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

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0
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0answers
52 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
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2answers
37 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 $z^c=e^{c\log z}$ ...
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
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2answers
56 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 $1/e^2$ ...
0
votes
2answers
51 views

If $f$ is an entire function then what about the set $S=\{Ref(z)+Imf(z) :z\in D\}.$

Let, $f$ be an entire function on $\mathbb C$ and let $D$ be a bounded open subset of $\mathbb C$. Let, $$S=\{Ref(z)+Imf(z) :z\in D\}.$$ Which of the following(s) is(/are) necessarily true ? (a) $S$ ...
3
votes
1answer
44 views

How many analytic functions are there on a given set

Consider the set $S=\{0\} \cup \bigl\{\frac{1}{4n+7}:n=1,2,...\bigr\}.$ Then the number of analytic functions which vanishes only on $S$ is (a) infinity (b) 0 (c) 1 (d) 2 I think, the answer is ...
1
vote
0answers
26 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
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, $\phi(z^w) = \phi(z)^{\phi(w)}$?
-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 $-4 \leq x \leq 4$
10
votes
6answers
1k views

Solve $z^4+1=0$ algebraically

I know the result and how to solve it using trigonometry and De Moivre. However, given that the complex number $z$ can be rewritten as $a+bi$, how can I solve it algebraically?
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 ...
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 ...
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
4answers
114 views

Show that $i^m + i^{m+1} + i^{m+2} + i^{m+3} = 0$ for all $m ∈ \mathbb N$

Here is the long answer I have come up with so far. m is a natural number If m is divided by 4 :- Let n be the quotient and r be the remainder Then m = 4n+r where 0<=r<4 im = i(4n + r) ...
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
34 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... ...
-1
votes
0answers
25 views

Complex Polynomial

If $z_0$ is a root of polynomial equation where all the coefficients of this equation are real numbers. I need to show that the conjugate of $z_0$ is also a root of this polynomial.
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 $u + v$ ...
2
votes
3answers
107 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= ...
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)- ...
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 ...
1
vote
1answer
27 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 ...
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
42 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; ...
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
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
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
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
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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 ...
0
votes
3answers
38 views

Type of singularity of a complex function

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.
2
votes
1answer
18 views

Show that for any $r>0$ and $z \in \mathbb{C}$ we have that $B_r(z)\subset S_r(z)$

Show that for any $r>0$ and $z \in \mathbb{C}$ we have that $B_r(z)\subset S_r(z)$ In other words, that any Ball of radius $r>0$ centred at $z \in \mathbb{C}$ is a subset of the square ...
4
votes
3answers
408 views

Modulus of a complex number

I am now encountering a problem regarding on complex analysis Lets say we have $w=u+iv$ What would it be for $$|w|^{2}$$ I check a lot of videos and lecture notes, and realize the answer is ...
0
votes
2answers
32 views

Complex number equation

How can I solve this equation? $$z^2 - 12 z - 25 i z + 150 i - 100 = 0$$ $z$ is a complex number and $i^2 = -1$. How many solutions should I get?
3
votes
4answers
415 views

When does $az + b\bar{z} + c = 0$ represent a line?

$a,b,c$ and $z$ are all complex numbers. My idea was to show that it passes through the point $\infty$ in the extended complex plane, but I'm not quite sure how to execute that. Update: It says in ...
1
vote
1answer
20 views

help on proving some property of cubic root of unity

Anyone can help to prove below expression $\omega^r$ and $\omega^{2r}$ satisfy the equation $x^3 - 1=0$ for any positive integer $r$, where $\omega$ and $\omega^2$ are non-real cube roots of ...
2
votes
2answers
27 views

If a, b are complex numbers then the maximum value of $\dfrac{a\bar b+\bar ab}{|ab|}$

If a, b are complex numbers then the maximum value of $\dfrac{a\bar b+\bar ab}{|ab|}$ is (A) 2 (B) 1 (C) the expression may not always be a real number and hence maximum does not make sense (D) ...
1
vote
2answers
37 views

Show that $e^\mathbf{iA} + e^\mathbf{iB} = 2e^\frac{i(A+B)}{2}\cos(\frac{A-B}{2})$

Where $i=\sqrt{-1}$ and $A,B\in \mathbb{R}$ are constants. I've tried already with Euler's formula, but cannot prove the equation above. Best Regards, Thanks.
2
votes
2answers
25 views

Set of Convergence for the following Series

What is the set of convergence for this series: $ \sum_{n=1}^{+\infty} \dfrac{3^{\sqrt{n}}(2+i-3z)^n}{\sqrt{n^2+1}} $ ? My initial thought was to use, $ \dfrac{1}{R} = \lim(|a_n|)^{1/n}$, but this ...
1
vote
1answer
32 views

Problem in a proof that we cannot order complex numbers

The order axioms of real numbers state 1) Either $x = y$ or $x < y$ or $x > y;$ 2) If $x < y,$ then $x+z < y+z;$ 3) If $x, y > 0,$ then $xy > 0;$ 4) If $x > y$ and $y > z,$ ...
0
votes
0answers
24 views

Indeterminacy of complex numbers

If $f(z)=\frac{|z|(1+z^2)^4}{(z+1)^2}$ calculate $f'(0)$ I did: $g(z)=|z|\rightarrow g'(z)=\frac{z}{|z|}$ and $h(z)=\frac{(1+z^2)^4}{(z+1)^2}\rightarrow h'(z)=\frac{(z^2+1)^3[6z^2+8z-2]}{(z+1)^3}$ ...
1
vote
1answer
40 views

Solve $\sqrt{5-12i}$ by square root definition

I KNOW it can be solved by the trig formula, but I want to solve it by the square root definition, so please don't just post an alternative way to do it. By the square root definition: $$z = 5-12i$$ ...
2
votes
1answer
31 views

Derivative complex function

Calculate the derivative of $f(z)=\frac{(1+z^2)^4}{z^2}$ I know that this function is discontinous at $z=0$, what I did is just calculate the derivative the same manner as is done with real ...