Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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1answer
19 views

Could someone point me in the right direction for this complex analysis equation?

I'm supposed to show that the maximum value of $|z^2+1|$ on the unit disk $|z|\leq1$ is 2. My teacher's hint was "triangle inequality". I've been racking my brain how to tie the triangle inequality ...
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1answer
23 views

How to factor polynomials with complex coefficients

Factor completely the polynomials $$p(x)=5ix^4-(9+2i)x^3+7x+6-i$$ and $$q(x)=9x^5-x^3+7x+6$$ First, I tried to use the Fundamental Theorem of Algebra but it did not work out. ThenI tried plugging in ...
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2answers
44 views

How would one prove this flavour of the triangle inequality?

I have to prove $|z_1 - z_2| \leq |z_1|+|z_2|$ where $z_1,z_2$ are in $\mathbb{C}$. What I wrote down is: $$|z_1| = |z_1+z_2-z_2| \geq |z_1-z_2|-|z_2|\implies |z_1|+|z_2|\geq |z_1-z_2|,$$as desired. ...
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0answers
19 views

How to slice a complex functions about an axis that is not though the origin and skew to the z-axis?

Graphing the real part of complex function $\frac{1}{1+z^2}$ colored according to the imaginary part yields: $$Re(\frac{1}{(1+z^2)})=\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}$$ ...
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1answer
47 views

Prove that if the functions$ f, g \in O(C)$ are such that $\lim_{z→∞} f(g(z)) = \infty$, then $f$ and $g$ are polynomials.

Prove that if the functions $f, g \in O(\mathbb{C})$ are such that $\lim_{z→∞} f(g(z)) = \infty$, then $f$ and $g$ are polynomials. First, notice that $\lim_{z \rightarrow \infty}g(z)=\infty$, ...
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2answers
31 views

finding the magnitude of an complex exponential

Let $u=6e^\frac{4\pi i}{7}$ and $v=11e^\frac{9\pi i}{5}$. Find $|u+v|$. So $|u+v|=|6e^\frac{4\pi i}{7}+11e^\frac{9\pi i}{5}|$ I know that $|x|$ is equal to sqrt of the conjuate of x times x but how ...
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1answer
20 views

An application of Schwarz Lemma to $g(z)=f(z)+zf'(z)$

Let $f(z)$ be holomorphic in the unit disk, with $f(0)=0$, and $|f(z)+zf'(z)|<1$. Show that $|f(z)|\leq |z|/2$. So this should be an easy problem, but I got stuck. I defined $g(z)=f(z)+zf'(z)$ ...
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1answer
42 views

Show that function has removable singularity

Let $a\in\Omega$ be an isolated singularity of function $f$ that is holomorphic on $\Omega \setminus \{a\}$. Show that if $\mathbb{Re}f(z)>0$ in some neighbourhood of point $a$, then $a$ is ...
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3answers
151 views

Finding the limit using Euler's Formula [closed]

I need to find the limit of the following, by using Euler's Formula. $$\lim_{n \to \infty} \left( 1 + \frac{1}{2} \cos{x} + \frac{1}{2^2} \cos{2x}. . . . + \frac{1}{2^n} \cos {nx}\right)$$ Thanks
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1answer
95 views

Limit of complex numbers

What would be the limit of following term? $$\lim_{n \to \infty} \frac{e^{inx}}{2^n}$$ I tried to convert the $e^{inx}$ into trigonometric form and tried to do some simplification but got stuck ...
2
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3answers
52 views

Suggestions for a Complex Number Proof

I am trying to work on a seemingly forward proof but I am not sure if I am taking the right approach. It is $$\mathrm ZW=0 $$ where Z and W are any complex numbers implies that $\mathrm Z=0$ or ...
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0answers
22 views

Extending a holomorphic map continously

Let $\Omega = \mathbb{C} \setminus [-1, 1]$. a) Give an example of a non-constant bounded holomorphic function on $\Omega$. b) Show that such a function cannot be extended continuously to all of ...
4
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1answer
65 views

There does not exist a holomorphic map between torus and Riemann sphere

So the question is as follows. Prove that there is no meromorphic function $f$ such that at every $z\in \mathbb{C}$ we have $f(z)=f(z+1)$ and $f(z)=f(z+i)$ with only simple poles at the points $m+ni, ...
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3answers
48 views

How to compute $(1 − i \sqrt{3})^3\cdot(1 + i)^2$ using the trigonometric form of complex numbers?

I need to compute it using the trigonometric form of complex numbers: $$(1 − i \sqrt{3})^3\cdot(1 + i)^2$$ I computed it using the standard method: $-16i$
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1answer
29 views

complex algebra

Calculate: \begin{align*} &\left( 1+\frac{1+i}{2} \right) \left( 1+ \left( \frac{1+i}{2} \right)^2 \right) \left( 1+{\left( \frac{1+i}{2} \right)^2}^2 \right) \ldots \\ &\left( 1+{\left( ...
1
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3answers
64 views

Equation in complex numbers

Find all complex numbers such that: $$|z_1|=|z_2|=|z_3|$$ $$z_1+z_2+z_3=1$$ $$z_1\cdot z_2 \cdot z_3=1$$ There is solution with vectors or Vietes formulas. Can we solve this problem with using only ...
2
votes
1answer
33 views

How to understand $Re(z) + Im(z) = 2$ & $Re(z) - Im(z) = 0$, $z \in \mathbb{C}$

So I got stuck on a question asking me to plot in $\mathbb{C}$ the following two expressions ($z \in \mathbb{C}$): $$Re(z) + Im(z) = 2$$ $$Re(z) - Im(z) = 0$$ Now this should be easy enough... If we ...
3
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1answer
41 views

Prove that there is no entire function such that for every $w$ in the complex plane $f^{-1}(w)$ consists of exactly 2 points.

So I came across the following problem. Prove that there is no entire function such that for every $w$ in the complex plane $f^{-1}(w)$ consists of exactly 2 points. So here is what I was thinking. ...
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2answers
60 views

Simplify the following fraction $\frac {2+3i}{6-i}$ [closed]

Simplify the following fraction: $\;\dfrac {2+3i}{6-i}.$ Please show all of your steps so I can understand the full method of solving such problems. Thank you!
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1answer
25 views

Find general values of $\sinh^{-1}(1+i)$ and $\ln(1+i)$.

How to find the general values of $\sinh^{-1}(1+i)$ and $\ln(1+i)$. I really can't get my head around it.
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0answers
22 views

Argument of $0$ in Complex Numbers

What is the proof behind the inexistence of $\arg(\theta)$ ? Is there anything besides the inability of $\cos\theta$ and $\sin \theta$ to be $0$ simultaneously?
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1answer
28 views

Find a quartic (degree 4) polynomial with integer coefficients whose roots are the primitive 12th roots of unity

Find a quartic (degree 4) polynomial with integer coefficients whose roots are the primitive 12th roots of unity How to even approach this?
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2answers
63 views

Complex numbers

Express $\cos^7\theta$ as a polynomial of $\cos\theta$. Hence show that $\sec^2\frac{\pi}{14} + \sec^2 \frac{3\pi}{14} + \sec^2 \frac{5\pi}{14} = 8$. I'm new to complex numbers and I'm unsure about ...
1
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1answer
23 views

fundamental definition of complex used to solve problem

Suppose $z^2$ is real and negative; that is $z^2 = (a,0), a<0$. Show that $z =(0,b)$ and find $b$ in terms of $a$. Now taking the square of $z^2 = (a,0)$ won't work especially since $a$ is ...
0
votes
1answer
22 views

Complex numbers working with real definition

suppose that $z = (x,y)$ and $z^2 = (-1,0)$. Show that $z = i$ or $z = -i$ Now in this form the ordered pairs are defined for the real numbers, Sonmy idea was by definition of multiplication: $Z^2 ...
8
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4answers
834 views

Confused about complex numbers

I am confused about something: \begin{eqnarray} (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray} e^{2 i \pi}=1~ and~ 1^{0.5}=1 \end{eqnarray} ...
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1answer
33 views

Characteristic function, how to integrate?

Find the characteristic function $\phi_X(t)$ of an absolute continious r.v. $X$ with density: $$f_X(x) = \frac{a}{2}e^{-a|x|} \qquad (a>0; x\in \mathbb{R})$$ Notation I have some ...
0
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2answers
23 views

convert complex number to polar form

i have this complex number $\sqrt x/2 + \sqrt x/2 i$ i am trying to convert it to polar form. I know that $r = \sqrt (x^2 + y^2)$ but what are the x and y, $1/2$ ?
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0answers
29 views

Complex number - prove an inequality

Question: Given that:$$z^n\tan\theta_0 + z^{n-1}\tan\theta_1 + z^{n-2}\tan\theta_2 + ... + \tan\theta_n = 3$$ And that $\theta_i \in (0, \frac{\pi}{4})$, prove that: $$|z| > \frac{2}{3}$$ ...
2
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1answer
41 views

Complex numbers - minimum value proof

Question: For:$$|z - z_1|^2+|z - z_2|^2+|z - z_3|^2+\cdots+|z - z_n|^2 = S$$ Prove that the minimum value of $S$ is when:$$z = \frac{z_1+z_2+z_3+\cdots+z_n}{n}$$ I have no idea how to even ...
0
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1answer
23 views

Mumford's proof of theta function convergence

On Dave Mumford's Tate Lectures on Theta I, he begins by proving that $\theta(z,\tau)$ converges. It begins something like: Let $|Im(z)|<c$ and $Im(\tau)>\epsilon$, then: $|e^{\pi i n^2 ...
0
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1answer
23 views

Help with Complex Number (Locus/Argand Diagram)

Hi please help with the following question: "The complex number $\ z $ is given by $\ z=t+1/t $ where $\ t=r(cos\theta + isin\theta) $ find the equation of the locus of the point $\ P$ which ...
0
votes
1answer
30 views

Assume $\arg(z_1)-\arg(z_2)=2n\pi$. Show that this implies $|z_1+z_2|=|z_1|+|z_2|$

I am really lost here. Can anyone please give me a hint or two?
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3answers
35 views

Assume $|z_1+z_2|=|z_1|+|z_2|$. Show that this implies $\arg(z_1)-\arg(z_2)=2n\pi$

The hint I am given is that the relationship of $|z1||z2|$ implies $\arg(z1)-\arg(z2)=2n\pi$ is to be used somewhere. I think the only way this can be done is to square it but after that I'm not ...
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1answer
26 views

Assume $\theta_1-\theta_2=2n\pi$. Prove that $\text{Re}(z_1 \bar z_2)=|z_1||z_2|$

I proved the other way around already but I can't seem to prove this way. Both $\theta_1$ and $\theta_2$ are the arguments of $z_1$ and $z_2$ respectively. Can anyone help me out here? Edit: The ...
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3answers
80 views

What Vector Operation Performs $(a,b)*(c,d)=(ac-bd,ad+bc)$?

When you multiply two complex numbers, you get \begin{equation} (a+bi)\times(c+di)=(ac-bd)+(ad+bc)i \end{equation} As betterexplained.com points out, this multiplication of two complex numbers can be ...
5
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1answer
75 views

convergence of $\prod_{n=1}^\infty (1-\frac{z}{n!})$

I want to show that $\prod_{n=1}^\infty (1-\frac{z}{n!})$ is convergent (or uniformly convergent) (z is complex) Can I use the Theorem: The infinite product $\prod_{n=1}^{\infty} (1+a_n)$ converges ...
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2answers
43 views

Complex Solution

I was looking at a simple complex problem and I came by this: Solve: $z^2 = 3 - 4i$ Let $$z= x + yi$$ Rewrite as $$(x + yi)^2 = 3 - 4i$$ Expand $$x^2 +2xyi + y^2i^2 = 3 - 4i$$ Simplify $$x^2 - y^2 + ...
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1answer
42 views

Classification of Discrete Subrings of $\mathbb C$

I am interesting in classifying the subrings of $\mathbb C$ which are discrete with respect to the standard topology (that is, the topology induced by the standard absolute value). Here, I am using ...
11
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3answers
756 views

How to raise -1 to non-integer powers

How do you calculate $(-1)^x$ where $x$ is some real number. For example, what is $(-1)^{\sqrt{5}}$. This question came as I was trying to computer $e^{i\pi a}$ where $a$ is irrational.
0
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1answer
35 views

Proving imaginary number lies parallel to the real axis.

Define $$z \equiv 2e^{i\theta}$$We are to obtain the imaginary and real parts of $$w=\frac{z-2}{z+2}$$ I ended up getting $w=\frac{i\sin\theta}{1+\cos\theta}$ I got this by multiplying by ...
0
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2answers
38 views

Lower bound an expression given a relationship between three variables

Given the following relationship between three variables $$ |x|^2 \leq 2|y|^2 \leq |z|^2$$ I would like to lower bound the function $F$ below and end up with function of $z$ only. I think this can ...
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1answer
44 views

Cauchy Residue Theorem and Cauchy integral formula

Is it true that you can use the Cauchy Residue Theorem and the Cauchy integral formula interchangeably? I believe that the functions that satisfy the conditions of one, will indeed satisfy the ...
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2answers
44 views

Complex number - locus of a point

Question: If argument of $\frac{z - z_1}{z-z_2}$ is $\pi\over4$, find the locus of $z$. $$z_1 = 2 + 3i$$$$z_2 = 6 + 9i$$ Approach: I tried to solve the equation using diagram, basically ...
0
votes
1answer
10 views

Classifying singularities and finding their residues

How would one find the residues of: $f(z)=z/cos(z)$ I believe that the singularities are $z=\pi/2 + 2k\pi$ where k is an integer, but I'm not sure how to go about classifying them and then finding ...
2
votes
3answers
47 views

Arctan in complex logarithm form

I tried to formulate the arctan function in a complex logarithmic form by integrating its derivative by using partial fraction decomposition. And I was wondering if my attempt is valid or not: Using ...
0
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4answers
71 views

Pre-Calculus Complex Number

My younger cousin asked for help on his math homework and I don't remember doing this, can anyone help please? The denominator of $w$ has $z^*+1$ where the $^*$ means to negate the $z$ term so $z^*= ...
3
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3answers
53 views

Closed form of product of complex numbers [duplicate]

I'm stuck in a proof where I want to get a closed form of something. This is the last thing I need to complete my proof: Apparently for small $n\geq2$, the product $\prod\limits_{k=1}^{n-1} ...
0
votes
3answers
61 views

Adding complex exponentials

Can somebody please explain $$e^{-\frac{3}{4}\pi i}+e^{-\frac{9}{4}\pi i}+e^{-\frac{15}{4}\pi i}+e^{-\frac{21}{4}\pi i}=0$$ WolframAlpha Computation.
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2answers
18 views

Lower bound this expression

The following expression is a logarithmic expression I am trying to put a lower bound on. Assuming $x,y$ are complex variables. $$F=\log \left( 1 + \big||x|-y\big|^2\right) $$ where the notation |.| ...