3
votes
1answer
10 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
0
votes
1answer
23 views

Applying Jensen's formula to polynomials?

Prove that $$\frac{1}{2\pi}\int_{0}^{2\pi}|f(e^{i\theta})|^2d\theta=\sum_{k=0}^n|c_k|^2$$ for each polynomial $f(z)=\sum_{k=0}^nc_kz^k$. The hint given by the homework is: show first that for integer ...
0
votes
1answer
20 views

Find a conformal map from the exterior of the closed unit disk to the unit disk

Question: Find a conformal map from the exterior of the closed unit disk to the unit disk. Also, prove that it is indeed a conformal map (bijective and holomorphic along with its inverse). I missed ...
1
vote
0answers
34 views

Family of holomorphic functions in unit disk with all derivatives bounded by $n!$.

Let $\mathcal{F}$ be the family of holomorphic functions in the unit disk satisfying $|f^{(n)}| \leq n!$ for all $n \geq 0$. Show that $\mathcal{F}$ is a normal family. Is it not enough to fix a ...
0
votes
0answers
18 views

Discontinuity of principal argument in nonpositive real axis

Let $Arg(z)$ be principal argument function defined in branch $(-\pi, \pi]$. Prove that $Arg(z)$ is discontinuous in every point in nonpositive real axis. "Solution": Let $z_0$ be a point on the ...
0
votes
3answers
31 views

How to find the $n$ zeros of $\displaystyle1+z^n$?

How to find the $n$ zeros of $1+z^n$?
0
votes
2answers
24 views

How to compute $f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$

How to compute this serie : $$f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$$ The serie is convergent if $|z| < \sqrt{2} $ I can find that $$f(z) = \sum_0^{\infty} 3^{-2k}(1+i)^{-2k}z^{2k} + ...
2
votes
1answer
40 views

Absolute value of derivative of complex analytic function

Let $f(z)$ be complex function analytic at point $z_0$ and $f'(z_0) \neq 0$ Prove: $\lim_{z \to z_0}\frac{|f(z) - f(z_0)|}{|z - z_0|}$ = $|f'(z_0)|$ Solution: $f(z)$ analytic at $z_0$ therefore ...
0
votes
1answer
44 views

How do i prove this function satisfies Cauchy-Riemann equation?

This is the first time i'm trying to apply abstract complex analysis to an explicit function . So, i need to see what a correct approach looks like. In the class, professor had shown easy examples ...
0
votes
1answer
33 views

How to compute this integral : $\oint \bar{z}^n dz$

How to compute this integral : $$\oint_{|z|=a} \; \bar{z}\;^n dz$$ I choose $z = ae^{i \theta}$, and so $\bar{z}\;^n = a^n e^{-i\theta}$ And $$\oint_{|z|=a} \; \bar{z}\;^n dz = ...
-2
votes
0answers
38 views

Determine the number of zeros of the upper half-plane [closed]

$$z^4 + 3iz^2 + z - 2 + i$$ Can anyone please help me????
1
vote
0answers
33 views

Find the number of zeros of the polynomial in the first quadrant [closed]

$$p(z) = z^6+9z^4+z^3+2z+4$$ Please help! I have an exam coming up and I don't completely understand!!!
2
votes
3answers
61 views

Determine the number of zeros in the first quadrant $f(z) = z^4- 3z^2 + 3$ [closed]

Determine the number of zeroes of the following function which are in the first quadrant: $$f(z) = z^4- 3z^2 + 3$$ Help please!!! I'm not that good at complex variables!
1
vote
2answers
109 views

integral of sin(x) to the power 2014

For a course in Complex Analysis we're tasked to find the integral of \begin{align*} \int_0^{2 \pi} (\sin\theta)^{2014} d \theta \end{align*} but I'm a bit stumped so far on how to do this. What I've ...
1
vote
1answer
25 views

Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
1
vote
1answer
65 views

Laurent series expansion, principal part

I need to find the principal part of the Laurent series for $f(z) = \frac{e^{2z}}{1-\cos(z)}$, around $z = 0$. Also, I have to use the undetermined coefficient method. I don't know how to proceed. ...
1
vote
1answer
29 views

Canonical isomorphism between complexified tangent space of submanifold fixed by antiholomorphic involution and tangent space of complex manifold

I haven't really studied complex manifolds and I am at a bit of a loss in regards how to approach this problem: Let $M$ be an $n$-dimensional complex manifold, and let $\phi:M\rightarrow M$ be an ...
2
votes
2answers
186 views

Problem with Cauchy integrals

Hello everybody I need to solve some integral with the help of the Cauchy Integral Formula (CIF). I'll post near each integral the job that I've done and the question that I can't answer. let $\kappa ...
0
votes
1answer
30 views

Show two complex functions are conjugate

I am stuck on a homework problem that asks Show that the functions $f(z) = \frac{z^2}{z^2 + 1}$ and $g(z) = z^2 + 1$ are conjugate. Two functions $f$ and $g$ are conjugate if there is a ...
1
vote
1answer
28 views

Integrating a complex function with Cauchy formula

We have I =$\oint_{C}^{} \frac{(z-1)\sin(z)}{z^2 - 2z - 3}$, C is a circle for which $|z-2| = 2$. I wrote $I = \oint_{C}^{} \frac{(z-1)\sin(z)}{4(z-3)} - \oint_{C}^{} \frac{(z-1)\sin(z)}{4(z+1)}$ ...
2
votes
1answer
26 views

Showing that a function is nowhere analytic

I have to show that $|z|^2$ is nowhere analytic, where $z=x+iy$. I took $f(z) =|z|^2 = x^2+y^2$ =>$u(x,y)=x^2+y^2$ and $v(x,y)=0$ Then I started off by checking whether the Cauchy Riemann equations ...
1
vote
1answer
29 views

Finding all harmonic functions satisfying initial conditions in the unit disk

Find all harmonic functions $\phi$ in the unit disk $D= \{\ z \in \mathbb{C} : |z|<1 \}\ $ that satisfy $\phi(\frac{1}{2})=4$ and $\phi(z)\ge 4$ for all $z \in D$. Through $\phi$ being harmonic, ...
0
votes
1answer
25 views

Complex Weierstrass M-test question.

Use the Weierstrass M-test to show $\forall\epsilon>0,\sum_{n=1}^{\infty} a_nn^{-z}$ converges uniformly if $Re(z)>=1+\epsilon$, where $a_{n}$ is bounded. This is what I've done: ...
0
votes
1answer
27 views

Determining a branch of logarithm

The question I have is that what is the explicit mapping that takes the value $-i \pi/2$ at $-i$ where the mapping is a branch of the logarithm in the slit plane $\mathbb{C}- [0,\infty)$? I'm familiar ...
4
votes
2answers
47 views

Prove Converse continuity using the Preimages

I would like to prove that if pre images $f^{-1}(U) \subset D $ of open subsets $U\subset \mathbb{C}$ are open in $D$ implies a function $f:D \to \mathbb{C}, D\subset \mathbb{C}$ is continuous. ...
0
votes
2answers
20 views

describe and sketch complex set

$f(D)$ where $D = { z : |z|<1}$ and $f(z) = \frac{z+i}{z-1}$. Am I right on saying this set can be described as a translation by $i$ and dilation by $\frac{1}{z-1}$?
3
votes
0answers
32 views

Has anyone used Complex Analysis in the Spirit of Lipman Bers as their textbook?

I have free access to many Springer books from my library, which includes Complex Analysis In the Spirit of Lipman Bers. From what I've seen, it's a decent book that introduces the subject. ...
1
vote
2answers
54 views

evaluate integral by complex method

Can you guys help me how to evaluate this integral by complex analysis method?
0
votes
0answers
23 views

Using Harmonics to find a solution to a boundary value problem

Consider a boundary value problem with two given level sets of phi. One set is in the imaginary plane with center (1,i) and radius 1. This set has level set phi = 0. Another set is in the imaginary ...
2
votes
2answers
65 views

Find $\int_\gamma \frac{dz}{z^2}$ wihtout explicit calculations

Evaluate the following integral without doing any explicit calculations: $\int_\gamma \frac{dz}{z^2}$ where $\gamma(t) = \cos(t) + 2i\sin(t)$ for $0 \le t \le 2\pi$. This exercise comes along with ...
0
votes
0answers
29 views

Example of holomorphic function from unit disc to itself

let $f:\mathbb{D} \to \mathbb{D}$ be analytic function with $f(0)=0$,where $\mathbb{D}$ is the open disc $\{z \in \mathbb{C}:|z|<1 \}$ then $1.|f'(0)|=1$ $2.|f(\frac{1}{2})|\leq \frac{1}{2}$ ...
2
votes
1answer
66 views

Show that two sets are countable infinite

I have to solve the following problem and I was hoping some of you could give me some hints on how to procede. Show that for all $w\in \mathbb{C}$ the sets $$ \{z \in \mathbb{C}: \sin(z) = w\} ...
1
vote
0answers
36 views

Expansion of a function analytic at infinity

Having a bit of trouble with a problem in my complex variables class: Prove that if f(z) is analytic at infinity, then it has expansion of the form $$f(z) = \sum_{n=0}^{\infty} ...
2
votes
1answer
24 views

Applying the Cauchy Integration Formula to $\int_{\left|z\right|=4}{\frac{8\sin(z)}{(z-6)z^2}}dz$

In a section on the Cauchy Integration Formula in my complex analysis text, this problem is an exercise: Evaluate$$\int_{\left|z\right|=4}{\frac{8\sin(z)}{(z-6)z^2}}\,dz$$ I'm failing to see how I ...
1
vote
0answers
22 views

zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$ \frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
1
vote
0answers
26 views

finding fix points on mobius transformation

I need to find the fixed points of the following: dilation on $\mathbb{C}_{\infty}$ translation on $\mathbb{C}_{\infty}$ inversion on $\mathbb{C}_{\infty}$ I'm thinking using mobius ...
2
votes
3answers
68 views

Show that there are no analytic functions $f=u+iv$ with $u(x,y)=x^2+y^2$.

I was trying to prove this using the method of contradiction, so, I assumed on the contrary that their is such an analytic function $f=u+iv$ with $u(x,y)=x^2+y^2$. Now since $f$ is analytic ...
0
votes
1answer
29 views

Complex polynomial decomposition - Residue Theory

I am given the following function: $R(z) = (z^2-9)/(z^2+9)^2 $ I need to let $R = P/Q$ be a rational function with $deg P < deg Q$. I will let $ξ$ be a pole of $R$ and the coefficient of $1/(z-ξ)$ ...
0
votes
0answers
25 views

If $Z$ is an admissible function, does $f(z) = f(|z|)$?

If $Z$ is an admissible function, does $f(z) = f(|z|)$? For example, if $f(z) = x^3 + 1$ and I am given $2$ points $z_1 = -1+i\sqrt3$ and $z_2 = -1-i\sqrt3$, can I just find the moduli and use that ...
1
vote
1answer
66 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let $2$ points $z_1 = (-1+i\sqrt3)/2$ and $z_2 = (-1-i\sqrt3)/2$ I am trying to show that there is no point $w$ on the line segment from $z_1$ to ...
0
votes
2answers
43 views

Complex function mapping the unit circle onto an interval

Show that the function $f(z) = z^2 + z^{-2}$ maps the unit circle onto the interval $[-2, 2]$. Okay so far, doing previous questions I firstly try and find the inverse mapping. Here I considered the ...
0
votes
2answers
33 views

Find the image of the set under the exponential function

Find the image of $\{z: |\Im(z)| < \frac{\pi}{2}\}$ under the exponentional function. So, i've set $e^{z} = e^x(\cos y + i\sin y)$ where $\Im(z) = e^x \sin y$ So I have to find the "image" such ...
2
votes
0answers
26 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
0
votes
1answer
25 views

show $f(z)=Az^2+Bz+C$

Let $f(z)=u(x,y)+iv(x,y)$ be an entire function where $z=x+iy$. Suppose there exist real-valued differentiable functions $\phi(x)$ and $\psi (y)$ s.t. $u(x,y)=\phi(x)+\psi(y) \forall x,y\in ...
0
votes
1answer
35 views

Summation using complex methods

Find $\sum_{-\infty}^{\infty}\frac{1}{n^3+2} $ using complex methods. This last one on my massive homework assignment is a real doozie! I'm going to have to finish this in the morning, but I'm ...
0
votes
0answers
12 views

showing solution to kummer differential equation

struggling to solve kummer's differential equation and show that the confluent hyper geometric series is a solution. I have simplified the problem to showing that the sum over j to infinity of ...
3
votes
1answer
44 views

Real integral using complex methods

Evaluate $\displaystyle\int_0^\infty \frac {x^\frac{1}{2}}{1+x^4}dx$ using complex methods. I'm totally locked up on this one and have thrown in the towel. My strategy was to integrate around a ...
0
votes
2answers
29 views

Prove a function is entire

If $g(z)=u(x,y) + i\,v(x,y)$ and $h(z)=a(x,y) + i\,b(x,y)$ are entire prove that for any $\alpha, \beta \in \Bbb C$ - Complex constants. $f(z)= \alpha*g(z) + \beta*h(z)$ is also entire $f(z)= ...
1
vote
1answer
45 views

Harmonic functions on $\mathbb C-\{0\}$

Find all real-valued $C^2$ differentiable functions $h$ defined on $(0,\infty)$ such that $u(x,y)=h(x^2+y^2)$ is harmonic on $\mathbb C-\{0\}$. This is one of my homework problem. As I understand I ...
0
votes
0answers
54 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...