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)

0
votes
1answer
18 views

laurent series expansion about $z=0$

using the Laurent expansion i got the answer to be $$-(z+1)\sum_{n=0}^\infty \frac{z^{n-1}}{2^{n+1}}$$ however, I've got a feeling I've made a mistake somewhere?
2
votes
1answer
17 views

Finding the order and computing the residue of a pole

Find the poles, indicate their order and compute their residues for the following functions: $$g(z)=\frac{e^z}{\sin z}$$ I have a singularity at $z=0$ where the residue would be $1$ ... however, ...
0
votes
2answers
32 views

How to solve this complex logarithm equation?

define $Log z := ln|z| + i Argz$ and solve the equation $Log(z^2-1)=i \pi/2$, for all possible value I've try that let $w=z^2-1$and $Log\ w = i\pi/2$, then $|w|=1$and$Arg\ w=\pi/2$ ...
0
votes
1answer
15 views

Characteristic Function and Density Function

Consider a random variable $X$ with density function $f(x)$, moment generating function $M(t):= \int e^{tx}f(x) dx$ (existing in an interval containing $0$), cumulant generating function $K(t):=\log ...
2
votes
1answer
44 views

Prove that $f$ has a simple pole at $z=0$

Let, $f:\{z\in \mathbb C:0<|z|<1\}\to \mathbb C$ be analytic such that $n\le |f(1/n)|\le n^{3/2}$ for $n=2,3,...$. Assume that $z^2f(z)$ is bounded in $|z|<1$. Show that $f$ has a pole of ...
1
vote
1answer
46 views

If $ g \circ f$ is real analytic and $g$ is a real analytic immersion, then $f$ is real analytic

Let $M$ $N$ $P$ be complex manifolds, and let $$f: M\rightarrow N, g: N\rightarrow P$$ be $C^\infty$ maps with $g$ and $g\circ f$ holomorphic, and with $dg$ never degenerate. It's easy, then, to see ...
1
vote
1answer
19 views

set of arbitrary positive measure conformally equivalent to unit disk

Show that for any  $\epsilon$ > 0, there is a dense subset of $\mathbb{C}$ with measure less than $\epsilon$ and which is conformally equivalent to the unit disc. To make a dense set that has ...
0
votes
1answer
18 views

Winding number is locally constant

Let $\gamma$ be a closed path in the plane $\mathbb{C}$ and let $a\in \mathbb{C}$ which does not belong to the image of $\gamma$. The winding number (or index) is defined as $$I(\gamma, ...
0
votes
1answer
18 views

How transformation of co-ordinates system relates to its vectors?

Consider a positive definite matrix. Can we consider that it has a underlying co-ordiante system? If we transform that co-ordinate system how the the vectors are transformed? Is this question even ...
0
votes
1answer
37 views

Radius of convergence and sum of alternating series $1 - z + z^2 - z^3 + \ldots $

I have a (complex) function represented by the power series \begin{equation*} L(z) = z -\frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} \ldots \end{equation*} which I have tried to represent (perhaps ...
3
votes
1answer
101 views

Complex Analysis proof of multinomial expression

I've recently come across the following identity $$ \displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n} $$ A nice complex analysis proof (by Felix Marin, here) follows as: ...
1
vote
1answer
14 views

Meaning of co-ordinate system of Covariance matrix

Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the ...
0
votes
0answers
20 views

Proving a version of maximum modulus principle elementarly.

There is this version of maximum modulus principle: If $P$ is a non-constant polynomial, then $|P|$ doesn't have a local maximum. I know that if $P$ is non-constant, then $|P(z)| ...
5
votes
2answers
170 views

What's wrong in this equation? (Regarding Euler's eqn)

I got an idea, but that doesn't match with Euler's theory.. So What's wrong?! $$e^{jx} = (e^{j 2\pi})^{x/2\pi} = 1^{x/2\pi} = 1$$
1
vote
0answers
23 views

Help for solving limi of the Complex Fourier Series

I need help for this exercise. Let: $ f:\left[ -T /2, T/2 \right]\rightarrow \mathbb{R}. $ I need show that $$\lim_{N \to \infty} \int_{-T/2}^{T/2} \vert f(t)-f_{N}(t) \vert^{2} dt = 0 $$ ...
0
votes
0answers
20 views

Computing an integral using residues

I am trying to find an integral $ \int_{-\infty}^{+\infty} \frac{ e^{-(x^2 + 1)}}{(x^2 + 1)^2} dx $. I went about applying contour integral over a semicircle with diameter along $ x = +\infty$ to ...
1
vote
0answers
27 views

Existence of certain Analytic functions on the open unit disc ( possible application of Schwarz lemma )

Let $D$ be the open unit disc. Then can there be analytic functions with the property (1) $f(\frac{3}{4})=\frac{3}{4}$ and $f'(\frac{2}{3})=3/4$ 2) $f(\frac{3}{4})= -\frac{3}{4}$ and ...
0
votes
0answers
16 views

2nd order pole inwhile computing residue in a complex integral

I was wondering - how does one deal with finding a residue of a contour integral when you introduce a fresh pole while computing the residue. For example: $$ \int \, \frac{ \frac{e^{\sqrt{x^2 + ...
2
votes
1answer
20 views

For fn(z)= 1/nz, If we make fn(0)= 1, does that make the family of functions bounded?

I have a problem that requires me to use a theorem requiring a bounded family of functions. The family provided that I am supposed to use this theorem for is $f_n (z) = \frac 1 {nz}$ when $z \neq 0$ ...
0
votes
1answer
23 views

How is the boundary in product spaces defined?

The general question: how is the boundary defined in product spaces? Given two topological spaces $X,Y$, I'd say that $\partial(X\times Y)=\partial X\times\partial Y$. But looking at what follows it ...
1
vote
1answer
42 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
1
vote
2answers
51 views

What does the graph of $5e^{it}$ look like on the complex plane?

I know that $5e^{it} = 5(\cos(t) + i\sin(t))$, but that doesn't really help me. What other information can I use to visualize this graph besides plotting many points and seeing what type of graph it ...
0
votes
1answer
18 views

How to use Cauchy's integral formula with more than one pole?

$\int\limits_{\gamma} \frac{z^2}{z(z-2)}$ $\gamma(\theta) = 3e^{i\theta}$, $0 \leq \theta \leq 2\pi$ Cauchy's integral formula is given by: $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = ...
6
votes
4answers
179 views

Integral by residue - “dog bone”

Let $I=\int_{-1}^{1}\frac{x^2 dx}{\sqrt[3]{(1-x)(1+x)^2}}$. I used complex function $f(z)=\frac{z^2}{\sqrt[3]{(z-1)(z+1)^2}}$, which we can define such that it is holomorphic on ...
1
vote
0answers
28 views
+100

How to prove $\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $ around a phase singularity/over a cut

How would you prove that $$\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $$ We know that $\theta\in(-\pi,\pi)$, suppose that $\theta$ is continuous in the region bounded by and along $\Gamma$ apart ...
-5
votes
1answer
52 views

Is $pi=ln(-1)/sqrt(-1)$, and if so what does this mean?

Using the complex integral $z=\cos(x)+i\sin(x)$ $\frac{dz}{dx}=-\sin(x)+i\cos(x)$ $dz=i[\cos(x)+i\sin(x)]dx$ $dz=iz\cdot dx$ $\frac{1}{z}dz=i\cdot dx$ $\ln(z)=ix$ $z=e^{ix}$ ...
-1
votes
0answers
35 views

Let $f$ be an entire function with $g(z)=\overline{f(\bar z)}$ [CSIR-NET-2014] [closed]

Let , $f$ be an entire function. Let, $g(z)=\overline{f(\bar z)}$. Let, $D=\{z:Im(z)=0\}\cup\{z:Im(z)=a\}$ for some $a>0$. Then which are correct ? (A) If $f(z)\in \mathbb R$ for all $z\in \mathbb ...
1
vote
1answer
38 views

Why is $\int\limits_{\gamma} \frac{1}{z-1} \neq 2\pi i$, $\gamma = \{z : \lvert z \rvert = 1\}$?

$\int\limits_{\gamma} \frac{1}{z-1}$ $\gamma = \{z : \lvert z \rvert = 1\}$ I use Cauchy's integral formula, which says $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = \frac{2\pi i}{n!} ...
0
votes
3answers
24 views

$\int\limits_{\gamma} \frac{1}{z-1}$, $\gamma(\theta) = 2e^{i\theta}$, $0 \leq \theta \leq \frac{\pi}{2}$

$\gamma(\theta) = 2e^{i\theta}$ is a circle centered at $(0,0)$ with radius $2$, so $z = 1$ is inside this path and thus we have to use Cauchy's integral formula for $\int\limits_{\gamma} ...
0
votes
1answer
26 views

Evaluate an integral along a semicircle.

Let $\gamma$ be the semicircle $[-R,R]\cup\{z\in\mathbb{C}:|z|=R\ and\ Im{z}>0\}$ traced in the positive direction, and let $R>1$. Evaluate $$\int_\gamma\frac{dz}{(z^2+1)^2}.$$ I want to say ...
1
vote
2answers
65 views

Let f be an analytic function in the disk … Prove that

Let $f$ be a function analytic in the disk $D={z:|z-1-i| \leq 4}$ that satisfies $|f(z)| \leq 1$ for all $z$ in $D$. Prove that $|f^{(3)} (1-i)| \leq \frac{3}{4}$. Hint: apply Cauchy's Integral ...
1
vote
3answers
77 views

Finding $\int_0^\infty\frac{\sin^{2}x}{1+x^4}dx$

I am trying to evaluate $$\int_0^\infty\dfrac{\sin^{2}x}{1+x^4}dx$$ and I am stuck on how to start. I am thinking the first step would be to substitute $$\dfrac{(1-e^{2ix})+(1-e^{-2ix})}{4}$$ for ...
0
votes
1answer
17 views

Quick Question on Zeroes of Transfer Function

Sorry for not providing context here. Suppose I have an output $Y(z)=\frac{z-1}{z}$ and input $X(z)=\frac{z^2+3z+2}{z^2}$ to yield a transfer function ...
1
vote
1answer
27 views

Path dependence of integrals

Are the integrals of the function $ \Large f(z)=e^{1/z}dz$ path independent in the domain $D= \{Re z >0\}\setminus\{3\}$?
1
vote
1answer
24 views

Circular Contour Integration .

Doing some revision for an upcoming exam I have stumbled across the following problem: Evaluate the integral $\int_{C}\log(z)$ where $C=C(2,1)$ the positively oriented circular contour, centre 2, ...
2
votes
1answer
26 views

Classify the singularities of the function .

Classify the singularities of the function $\frac{1-\cos(z)}{z^2(z-1)}$. I think my answer may be that I have a simple pole at $z=0$ and a removable singularitie at $z=-1$ however i am not too sure. ...
1
vote
1answer
32 views

Evaluate integral using Residue Theorem

Let $\gamma$ be the semicircle $[-R,R]\cup\{z\in\mathbb{C}:|z|=R\ and\ Im{z}>0\}$, traced in the positive direction, and $R>1$. Evaluate $$\int_\gamma\frac{dz}{z^4+1}.$$ I note that ...
1
vote
1answer
33 views

Prove a union is a domain

Prove that if S and T are domains that have at least one point in common, then S union T is also a domain I wrote: A domain is a set that is open and connected. The union of open sets is easily open. ...
0
votes
1answer
55 views

integral of $ \int_{\gamma}e^{1/z}dz$ [on hold]

How do you find the integral of $$ \int_{\gamma}e^{1/z}dz$$ in the domain $ D= \{Re z >0\}$
16
votes
2answers
1k views

If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?

If two analytical functions of $\mathbb{C}$ f and g are equal on an infinite number of input values, than they are equal. I can't seem to find a counterexample, but I haven't seen this anywhere ...
0
votes
0answers
18 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
0
votes
0answers
25 views

Uniform convergence of $\sum^n_{k=-n} \frac{1}{z+k}$

Let $D=\mathbb C \setminus \mathbb Z$ and define $$f_n(z)=\sum^n_{k=-n}\frac{1}{z+k}$$ I have to prove that $\{f_n\}^\infty_{n=0}$ is locally convergent on D. We are given the hint to write $f_n$ as a ...
1
vote
1answer
317 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
0
votes
0answers
10 views

Prove that an entire function of exponential type is of order at most $1$.

By Entire functions theory, the order (at infinity) of an entire function $f(z)$ is defined using the limit superior as: $$\rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln\Vert f \Vert_{\infty, B_r} ...
1
vote
2answers
35 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
0
votes
1answer
19 views

path dependence of the integral $f(z)=\frac{1}{(z-4)^2} + \sin z$

Are the integrals of the $$f(z)=\frac{1}{(z-4)^2} + \sin z$$ path independent in the following domain $$D= \{\operatorname{Re} z >0\}\setminus\{4\}$$ My thought is that since ...
1
vote
1answer
27 views

Finding $a_{-n}$ where $\cot (\pi z)=\sum_{n=-\infty } ^\infty a_nz^n$

The following is problem 5.11.2 of Berkeley Problems in Mathematics. Let $\cot (\pi z)=\sum_{-\infty} ^\infty a_nz^n$ be the Laurent expansion for $\cot (\pi z)$ on the annulus $1<\vert z \vert ...
1
vote
1answer
99 views

On the construction of hyperelliptic Riemann surfaces.

I have seen two ways to construct hyperelliptic curves, and it seems to me that the intuition behind the change of coordinate is not the same. I like better the second construction (which is pretty ...
1
vote
0answers
38 views

$\tan z=az+b$ has infinitely many solutions

I try to prove following questions. Prove that, for all complex numbers $(a,b)\neq (0,\pm i)$, the equation $\tan z = az + b$ has infinitely many solutions. By assuming $(a,b)=(1,0)$, I tried ...
1
vote
1answer
49 views

infinite product expansion $\frac{1}{\sin(z)}-\frac{1}{z}$

I have successfully solved that $$\frac{1}{\sin(z)}-\frac{1}{z} = \sum_{n=-\infty, n \neq 0}^\infty (-1)^n\left(\frac{1}{z-n\pi}+\frac{1}{n\pi}\right)$$ and am now attempting to integrate both sides ...