent series for $\frac{2}{(z)(z-1)(z-2)}$

! So I think I am getting the hang of Laurent Series, but having a bit of trouble with one of the fractions for part a). So I split this up in to partial fractions: $\frac{1}{z} - ...