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
29 views

Fail to see the mistake when applying Cauchy's integral formulae

We need to find $\;f(z)$ with the property that $$f''(a)=\oint_{\partial C_1(0)}{\sin^2z\over (z-a)^3}dz, \quad \forall\;|z|<1$$ Could someone explain me why I cannot do it this way: ...
2
votes
1answer
40 views

Prove that $f$ has a removable singularity at $z_0$, and compute $\lim_{z\to z_0} f(z)$

I am again stuck on a qual question while I am preparing for my upcoming exam: Let $W$ be analytic in a domain $D$. Let $z_0\in D$ be such that $W'(z_0)\neq 0$. Define $$f(z) = ...
1
vote
0answers
14 views

Complex contour integral properties

Do I understand correctly, that for complex line integrals the properties of common integrals (e.g. Riemann-integrals) cannot be applied? Neither linearity: $$\int_\gamma \beta \;f(z)dz \not = ...
0
votes
1answer
36 views

Does there exist an analytic function that satisfies these properties?

Does there exist an analytic function $f:\{z\in\mathbb{C}:0<|z|<1\}\to\mathbb{C}$ such that $\displaystyle\lim_{z\to0}[z^{-3}f^2(z)]=1$? I'm assuming that there is not such a function, so I've ...
-8
votes
0answers
31 views

give the details answer [on hold]

for any integer n, let $f_n:[0,1]\to\mathbb{R}$ be defined by $f_n(x)=\frac{x}{nx+1}$ for $x \in[0,1]$ then (a) the sequence $f_n$ converges uniformly on $[0,1]$ (b) the sequence $f_n'$ of ...
2
votes
1answer
37 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
-1
votes
0answers
41 views

Infinite radius of convergence

If the root test limit tends to infinity, is that sufficient to say we have an infinite radius of convergence? Thanks
2
votes
1answer
73 views

Power series difficulty

How would I find the region of convergence of the series of $\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting $\frac{z+1}{z-1}$ as $\frac{2}{z-1}+1$ but I don't think that helps. Thanks
0
votes
2answers
24 views

Radius of convergence query

Find the radius of convergence of the series of $\frac{2^n(4z-8)^n}{n}$ My answer: $(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let $c_{n}=\frac{2^{3n}}{n}$. Then $\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$ so ...
2
votes
1answer
38 views

Complex analysis without Cauchy's theorem

Is there an approach to complex analysis that is fundamentally different from the usual route via Cauchy's theorem? For example, can one prove that a complex-differentiable function is given locally ...
-2
votes
1answer
16 views

differntiability of the following function [on hold]

Let f(x)=sinx/x,x≠0 =1 ,x=0, then f is a)discontinuous b)continuous but not differentiable c)differentiable only once d)differentiable more than once
2
votes
1answer
34 views

Computing a contour integral over curve not centered at origin

Consider the integral $$ \int_C \frac{1}{z} \, dz $$ where $C$ is the circle of radius $R$ centered at the point $z_0 \in \mathbb{C}$. We parametrize the curve by $z(\theta) = z_0 + Re^{i\theta}$ ...
-4
votes
0answers
26 views

real anaylsis for twice differentiable function [on hold]

Let f:[0,1] tends to [0,1] be any twice differentiable function satisfying f(ax+(1-a)y) less then or equals to a(f(x)+(1-a)f(y)) for all x,y belongs to [0,1] and any a belongs to [0,1].then for all x ...
0
votes
2answers
16 views

The proof of the Area Theorem for Conformal Maps

The Area Theorem: Suppose $f(z)$ is one-to-one and analytic on the punctured unit disk, and is given by $f(z) = 1/z + \sum_0^\infty a_nz^n$ Then $\sum_0^\infty n|a_n|^2 \le 1$ I'm reading the ...
1
vote
0answers
14 views

Prove that $|Im\{f(z)\}|≤ $$\frac{2}{\pi} \log \frac{1+|z|}{1 - |z|},$ $z \in \mathbb{D}$. [duplicate]

Let $f(z)$ be an analytic function defined on the unit disk $\mathbb{D} = \{z : |z| < 1\}$ so that $f(0)=0$ and $−1<Re\{f(z)\}<1$ for all $z \in \mathbb{D}$. Prove that $|Im\{f(z)\}|≤ ...
3
votes
2answers
78 views

Which contour is best for $\int_0^\infty\frac{1}{x^2 + x + 1}dx$

The following is a complex analysis problem. Does anyone have any idea what contour would be good to use? $$\int_0^\infty\frac{1}{x^2 + x + 1}dx$$ Its roots on the bottom are are $\frac{-1 \pm ...
0
votes
1answer
25 views

Entire functions satisfying $|\mathrm{Re}\, f(z)| \geq c|\mathrm{Im} \,f(z)|$ [on hold]

Suppose $f$ is entire and there exists a constant $c > 0$ such that $|\mathrm{Re}\, f(z)| \geq c|\mathrm{Im}\, f(z)|$ for all $z \in \mathbb{C}$. Must $f$ be a constant?
1
vote
0answers
56 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
4
votes
1answer
65 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
0
votes
0answers
13 views

Problem on Conformal transformation

An angular domain in the complex plane is defined by $0< \phi<\pi/4$. The mapping which maps this region onto the left half plane is....(fill in the blanks)... My thoughts: The transformation ...
0
votes
0answers
55 views

Complex derivative of log(z)/z

If we use the principal branch of the log function, at which points of $\mathbb{C}$ does $\frac{\log z}{z}$ have a complex derivative? What is its derivative at these points. This is what I have so ...
4
votes
1answer
36 views

let $f : U \rightarrow U$ be an analytic function, whose Taylor series at $0$ is $f(z) = z + a_2z^2 + a_3z^3 + …$

Let $U \subset \mathbb{C}$ be a bounded open set containing $0,$ and let $f : U \rightarrow U$ be an analytic function, whose Taylor series at $0$ is $f(z) = z + a_2z^2 + a_3z^3 + ...$ Prove that ...
3
votes
1answer
29 views

Let $P_N(z) = \sum_{n = 0}^N\frac{z^n}{n!}$. Calculate $\lim_{N \rightarrow \infty}\int_{|z|=2}\frac{1}{P^3(z) - 1}dz$.

Let $P_N(z) = \displaystyle\sum_{n = 0}^N\frac{z^n}{n!}$. Calculate $\displaystyle\lim_{N \rightarrow \infty}\int_{|z|=2}\frac{1}{P^3(z) - 1}dz$. I am not sure how to do this. If I could figure ...
1
vote
1answer
43 views

Derivative calculation $\frac{d}{dz}\cos(yz)$

Perhaps the question that I am about to write may seem trivial, but I just started to study the course of Complex Analysis. The question is the following. I have to calculate the derivative ...
0
votes
0answers
33 views

Real-valued Fourier series representation

I have got stuck on the following task: Find the value of the series $${4\over \pi^2}\sum_{k=1}^\infty {1\over k^2}-{1\over \pi^2}\sum_{k=1}^\infty{(-1)^k\over k^2}$$ using real-valued Fourier ...
4
votes
0answers
51 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
3
votes
2answers
45 views

Types of singularities of a function

How can I determine the type of a singularity of a function $$f(z)={e^{1/z} \over z-1}+{\pi z \over 2\sin(\pi z)}$$ at $z_0=0$? I don't see an easy way to represent it using Laurent series, neither ...
0
votes
1answer
21 views

Euler's Formula, from Needham's Visual Complex Analysis

I'm having trouble understanding this passage. http://imgur.com/a/ao68Q#2 I don't understand the last paragraph. Why is it that $|Z(t)|$ remains equal to 1 throughout the motion?
1
vote
1answer
28 views

Ratio of convergence

Let $ \phi(z)= \log (1 + \sin z) $ for a small disk, (the origin is the center of the disk) Find the ratio of convergence of Taylor series? If we consider that $\log(1 +z) = \sum_0^\infty ...
3
votes
0answers
51 views
+50

When functional equations determine a unique global analytic function?

I'm learning about analytic continuations and global analytic functions which were seen to be connected components of the sheaf of analytic germs. Sometimes we get problem sets in which we are ...
0
votes
0answers
27 views

Confused by a Laplace transform of $f(t)=t^ne^{at}$

Having looked at my lecture notes I was confused by the following part of a derivation of a Laplace transform for the function $\;f(t)=t^ne^{at} ,\quad n\ge0,\; a \in \mathbb{C}, \; f(t)=0 \;\forall ...
0
votes
3answers
47 views

Showing that $\frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right)$

I'm trying to show that $$ \frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right) $$ but am having trouble figuring out exactly how to approach the ...
1
vote
1answer
58 views

Why is the algebraic number a whole number.

Assume the function $f$, analytic on some domain has a non-essential singularity $a$. Define the algebraic order $h$ of $f$ at $a$ to be the real number such that $\lim_{z\to a}|z-a|^k|f(z)|=0$ for ...
0
votes
0answers
11 views

If an analytic function has an algebraic order $h$ at infinity then $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity

Assume infinity is not an essential singularity of the analytic function $f$. Then how is $h$, the algebraic order of $f$ such that $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity? p.s. ...
3
votes
3answers
57 views

Bound in Complex Analysis

Can someone direct me towards the right way to approach this problem? Show $$\displaystyle \left|\int_{|z|=R} \frac{Log{z}}{z^2} dz\right| \leq 2\sqrt{2}{\pi}\frac{\log{R}}{R},\; \text{ for } ...
0
votes
1answer
21 views

Images of Regions Under Cayley's Transformation

I'm working on the following problem for my complex analysis course: I can't seem to find Cayley's transformation anywhere in our textbook - could someone clarify to me what it is? I've done a ...
7
votes
1answer
131 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
3
votes
1answer
33 views

How does Ahlfors define derivative on a Riemann Surface?

I'm reading a passage in Ahlfors (3rd Edition page 298) and he says the following: He has previously defined $G_0(f)$ to be the connected component of any germ generated by $f$. Then he wants to ...
2
votes
2answers
65 views

Goursat's theorem and residue theorem understanding

It seems to me that Goursat's theorem doesn't align with the residue formula, because with the residue formula we end up with a number different than zero. Could you help me find what I understood ...
1
vote
0answers
11 views

some questions on the soluction of the Dirichlet's problem in the unit disk

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
2
votes
3answers
48 views

Parametrizing curve for complex analysis integral

I'm trying to show that $$ \int_{|z-z_0| = R} (z-z_0)^m \, dz = \begin{cases}0, & m \neq -1 \\ 2\pi i, & m =- 1. \end{cases} $$ Here's my attempt at a solution: We parametrize the curve at ...
0
votes
1answer
35 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
0
votes
1answer
23 views

Isometries of the plane and fixed lines

I am given that for all reflections $g$ there are infinitely many lines $L$ satisfying $g(L) = L$ which makes perfect sense (just take lines perpendicular to the axis of reflection). I am asked to ...
0
votes
0answers
33 views

Analytic continuation of a function

Let $$f(z) = A_0 + A_1(z-a) + A_2(z-a)^2 + \cdots$$ converge in some disk $|z - a| < r$. Following Weyl, we magically re-arrange this power series at point $b$ in this disk and the power series ...
2
votes
4answers
58 views

Showing that Gaussians are eigenfunctions of the Fourier transform

I'm having a bit of trouble on this problem: I've tried to evaluate the integral directly (using the trick from multivariable calculus where you "square" the integral and convert to polar ...
1
vote
1answer
61 views

$|f(z)| ≤ 16$ for $|z| = 4.$ Prove that $|f(3i)| ≤ 9.$

(a) Suppose that $f(z)$ is analytic for $1 ≤ |z| ≤ 4.$ Assume that $|f(z)| ≤ 1$ for $|z| = 1$ and $|f(z)| ≤ 16$ for $|z| = 4.$ Prove that $|f(3i)| ≤ 9.$ (b) Prove that there is no non-constant ...
0
votes
1answer
34 views

functional equation of entire functions shall have only constant solutions

Given an entire function $f$ with $f'(0)=0$ and a function $g$ holomorphic (at least) in $\mathbb D:=\{z\in\mathbb C\ |\ |z|<1\}$ such that $f*g$ is entire as well and satisfies the functional ...
1
vote
0answers
42 views

Show that $f(z)= \sum_{n=0}^{\infty} z^n$ is analytic in $|z| <R$ [on hold]

Let $f(z)= \sum_{n=0}^{\infty} z^n$ with $|z| <R$ where $R$ is the radius of convergence of $f$. Then show that $f$ is analytic in $|z|<R$.
0
votes
2answers
26 views

Cauchy integral formula or something else?

I need to determine the function $\;f(z)$ if $$f''(z)=\oint_{\partial C_1(0)}{\sin^2\xi \over\left(\xi-z\right)^3}\mathbb{d}\xi$$ with $C_1(0):\left|z\right|<1$ positive. Additionally ...
2
votes
0answers
35 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...