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

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-1
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2answers
29 views

Is function holomorphic for all complex numbers

pls for help/check. I have a function: $f(z)=3e^{i(z)}$. Is this function holomorhics for all complex numbers? $f(x+iy)=3e^{i(x+iy)}=3e^{-y}\cos (x)+i3e^{-y}\sin (x)$ so $Re=3e^{-y}\cos (x)$ and ...
0
votes
1answer
38 views

Find the line integral of $1/(z^2+4)^2$ over region $\gamma$

I have to find: $$I=\oint_{\gamma}\frac{dz}{(z^2+4)^2}.$$ $\gamma$ in this case is a circular curve defined by $|z-i|=2$, which is a circle centered at $i$ with radius $2$. It is clear that the ...
0
votes
0answers
21 views

Find Line integral of $e^{-z} /{z-\pi/2}$ on a region $\gamma$

Let $\gamma$ be the diamond connecting points $x=2, -2$ and $y=2, -2$. and its oriented positively (counter-clockwise, I believe?). I'm not so sure if we can use the Cauchy integral formula here and ...
2
votes
1answer
64 views

Weird problem $z^i=i$

Weird problem $z^i=i$: $$i\ln z=i$$ Then: $$\ln z=1$$ Therefore: $$e^1=z\implies z=\cos(1)+i\sin(1)$$ is that right? I was told it's not.
0
votes
0answers
43 views

What does it mean to raise a complex number to a complex power?

What does the formula $ (a+ib)^{x+iy}$ mean? All variables are real here, and $i $ represents $\sqrt {-1}$.
1
vote
2answers
17 views

how to find the argument of a multipication of complex numbers? [closed]

I know how to find Arg(z). I'm having problem with the following expression: $$Arg((\frac{\sqrt{3}}{2}+ \frac{i}{2})^{16}(\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i)^{10})$$
0
votes
1answer
35 views

Complex numbers - addition of two modulus help

Ok, so I got the answer to part i), but however, I'm not so sure how to get the answer to part ii). The answers say its an ellipse and they specified the equation, but I can't understand how they came ...
4
votes
2answers
52 views

How can “$y = \sqrt{-x}$ ” be sketched on the x-y plane?

I was reading James Stewart's book Calculus 5th edition (international student edition) and I came across an example that seemed wrong to me. In chapter 1 section 3, he talks about transformations of ...
1
vote
2answers
29 views

Inequality with absolution value for complex number

How to show that inequality: $|1-\bar{\alpha} z| \ge |z-\alpha|$ $z$ and $\alpha$ are complex number, $\alpha$ is constans and $|z|<1$, $| \alpha| < 1$ I can proof that by using substition ...
0
votes
1answer
21 views

Does Fermat's Theorem for Stationary Points hold for functions $f: \mathbb C \to \mathbb R$

Given a function $f: \mathbb C \to \mathbb R$ ($z = x+yi, \; x,y \in \mathbb R)$ Does this hold? $f$ has an extremum at $ z_0 = x_0 + iy_0$ $f$ is differentiable at $S$ and $z_0 \in S$ ...
1
vote
2answers
39 views

Finding the minimum value of magnitude of this complex number

|z|>2 then find the minimum value of "|z+1/2|" How can I solve this using circles?
-1
votes
2answers
54 views

If $\sum (2009-r)\cos(\frac{2\pi r}{2009})=-n/2;\quad 1\leq r\leq 2008,$ then the digits in the unit's …

$$\text{If}\;\sum (2009-r)\cos\left(\frac{2\pi r}{2009}\right)=-n/2;\quad 1\leq r\leq 2008$$ then the digits in the unit's place of $(9417709487)^n$ must be equal to? Well how do I proceed? Hints ...
2
votes
1answer
26 views

Let $A_1,A_2,..,A_n$ be the vertices of n sides of a regular polygon such that $1/A_1.1/A_2=1/A_1.1/A_3+1/A_1.1/A_4$ then value of $n$ must be?

Let $A_1,A_2,..,A_n$ be the vertices of n sides of a regular polygon such that $$\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$$ then value of $n$ must be? Any ideas on how to start? I'm having ...
0
votes
3answers
36 views

Trigonometry-Complex Numbers Based Problem

If $2^7\cos^5x * \sin^3x$=$a\sin8x- b\sin 6x +c\sin 4x + d\sin 2x$ where $x$ is real then what will be the value of $a^4 + b^4 + c^4 + d^4$? Even a hint will suffice... I don't know how to proceed! I ...
0
votes
0answers
55 views

Does there exist an entire function such that $f\left(n+\frac{1}{n}\right)=0$

Does there exist an entire function $f:\mathbb C \to \mathbb C$ such that $f\left(n+\frac{1}{n}\right)=0$ for all $n\in \mathbb N$ ? I tried through Taylor series expansion , also by contradictory ...
0
votes
2answers
38 views

Complex Number -A problem on conjugate

|$z_1$|=2,|$z_2$|=3,|$z_3$|=4 and |$2z_1+3z_2+4z_3$|=9 then the absolute value of $8z_2z_3+27z_3z_1+64z_1z_2$ must be equal to? ($z_1,z_2,z_3$ are complex numbers) I tried manipulating with the ...
3
votes
2answers
67 views

Complex number, how to solve

Calculate i)$(1+i)^i$ ii)$(-1)^{\frac{1}{\pi}}$ I did i)$(1+i)=\sqrt{2}e^{i\frac{\pi}{4}}$. Knowing that if $z$ and $c$ are complex numbers $z^c=e^{c\log z}$ ...
4
votes
1answer
181 views

Base conversion: How to convert between Decimal and a Complex base?

My motivation for this question is exploring beyond the ideas in Project Euler Problem 508. In that problem, it is helpful to know how to convert between a decimal number and a number in base ...
1
vote
1answer
24 views

Complex number, logarithm power proof

Proof that i)$Log(1+i)^2=2*Log(1+i)$ ii)$Log(-1+i)^2\neq2*Log(-1+i)$ What I did i)By definition $z^a=e^{a\log z}$, so if $z=(1+i)$ and $a=2$ $$Log(1+i)^2=Log(e^{2\log(i+1)})=2*log(i+1)$$ But I do ...
1
vote
1answer
10 views

Find all solutions for a complex logarithm

$\log z = 6i$ I am working on a problem very similar. What I am seeing $\log z = \ln|z| + i(\theta + 2\pi n)$ for $n\in\mathbb{Z}$ What I am curious about, as if seen obvious to me that $ \log ...
0
votes
1answer
20 views

Complex number, logarithm and exponential

Find i)$Log(-ei)$ ii)$Log(1-i)$ I'm not too sure about how to solve this, what I did is Take $z=-ei$ so $Log(z)=\log r + i\theta, \space r>0, \space -\pi<\theta\leq\pi$ ...
0
votes
1answer
31 views

Complex number, logarithm

Find i)log(e) ii)log(i) I do not know if these issues are of simple fact, that there is something behind. I did i)Since $log$ and $e$ are inverse functions so$$log(e)=log(e^1)=1$$ Knowing that ...
1
vote
0answers
25 views

Does there exists an automorphism of $\Bbb{C}$ that's also an exponential hom?

Is there an automorphism of the field $\Bbb{C}$ of complex numbers, $\phi$, such that for all $z, w \in \Bbb{C}$ we have in addition to being a ring hom, $\phi(z^w) = \phi(z)^{\phi(w)}$?
0
votes
0answers
27 views

Complex number, exponential

Find all values of $z$ such that i) $e^z=-2$ ii) $e^z=1+i\sqrt{3}$ iii) $e^{2z-1}=1$ What I did I know that $e^z=e^x(cosy+iseny)$, then i) $e^x\geq0 \forall x \in \mathbb{R}$ so I need to find ...
-1
votes
0answers
18 views

How do we solve a system of complex linear equations using matrices?

For instance, how would we solve the following equation: (-1-i)x - 2y = 0 and 1x + (1-i)y = 0
0
votes
0answers
26 views

Maximum value of the function $f=2e^z + e$ on the line $z=x+4i$

Q1 : Where does the complex function $f=2e^z + e$ get's maximum at line $z=x+4i$, where $-1 \leq x \leq 1$ . Q2 : Same question for the line $z=1+xi$ where $-4 \leq x \leq 4$
0
votes
1answer
34 views

Line integral confirmation and Geometric interpretation

I have $$\int_{C}(z - \bar{z})dz$$ where $C = \{z \; : \; |z-1| =2\}$ So I parametrize $C$ by letting $z = 2e^{it} + 1 = 2\cos(t)+ 1 + 2i\sin(t)$ and let $x = 2\cos(t)+1$ and $y = 2\sin(t)$, for $t ...
0
votes
2answers
57 views

Corrrect treatment of a limit approaching zero in complex plane?

How to (correctly) evaluate this limit? $$\lim_{k\to0}\left(\frac{\mathrm{i}+k}{\mathrm{i}-k}\right)^{-\frac{\mathrm{i}}{k}}$$ Here $\mathrm{i}$ is the imaginary unit. Mathematica gives $1/e^2$ ...
1
vote
2answers
46 views

Show that $\frac{z}{z+1} = 1 - \frac{1}{z+1}$

I'm doing some complex analysis work and I came upon this equivalency: $$f(z) = \frac{z}{z+1} = 1 - \frac{1}{z+1}$$ I was trying to find out how to go from $z/ (z+1)$ to the equivalent expression ...
0
votes
1answer
22 views

Prove that $|f^{(k)}(0)|/k! \leq M (e/k)^k$

Assume that f(z) is an entire function and $|f(z)| \leq Me^{|z|}$. Here's what I have gotten so far. I used the Cauchy estimate which is really simple math and I've gotten it down to something like ...
1
vote
2answers
37 views

Find the greatest value of $\arg z$ achieved on a circle in $\mathbb{C}$

Consider the circle $$|z-6i|=3;$$ its centre is $(0,6)$ and its radius is $3$. I want to find the greatest value of $\arg z$ achieved on this circle. My idea is that the tangents to the circle from ...
0
votes
2answers
18 views

Prove that for all $z\in\mathbb{C}$, if $|z| = 1$ and $z\neq−1$ then $Re((1-z)/(1+z)) = 0$

What I have so far: Assume $|z|= 1$ and $z\neq-1$, then $z=1$ or $z=i$ or $z=-i$. If $z=1$, $Re(1-1)= 0$ as needed, but for trying to prove $z=i$ and $z=-i$ I get $Re(1-i) = 1$ and vice versa... ...
4
votes
2answers
37 views

Euclidean norm of complex vectors

I am working on a proof: One has two vectors, $u,v \in \mathbb C^n$, such that $u \cdot v=0$ . I am trying to prove that $$|u + v|^2 = |u|^2 + |v|^2.$$ I am a little stuck on how to do $u + v$ ...
1
vote
1answer
16 views

Proving that $|a(z)^\alpha - b(z)^{\alpha}| \rightarrow 0.$

Related to an earlier question, I asked that Is it true that $|a^{\alpha} - b^{\alpha}| \leq |a-b|^{\alpha}$? I was asking this since I am given that two functions $a(z)$ an $b(z)$ satisfy $|a(z)- ...
4
votes
0answers
59 views

Writing circles as $|z-a| = \lambda |z-b|$ for the same $a,b$

My problem is in the context of the complex plane. I want to know if given two disjoint, not concentric circles $C_1,C_2\subset \mathbb{C}$, can you find $a,b\in \mathbb{C}$ such that $$C_1=\{z\in ...
1
vote
1answer
29 views

Complex integration confusion

I wish to compute $\int_{C}(x^2 - iy^2)dz$, where $C := \{z\mid |z|=1\}$ is positively oriented. I am a bit confused on what $dz$ actually is. I know I have $\int_{C}x^2dz - i\int_C y^2dz$, but I ...
1
vote
1answer
53 views

Solve $(z+2) /( z-3i) = 4+2i$ for $z$ in complex numbers

I'm having some trouble trying to isolate $z$. Can I multiply $(4+2i)$ by $(z-3i)$ without changing $z$ into standard form?
0
votes
3answers
48 views

prove that this complex function is the zero function

we know that $f$ is analytic in the ring $R=$ {$z: r_1 < |z| < r_2$}. moreover at the circle {$z:|z|=r_2$} $f$ is continuous and for every $z$ in that circle $f(z) = 0$ . We have to prove that ...
0
votes
2answers
16 views

is the following complex function is defined at deleted neighborhood of $z=0$

the function is: $\dfrac{2\pi z - i}{\sinh(\frac{1}{z})}$ of course the function is not defined at $z=0$, but what happen in a deleted neighborhood of that point ?
2
votes
2answers
44 views

Prove $\frac{1}{z^2}=\sum\limits_{n\ge0}(-1)^n(n+1)(z-1)^n$

Prove that for any complex number $z$ such that $|z-1|<1$, one has: $\frac{1}{z^2}=\sum\limits_{n\ge0}(-1)^n(n+1)(z-1)^n$ What I've done; ...
2
votes
3answers
108 views

Does $\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0 $ for all $n$ imply that $\lambda_1= \lambda_2= \dots= \lambda_k = 0 $?

Suppose $\lambda_1, \lambda_2, \dots, \lambda_k $ are complex numbers that $\forall n \in \mathbb{N}$ satisfy $$\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0.$$ Can we deduce that $\lambda_1= ...
0
votes
2answers
15 views

Positively oriented circle

I wish to have a positively circle centered around $z_0 = -1 +2i$ with radius $3$. Clearly, the circle is $|z - z_0| = \rho = 3$. And written out we have $ \sqrt{(x + 1)^2 + (y - 2)^2} = 3$. I am ...
0
votes
3answers
32 views

Basic Expansion

I know this is probably easy but how would I multiply out $$(z-(5+i))(z-(5-i))$$ Would I multiply inside both brackets by the "-" then the z part? Thanks
0
votes
2answers
32 views

different ways of calculating laurent series of $\frac{1}{1+z^2}$ at $z=0$

it seem to me that in that simple case there are different ways of doing this. I have to wonder on the general case of $\dfrac{1}{a+z^2}$ at $z=0$. edit: the interesting part of course is where ...
0
votes
2answers
28 views

Verify that Log$(z^{w}) = w$Log$z$ + $2\pi i n$

The symbol "Log" denotes the complex logarithm. Let $w$ be a complex number so that $w = u+iv$ for some reals $u, v.$ We have $$\mbox{Log}(z^{w}) = \log |z^{w}| + i\arg (z^{w}) = u\log |z| - v\arg ...
0
votes
2answers
46 views

Principle argument for $-1-i=-\frac{3\pi}{4} \ne \frac{\pi}{4}$

I want to find the principle argument for $-1-i$ Using the standard method for finding $\theta=\arctan\left({\frac{y}{x}}\right)$ $$\arctan(\frac{-1}{-1})=\frac{\pi}{4}$$ But: ...
1
vote
2answers
28 views

Proving a property of cubic root of unity.

Can anyone help on proving below expression? For $w$(complex cubic root of unity), $1+w^r+w^2 r = 0$ for any positive integer $r$, which is not a multiple of $3$.
0
votes
0answers
25 views

Write $A\cos\theta_1 + \cos\theta_2$ as product of cosines

I would like to write $A\cos\theta_1 + \cos\theta_2$ as product of cosines. If $A=1$ and $x=\operatorname{Re} z = \cos\theta_1 + \cos\theta_2$, where $z=e^{i\theta_1}+e^{i\theta_2}$. Then ...
2
votes
2answers
60 views

What is the correct value?

My confusion is: $(-9)^{2/3} = ((-9)^{2})^{1/3} = ((-9)^{(1/3)})^{2} = 4.32$ But my calculator shows math error, and google says: $(-9)^{2/3} = 2.16+3.74i$
0
votes
3answers
39 views

Type of singularity of a complex function [on hold]

what is the type of singularity: (if pole - what order) $$z\cos{\dfrac{1}{z-1}} \quad \text{at} \quad z=1$$ general tips to determine the type would be also appreciated.