1
vote
1answer
26 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
0
votes
1answer
19 views

Cauchy-Riemann and Analytic Functions

Using the Cauchy-Riemann conditions, tell if $f(z) = z^{*}$ is analytic I have tried this: $Z = x + iy$ $f(x + iy) = Z^{*} = x - iy$ $U(x,y) = x$ $V(x,y) = -y$ $U_x = 1$ Deriving respect to $x$ ...
3
votes
1answer
37 views

Conformal mapping of the domain bounded by a line segment and a circular arc

I am trying to construct a conformal map from the region $R$ which is the set of points in the complex plane bounded by the segment connecting $i$ and $1$ and the part of the unit circle in the first ...
2
votes
0answers
40 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
1
vote
1answer
23 views

Sketching regions is complex plane

When sektching the region $\left|\frac{2z-1}{z+i}\right|$$\geq$1 on the argrand diagram, how should we go about identifying the region, should we take $\left|2z-1\right|\geq\left|z+i\right|$ or ...
0
votes
3answers
75 views

Solving the equation $(z-2)^{4}+(z+1)^{4}=0$

$(z-2)^{4}+(z+1)^{4}=0$ I tried starting by solving $z^{4}=1$ with the solutions being , $1cis (\frac{n\pi }{2})$, where $n = -1, 0, 1, 2$ I am unsure about how to proceed from here, I tried to ...
1
vote
1answer
34 views

Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
1
vote
1answer
44 views

Find the set of $z$ which satisfies the given equation

Let $w \to w^{a}$ be the principal branch of the power function defined for $|\mathrm{Arg}(w)| <\pi$. Find the set of all values of $z\in \mathbb{C}$ such that the following identity holds for ALL ...
2
votes
1answer
44 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 ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
2
votes
1answer
57 views

Interesting examples of Cauchy's Integral formula [closed]

Question : What are some interesting and, albeit, counter intuitive examples of real integrals that are solved using Cauchy's integral Formula. Cauchy integral formula can magically transform some ...
1
vote
2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
3
votes
1answer
71 views

Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...
0
votes
1answer
38 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
0
votes
1answer
20 views

The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle

Let $z,z_1,z_2,z_3$ be four points on the extended plane. Their cross-ratio $(z,z_2,z_3,z_4)$ by definition is the image $Tz$ of $z$ under the Möbius transformation $T$ that sends $z_1,z_2,z_3$ to ...
2
votes
1answer
61 views

Singularities of complex functions.

How do I determine the singularities of a function? What is a singularity? In the functions below which are the singularities? a)$$f(z)=\frac{1}{(z^4+2z)}$$ b)$$f(z)={e^{1/z}}$$
1
vote
3answers
40 views

Determining Laurent Series expansion and residues

Determining Laurent Series expansion and residues of $f(z)=\frac{z}{(z+1)(z+2)}$ around $z = -2$. What is the validity of the expanded region? What is $res(f, -2)$??
1
vote
4answers
62 views

Which way will produce the following integral?

Which way $\gamma$ will produce the following integral? $$\int\limits_{\gamma}\frac{3+i}{z^5 - z}dz = 0$$
0
votes
3answers
40 views

Complex solutions of polynomial question

$2z^3-6z^2+mz+n = 0$ $m, n$ are real and $1+\sqrt{ 2} i$ is a solution. Find $m$ and $n$. Attempt to solve : Giving the known theorem $1-\sqrt{2}i$ is also a solution, so we can substitute each time ...
0
votes
1answer
17 views

Prove that a Möbius transformation $T$ sends the imaginary line to the circle $\{z: |z|=2\}$,

Problem Let $T:\overline{\mathbb C} \to \overline{\mathbb C}$ be a Möbius transformation such that $T(1+2i)=1$, $T(-1+2i)=4$ and $|T(0)|=2$. Show that $|T(bi)|=2$ for all $b \in \mathbb R$. The ...
2
votes
4answers
66 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
3
votes
1answer
77 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
1
vote
0answers
22 views

Curves composition with holomorphic function

Statement $(i)$ Let $\gamma:\mathbb R \to \mathbb C$ a $C^1$ curve. Let $v={\gamma}'(t_0)$ the complex number that one obtains from translating to the origin the tangent vector to $\gamma$ at ...
3
votes
1answer
50 views

Finding Möbius transformations that satistfy certain conditions

Problem Find Möbius transformations that send $(i)$ the circle $|z|=2$ to $|z+1|=1$, and $-2$ to $0$, $0$ to $i$. $(ii)$ the upper half-plane $Im(z)>0$ to $|z|<1$ and $\lambda$ to $0$ (where ...
0
votes
1answer
34 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
1
vote
2answers
27 views

Limit of complex numbers' sequence (related to Möbius transformation)

Problem Let $T(z)=\dfrac{7z+15}{-2z-4}$. Let $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$ Find $\lim_{z_n \to \infty}z_n$ I am having a lot of difficulties trying to solve this. I've tried to find a ...
1
vote
0answers
29 views

Controlling the Sum of a Set of Complex Numbers

Consider a set of N previously fixed angles $\phi_i$. Let $p$ be a positive integer. If $\sum^N_{i=1} e^{ip\phi_i} = 0$, what if any restriction does this place on the value of $p$? If $\phi_i = 2\pi ...
2
votes
1answer
33 views

Cross ratio and symmetric points exercise

Problem Let $C$ be a circle or a line belonging to $\overline{\mathbb C}$ and let $z_2,z_3,z_4$. Two points $z$ and $z^*$ are said to be symmetric with respecto to $C$ if ...
0
votes
1answer
40 views

Möbius transformations on $\space \overline{\mathbb R}$

Prove that a Möbius transformation $T(z)=\dfrac{az+b}{cz+d}$ maps $\overline{\mathbb R}$ to $\overline{\mathbb R}$ if and only if it can be written with real coefficients. If it can be written with ...
-1
votes
0answers
13 views

what is deformation of a contour and Fourier inversion in general?

I would like to to know how one can do contour deformation and Fourier inversion in general? I shall be very thankful to you if some one explain it with the help of example.
0
votes
4answers
66 views

complex numbers quadratic equation question

how to solve $z^2 +3|z| = 0 , z$ complex ? treating the complex number as $a+bi $ or anything similar didnt help much...also solving like simple algebric equations also didnt prove effective and ...
3
votes
1answer
38 views

Möbius transformation: proving the image of the unit circle is a line

Problem 1) Find the Möbius transformation which maps the points $0,i,-i$ to $0,1,\infty$ respectively. 2) Prove that the image of the circle centered at $0$, of radius $1$ is the line $\{Re(z)\}=1$. ...
5
votes
1answer
178 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
1
vote
1answer
12 views

Images of the stereographic projection's inverse

I am trying to solve a problem which states: Let $\phi: \bar{\mathbb C} \to S^2$ be the inverse function of the stereographic projection Calculate $\phi(Re(z))=0$ and $\phi(Im(z))=0$. I can guess ...
1
vote
1answer
42 views

Real part of Complex Function

I've this function $$f(k,\theta) = \frac{1}{k}\frac{1}{\cot\delta_0(k) -i }$$ and i know that $k\cot\delta_0(k) = -\frac{1}{a} + \frac{1}{2}r_ek^2 + \cdots$ it is an expansion. How can i get that ...
6
votes
1answer
63 views

Prove that $\cos(z)$ and $\sin(z)$ are surjective over the complex numbers. [duplicate]

I have an exercise that says: (a) Prove that $\cos(z)$ and $\sin(z)$ are surjective functions from $\mathbb C \to \mathbb C$. (b) Find the solutions of the equation $\cos(z)=\dfrac{5}{4}$. As far ...
1
vote
1answer
39 views

complex equations question

find all solutions of the equation: $w^4 = -8(1-i\sqrt{3})$ I dont wanna be that guy, but can someone tell me what the second solution to this equation is? cuz the solution manual says it's $-1 + ...
3
votes
3answers
35 views

Find the principle value and all other values of $i^{2/\pi}$

I'm a little confused about how to go about this? Any help would be appreciated, thanks.
-1
votes
1answer
39 views

Find all complex solutions to the equation

i) Find all complex solutions to the equation z^4 +1 -i*3^(1/2) = 0 I basically have no clue, any tips/advice/solutions would be great. I could also need some help with another question, this one ...
0
votes
1answer
37 views
+50

Prove that the functions $g_k(z) = f_k \circ h_k(z)$ form a normal family.

I am having a bit of trouble with the following complex analysis question which originates from a qual. Some help would be awesome. Let $f_k :\mathbb{D} \rightarrow \mathbb{C}$ be a normal family of ...
1
vote
3answers
61 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
1
vote
3answers
32 views

Query regarding Linear Transformation…

As we always read in Complex Analysis, Linear Transformation (L.T.) is a combination of Translation, Rotation and Magnification i.e. $T(z)=az+b$ is a L.T. in complex. However, It doesn't satisfy the ...
5
votes
1answer
110 views

Show $f$ is analytic if $f^8$ is analytic

This is from Gamelin's book on Complex Analysis. Problem: Show that if $f(z)$ is continuous on a domain $D$, and if $f(z)^8$ is analytic on $D$, then $f(z)$ is analytic on $D$. (I assume the ...
0
votes
1answer
39 views

Proving that $f(z)$ is polynomial [duplicate]

Given $R>0 , M>0$ Let $f(z)$ be a entire function such as $|f(z)|\leq M|z|^m$ for evey z such as $|z|>R$. Show that $f(z)$ is a polynomial of degree m or lower.
1
vote
1answer
47 views

Prooving that a complex function is constant

Let $f(z)=u(z)+v(z)i$, $u(z) \leq 0 , \forall z \in \mathbb{C}$, and $f(z)$ is entire. Then $f(z)$ is constant. The hint is "use the Liouville theorem". I tried , but i need prove which f is limited ...
0
votes
1answer
19 views

About the convergence or divergence?

Whether the following integral converge or diverge by comparison test. \begin{align*} ...
0
votes
0answers
34 views

Schwarz's Lemma and extensions in complex analysis

I was assigned this problem: (which I here present verbatim) Let $f$ be a holomorphic function of the unit disk unto itself. Prove that $|f'(0)|\le 1$. Isn't it also necessary to assume that ...
1
vote
1answer
50 views

Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.

I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks. Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb ...
3
votes
3answers
52 views

System of equations with complex numbers

This might seem quite trivial for people who are knowledgeable in complex analysis, but it is not so much to me. I am trying to find an efficient way to solve the following system of equations: $$ ...
1
vote
2answers
54 views

Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for all $z ∈ C.$ Prove that $|f(0)| ≤ 0.2.$

I am working on an old qualifying exam problem and I can't seem to really get anywhere. I would love some help. Thank you. Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for ...