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, ...

learn more… | top users | synonyms (2)

2
votes
1answer
8 views

Find $\int _{\Gamma} \frac{\cos(2z)}{(z-\pi/4)^2}dz$

Where $\Gamma$ consists of the sides of a triangle with vertices $i$, $-1-i$ and $\pi -i$. I think we use Cauchy's integral formula but I cant get it in the standard form of it. I don't think partial ...
0
votes
1answer
15 views

Largest $R$ value in domian $0<|z-1|<R$

Determine the largest real number $R>0$ such that the Laurent series of $$f(z)=\frac1{z-1} +\frac2{z-i}$$ about $z=1$ converges for $0<|z-1|<R$. The singularities are $1$ and $i$. But in the ...
0
votes
2answers
27 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
0
votes
1answer
17 views

Laurent series in domain $|z|>0$

Find Laurent series, in powers of $z$, of $$f(z)=\frac{\sin(2z)}{z}$$ valid in the region $|z|>0$. The singularity is $0$ but $0$ isn't inside the region of the domain so what do you exactly ...
0
votes
0answers
22 views

what would be the formula of $\phi$ in this question?

Suppose $\phi:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ be an entire map (i.e, the components of $\phi$ are entire in each variable separately) with $\phi_1$,$\phi_2$ as its components satisfying ...
0
votes
1answer
21 views

Calculating residues of multiple poles?

How would I calculate $$\mathrm{Res}\left(\frac{\pi}{\sin(\pi z)(2z+1)^3}\right)?$$ I understand it has singularities at $z=n$ and $z=-1/2$, I'm interested in the residue when $z=-1/2$. I know that ...
2
votes
1answer
29 views

From $\mathbb{H}$ to Poincaré disc?

What is the mapping that takes one from the Poincaré upper half plane $\mathbb{H} = \{ z\in \mathbb{C} \mid \operatorname{Im}(z)>0 \}$ to the Poincaré disc? Here $z=x+i y$.
0
votes
2answers
34 views

Contour integral, f(z)=$ze^{z^2}$

For part $(a)$ is the answer just $0$? Using Cauchy-Goursat theorem? For part $(b)$ I am confused. Do I use ? It seems very complicated. Am I missing a trick?
0
votes
1answer
17 views

Continuous Choice of Argument

Since $\arg(z)$ is a set, if we define it with a specific branch, there will be discontinuity at the branch line. However, suppose $z:[a,b]\to \mathbb C\backslash\{0\}$ is continuous (it is a curve ...
0
votes
0answers
15 views

Let $F(r)=\sum_{k=1}^m{|P(rz_k)|^2}$ for $r>0$. Prove that the function $F(r)$ is increasing if $m>n>0$.

Let $P(z)$ be a polynomial of degree $n$ with complex coefficients. Further, let $$z_k=e^{\frac{2 \pi i k}{m}}$$ for some $m$ and $k=1,2,...,m$. In other words, $z_1,\cdots z_m$ are the $m$th roots of ...
1
vote
1answer
16 views

If $\forall f \in \mathcal{H}(\Omega)$ such that $f(z)\neq 0$ exists a square root then $\Omega$ is simply connected

If $\forall f \in \mathcal{H}(\Omega)$ such that $f(z)\neq 0$ for all $z\in \Omega$ $\exists$ $\varphi \in \mathcal{H}(\Omega)$ such that $\varphi^2=f$ $\implies$ $\Omega$ is simply connected. Is ...
0
votes
0answers
10 views

Proof of the Lindelof theorem related to the radial limit of an analytic function in the unit disc

Hi I am looking for the proof of this theorem here by Lindelof: "Suppose $\Gamma$ is a curve with parameter interval $[0,1]$, such that $|\Gamma(t)| < 1$ if $t < 1$ and $\Gamma(1)=1$. If $g \in ...
0
votes
0answers
34 views

The Coin-Exchange Problem (Application of the Residue Theorem)

These day, I have met a problem about application of the Residue Theorem, see section 10.4 of enter link description here.Could anybody help me solve it? (The Coin-Exchange Problem) Suppose $a$ and ...
0
votes
0answers
21 views

Integrating $\operatorname{Log}(z+2)$ along the unit circle [duplicate]

For the function $f(z) = \operatorname{Log}(z + 2)$, where we choose the principal branch of logarithm (namely, $−\pi < \operatorname{Arg}(z) < \pi$), and the contour $C := \{z \in ...
2
votes
4answers
226 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! ...
1
vote
0answers
38 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 $\lim_{n \to \infty} |f(Z_n)| = \infty$ Here two cases arise If there exist a nbd ...
0
votes
0answers
20 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 ...
-1
votes
1answer
21 views

A question about zeroes and poles of complex functions. [on hold]

Let $f (z)=\frac {z}{z} $ be a complex function. Is 0 a zero, a pole, or neither of these?
2
votes
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 ...
0
votes
1answer
63 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 ...
0
votes
2answers
36 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...
0
votes
0answers
34 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.
0
votes
2answers
25 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 ...
-3
votes
0answers
15 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
votes
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 = ...
0
votes
1answer
27 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 ...
-2
votes
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 $$ \frac{z}{(z+1)(z+2)} $$
0
votes
1answer
22 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 ...
1
vote
1answer
24 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 ...
0
votes
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 ...
1
vote
1answer
66 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 ...
0
votes
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
votes
1answer
29 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
votes
1answer
41 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 ...
1
vote
0answers
42 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 $\cos(2z)$ ...
1
vote
1answer
53 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 ...
1
vote
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 ...
0
votes
0answers
21 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*} ...
0
votes
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 ...
0
votes
0answers
23 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 $U$ ...
1
vote
0answers
30 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 ...
5
votes
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 ...
1
vote
1answer
15 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
votes
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 ...
1
vote
1answer
23 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 ...
0
votes
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. ...
0
votes
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 ...
0
votes
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
-1
votes
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 $tan(z)$
1
vote
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: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$