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
0answers
292 views

Need help with the convolution of two complex functions

Could someone start me off with how to find the convolution of these two functions? Using the normal equation for convolution seems impossible as a common overlap interval is required for ...
1
vote
1answer
50 views

Compact convergence of series

I am trying to show that $$f(z) = \sum_{k=1}^{\infty} \frac{(-1)^{k}}{z+k}$$ converges compactly over $\mathbb{C}$ and starting to think that this statement is false after several attempts. If I ...
2
votes
1answer
35 views

Are the projective transformations between two planes Möbius Transformations?

Given two planes(infinity included) and one point as centre of projection, there is a transformation between the planes geometrically; regard the two planes as complex planes, is the former ...
1
vote
2answers
75 views

Contour integration question help

I am trying to evaluate the following integral using complex analysis: $$\int^{2\pi}_0\frac{\cos(2x)}{5-4\cos(x)}dx$$ I have written it in the following form: ...
0
votes
2answers
128 views

Simply connected and homotopic

In complex plane, if $C$ is a closed curve that is homotopic to a point, and $C$ is the boundary of a domain $E$, is $E$ simply connected?
2
votes
1answer
121 views

Essential singularity HW question

Show that $f(z)=ze^{\frac{1}{z}}e^{\frac{-1}{z^2}}$has an essential singularity at $z=0$. This one should be straightforward, as I should be able to tackle it by use of expanding the power ...
0
votes
1answer
55 views

Complex Integrals

Integrate: $$ \int_A^B z^3-5~dz $$ Where A = 2+i and B = 1-i are two points in complex plane. How can this complex integral be solved?
1
vote
1answer
32 views

Finding the Power Series of a Complex fuction.

Find a power series expression $\sum_{n=0}^\infty A_n z^n $ for $ \frac{1}{z^2-\sqrt2 z +2} $ I'm completely stuck on this question. I know how to manipulate power series but I've never had to find ...
2
votes
0answers
66 views

If $f(z) = \sum a_n z^n$, what is $\sum n^3 a_n z^n$?

Problem: If $f(z) = \sum a_n z^n$, what is $\sum n^3 a_n z^n$? Attempt: First we note that $$ f'(z) = \sum_{n=1}^\infty n a_n z^{(n-1)} $$ so that $$ z f'(z) = \sum_{n=1}^\infty n a_n z^n $$ ...
0
votes
1answer
224 views

Principal Value of complex log

Find the principal value of $(Log(1-i\sqrt3))^i$. Now $r=2$ and $\theta = tan^{-1}(-\sqrt3) = -\pi / 3$ so we have $log(1 + i\sqrt3) = ln2 + i(\frac{-\pi}{3} + 2\pi n)$. So PV of $(Log(1-i\sqrt3))^i$ ...
0
votes
1answer
38 views

Showing $\lim\frac{\left| a_{n+1} z^{n+1} \right|}{\left| a_n z^n \right|} = |z| \lim \frac{ a_{n+1} }{ a_n }$

Let $\sum a_n z^n$ be a complex power series. I've seen it asserted without explanation in a text that $$ \lim\frac{\left| a_{n+1} z^{n+1} \right|}{\left| a_n z^n \right|} = |z| \lim \frac{ a_{n+1} ...
0
votes
1answer
52 views

Showing the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$

Problem: If $\sum a_n z^n$ and $\sum b_n z^n$ have radii of convergence $R_1$ and $R_2$, show that the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$. Is the following proof ...
0
votes
2answers
97 views

Relationship Between Ratio Test and Power Series Radius of Convergence

Let $ \{a_k\} $ be a sequence of positive real numbers. Why does it hold that $$ \liminf \frac{a_{k+1}}{a_{k}} \leq \liminf (a_k)^{\frac{1}{k}}\leq\limsup (a_k)^{\frac{1}{k}} \leq \limsup ...
2
votes
0answers
82 views

Who proved Fundamental Theorem of algebra using Liouville's theorem?

One of the most famous proofs of the Fundamental Theorem of Algebra involves Liouville's theorom stating that a bounded entire function in constant. Who first came up to the idea of deriving FToA ...
0
votes
2answers
46 views

need to show a complex function is continuous on C (complex plane)

Prove $$f(z) = \sum\limits_{n=0}^\infty \frac{z^{2n}(-1)^{n}}{(2n)!}$$ is continuous everywhere on $\mathbb{C}$ I want to show, for each $\epsilon > 0$, there is some $\delta > 0$ such that ...
0
votes
0answers
86 views

Branch Cut of $\log{\log{z}}$

I'm wondering if I could get tips on how to get a definition of a branch of $\log \log z$, and how to prove that this branch is analytic. As I understand it, the usual definition of the principal ...
3
votes
2answers
68 views

Dependence of roots on parameters

Let a function $f$ be holomorphic in a polydisk $U=U'\times U_n$,and suppose that for each fixed $z'\in U'$ it has a unique zero $z_n = \alpha(z')$ in the disk $U_n$. Then the function $\alpha(z')$ is ...
7
votes
1answer
353 views

Finding the order of an entire function defined from an integral

The following problem is posed in Greene and Krantz, page 297, Problem 11. Let $g: \mathbb{R} \to \mathbb{C}$ be a continuous function, $\alpha \in \mathbb{R}$, and define $f: \mathbb{C} \to ...
0
votes
3answers
37 views

About the Scalar product

These are the lecture notes of my teacher and I am getting confused how he reached at $V_1$.$V_2$= Re($z_1$$z_2$). Can anyone help me to understand this.
1
vote
0answers
56 views

Limit of Blaschke product on the boundary of the unit disk

The following question was posed in Greene and Krantz's textbook 3rd edition, page 296 Problem 5: Let $\{a_n \}$ be a sequence in $\mathbb{D}$, the open unit disk s.t. $\sum_{n \in \mathbb{N}} 1 - ...
2
votes
1answer
52 views

Injective analytic function

Suppose $f: B(0,1) \rightarrow \mathbb{C}$ is an analytic function such that $f'(0)=1$ and $f'(z) \in B(1,1)$ for all $z \in B(0,1)$. Show that $f$ is injective.
2
votes
2answers
63 views

Why $\sin\,z$ does not have a pole at $z=\infty$

I believe that $\sin\,z$ does not have a pole at $z=\infty$. But how to prove it? I think we can use the fact that $\sin\,z$ have a pole at $z=\infty$ if and only if $\sin(1/z)$ has a pole at $0$. But ...
0
votes
1answer
74 views

absolute convergence of infinite product

We know if $\sum_{n=1}^\infty|z_n|$ converges then $\sum_{n=1}^\infty z_n$ converges absolutely. (kind of trivial) I wonder whether it holds for infinite products, that is, if ...
1
vote
2answers
24 views

Find complex solutions

Find all solutions of the equation $4\sin(z) + 5 = 0$. So $4\sin(z) + 5 = 4\sin(x+iy) + 5 = 0$. Hence $\sin(x+iy) = \frac{-5}{4}$. So $x+iy = \sin^{-1}(\frac{-5}{4})$ which gives us $x+iy =$ ... I ...
1
vote
1answer
36 views

Trigonometric functions (complex)

I have to find $sen^3{5a}$ and $cos^2{5a}$ considering that $sen{a}=\displaystyle\frac{1}{2}$ and $a$ belong to the first positive quadrant. I tried to apply De Moivre formula to find ...
3
votes
0answers
72 views

The Schwarz reflection principle and harmonic function (Big Rudin chapter 11)

In his book page 250 Exer 11: Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$. If I follow the ...
1
vote
0answers
20 views

Restoring an Analytic Function

Restore the analytic function $f(z)$ if its imaginary part $v(x,y) = -\frac{y}{(x-1)^2 + y^2}$ and $f(2)=2$. Now, $v_y = \frac{-[(x-1)^2 + y^2] - (-y)(2y)}{[(x-1)^2 + y^2]^2} = ...
0
votes
0answers
33 views

Checking where Differentiable and Analytic

Describe the set of the points in the complex plane where the following functions of complex variable $z$ are differentiable and the sets of points where the functions are analytic. a) $f(z) = ...
2
votes
3answers
100 views

Showing $f$ is constant in $D$ if $v=u^2$.

I am working on a group worksheet and none of us know how to approach this problem: Suppose that $f = u+iv$ is analytic in the domain $D$ and $v=u^2$ in $D$. Show that $f$ must be constant in $D$. ...
0
votes
1answer
61 views

Geometric interpretation of $\partial/\partial z$

My understanding is that analytic derivative ,$\partial\phi/\partial z$, and anti-analytic derivative ,$\partial\phi/\partial\bar{z}$, are resp. tangent and normal to the curve $\phi$. Am i right?can ...
2
votes
1answer
58 views

Covering four points by the vertices of a regular tetrahedron.

I’m looking for an explicit formula for the vertices of a regular tetrahedron that covers four given points. In particular, I have these two questions: 1) Given four distinct real numbers $a_1, ...
2
votes
1answer
41 views

Complex Analysis Proof: Derivative

I'm not quite sure what the question is asking me to show or even to show this. Please provide some guidance, because I think I need some kind of proof: Fix $c \in \mathbb{C} \setminus \{0\}$. ...
1
vote
1answer
63 views

Convergence question and degree of polynomial

I'm currently teaching myself power series and Taylor's theorem for complex analysis and I'm having trouble answering questions of the following form: $1)$ Suppose the power series ...
0
votes
1answer
106 views

How does the complex exponential function transform the unit circle?

I know you can write every complex number on the unit circle as $e^{i\theta} = \cos(\theta)+i\sin(\theta).$ But what does it look like when you raise $e$ to the values? You get ...
0
votes
0answers
50 views

$f$ is analytic with range as a circle

I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...
0
votes
1answer
34 views

Showing that $\underset{n \rightarrow \infty}{\lim} |a_n| / |a_{n+1}| = R$ implies that the radius of convergence of $\sum a_n z^n$ is also $R$

Hypothesis: Suppose that $\underset{n \rightarrow \infty}{\lim} |a_n| / |a_{n+1}| = R$. Goal: Show that $\sum a_n z^n$ has radius of convergence $R$. Attempt: The radius of convergence of $\sum ...
0
votes
1answer
35 views

Multi part problem to prove functional relation of the exponential function

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that $\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = ...
1
vote
2answers
152 views

Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$

Setting: Let $\sum a_n z^n$ have radius of convergence $R$. We have that $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$ via Hadamard's formula for the radius ...
1
vote
3answers
240 views

Using Hadamard's Formula to show that the radius of convergence of $\sum z^{n!}$ is $1$

Background: Recall that Hadarmard's formula for the radius of convergence of a complex power series $\sum a_n z^n$ is as follows: $$ R = \frac{1}{\underset{n \rightarrow \infty}{limsup} \left| a_n ...
1
vote
1answer
68 views

Finding the radius of convergence for $\sum n^p z^n$ (Proof Verification)

Goal: Find the radius of convergence for the following complex power series: $$ \sum n^p z^n $$ Attempt: We have by Hadamard's formula for the radius of convergence that the complex power series ...
0
votes
1answer
48 views

A question about complex analysis?

Given that $$u = e^{-x} (x \sin y- y \cos y)$$ is harmonic, that is, it holds that $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0$$, find $v = v(x, y)$ such that $f(z) = u + ...
2
votes
1answer
86 views

Find the maximum value of $u$ on a closed disk $D$

"Suppose that $u$ is a harmonic function in the disk $D = {z \in \mathbb{C} : |z|<2}$ and that $u(2e^{it}) = 3 \sin{2t}+1$ on the boundary circle. Without calculation find the maximum value of $u$ ...
1
vote
0answers
64 views

Show that $\sin z$ en $\cos z$ are analytic.

Show that $\sin z$ en $\cos z$ are analytic. To clarify, in my book, they say that analytic means that you can write it as a power series centered at any abritray $a \in \mathbb {C}$. It is not ...
8
votes
4answers
415 views

Factor $x^4+1$ over $\mathbb{R}$

Factor $x^4+1$ over $\mathbb{R}$ Well, I read this question first wrongly, because the reader is about complex analysis, I did it for $\mathbb{C}$ first. I got. $x^4+1=(x-e^{\pi i/4 })(x-e^{3 ...
0
votes
1answer
46 views

Show that $|\frac{e^{z}-1}{z}|$ is bounded on every half plane

Show that $|\frac{e^{z}-1}{z}|$ is bounded on every half-plane $\{z\in \mathbb{C}:\text{Re }z\leq c\}$. It is clear that the numerator alone is bounded. How do we deal with the $\frac{1}{z}$ term? ...
0
votes
1answer
55 views

Example of a simply connected domain

I'm a bit stuck trying to think of an example of a simply connected domain $D$ whose complement $\mathbb{C} \setminus D$ is a disjoint union of infinitely many closed connected sets. Could I be given ...
3
votes
2answers
114 views

Elimination of Trigonometric Functions

Is there a simple way to eliminate the trigonometric functions here? $$ \begin{array}{lcl} A\cos(3\omega\tau)+B\sin(3\omega\tau)+C\cos(\omega\tau) &=& D\\ ...
0
votes
1answer
55 views

Solution of this ODE

How to solve the following ODE? $$i\partial_tv=t^q|v|^pv$$ where $i$ is the imaginary unit and $v$ is complex valued? I think that separation of variables is to be used.
0
votes
1answer
37 views

About complex sum

Let $\left(c_{n}\right)_{n},\,\left(d_{n}\right)_{n}$ two successions of complex numbers and let $N$ a large natural number.Is it true that ...
0
votes
2answers
54 views

A Complex Analysis question?

Show that $$u = e^{-x} (x \sin y - y \cos y)$$ is harmonic, that is, it holds that $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0.$$ Not really sure how to go about this.