Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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3answers
166 views

Complex logarithm $\log(zw)\neq\log(z)+\log(w)$

Can anyone help me out with explaining why $\log(zw)\neq\log(z)+\log(w)$?
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3answers
344 views

Complex numbers modulus/argument question

$z=\cos x+i\sin x$ Find the modulus and argument of: $ \dfrac{1+z}{1-z} $ I wrote $z=e^{ix}$, so $$\frac{1+z}{1-z}= \frac{1+e^{ix}}{1-e^{ix}}= \frac{1+2e^{ix}+e^{2xi}}{1-e^{2xi}} = \frac{1+2(\cos ...
1
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1answer
88 views

Calculus $T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$

Calculus: $$T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$$ and $$T_2=\prod_{k=1}^{n-1}\sin\frac{k\pi}{2n}$$ My tried: I use Euler's formal: $$z_k=e^{i\frac{k\pi}{2n}}=\cos\frac{k\pi}{2n}+i\sin ...
8
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1answer
254 views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
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4answers
73 views

How does $e^{i\ln(2)}=2^i$

How does $e^{i\ln(2)}=2^i$? I apologise in advance for the simplicity of the question, but I just can't seem to realise why... My mind has gone blank! Thanks in advance.
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2answers
179 views

Proving entire function constant

$f$ is entire and $f(z)=f(z+1)=f(z+i)$ prove $f(z)=const$ I have no clue how to solve it
2
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2answers
141 views

Find all solutions of $z^5+a^5=0$

The task is as follows: Find all solutions of $z^5+a^5=0$, where $a$ is a positive real number. My initial attempt (which leads nowhere) My guess is that i'll have to find the 5 5th roots of ...
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1answer
206 views

How to work out phase of complex number

This one follows on from the question I just asked about logarithms.. Turns out 1/x questions confuse me (sorry for bombarding your exchange with questions, this isn't homework or anything I am just ...
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2answers
139 views

How do I solve this integral?

As stated the title, I get to a point which I can't do anything, and I'm sure I've made a mistake some where, here is my full working out: $$ \int e^{ix}\cos(x)dx \\ u = e^{ix} \text{ | } u'= ie^{ie} ...
2
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2answers
42 views

The pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfies $|w1| =1$, $|w2| = 1$ and $\Re\left(w_1\overline{w_2}\right) =0$

If $z_1=a + ib$ and $z_2 = c + id$ are complex numbers such that $\left|z_1\right| = \left|z_2\right| = 1$ and $\Re\left(z_1\overline{z_2})\right)=0$, then Prove that the pair of complex numbers $w_1 ...
3
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1answer
166 views

Finding the roots of a complex number (de moivre's formula)

2 from 1.2 of basic complex analysis 3rd edition (marsden & hoffman) just wanted to make sure I'm doing this right solve the following equations a) $z^6 + 8 =0$ first I write out the ...
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1answer
94 views

Complex De Moivre's theorem question

Express this in terms of multiple angles. $\cos^3x \sin^4x$ I've used the relationships $$\cos(nx) = \frac{z^n+z^{-n}}{2}$$ $$\sin(nx) = \frac{z^n-z^{-n}}{2j}$$ And end up with $$\cos^3x \sin^4x = ...
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1answer
36 views

every complex number can be put in this form?

I am reading a proof of a theorem and I see this: "if we put $F(x)$ in this form: $F(x)= r e^{i\theta}$" $F(x)$ takes values in $\mathbb{C}$, so the question is: In $\mathbb{C}$, every element can ...
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1answer
194 views

Optimization of Complex Functions, No Use?

During reading the appendix to an engineering text, I came across the following remark: "Complex cost functions are of no interest, because in the field of complex numbers no ordering (relations ...
6
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2answers
181 views

Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ?

I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$ But apparently the answer is $$i\cosh(x)+C$$ I was just wondering, is this ...
2
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3answers
59 views

Quaternion identity proof

If $q \in \mathbb{H}$ satisfies $qi = iq$, prove that $q \in \mathbb{C}$ This seems kinda of intuitive since quaternions extend the complex numbers. I am thinking that $q=i$ because i know that $ij = ...
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4answers
168 views

factor $z^7-1$ into linear and quadratic factors and prove that $ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$

Factor $z^7-1$ into linear and quadratic factors and prove that $$ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$$ I have been able to prove it using the value of $\cos(\pi/7)$. Given here ...
0
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2answers
50 views

Recurrence relation - Show that a sum of a sequence is zero

We are given the following sequence: $f(n)=4f(n-1)-5f(n-2)$, $f(0)=f(1)=a$ where $a$ is some value in $\mathbb C$. We are asked to show that $$\sum_{n=0}^{\infty}\frac{f(n)}{3^n}=0$$ First thing I ...
4
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3answers
193 views

Geometry of the dual numbers

A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to ...
2
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4answers
94 views

How to find the value of $\arctan(\frac{1}{1-x}) + \arctan(1-x)$?

I'm reading a book on complex analysis. In one step while evaluating a path integral, the author makes the following substitution: $$\arctan \left(\dfrac{1}{1-α} \right) + \arctan(1-α) = ...
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4answers
92 views

Integration $1/x$ - complex number

Why there is no integral $$\int_{-e}^{e}\frac{1}{x}$$ And why integral $$\int_{-e}^{-1}\frac{1}{x}= -1$$ and not $$\int_{-e}^{-1}\frac{1}{x}=(-1 + i\cdot\pi)$$ E.g. ...
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1answer
74 views

logarithm of a complex number?

I have a task to study a function like this one: $$F(z) = \frac{\ln(e^{iz^4})}{z^3}$$ I'm trying to simplify this: since the exponential is the inverse function of $\ln()$ can we simplify it to ...
2
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2answers
158 views

Show that if $z = e^{i\theta}$ is a solution to $0 = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$, …

Show that if $z = e^{i\theta}$ is a solution to $0 = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$ [1] where all $a_i$ are real, then $0 = a_{n-1}\sin\theta + a_{n-2}\sin2\theta + \cdots + a_0 \sin ...
2
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2answers
224 views

How to compute $i^i$? [duplicate]

My question is a bit straightforward. How can I solve $i^i$? Do I have to work it out based on polar form of complex number? Even that doesn't seem to help!!
3
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5answers
440 views

I'd like to get explain about complex roots

If $x^6+1=0$ so $x^6=-1$, then we have to find the roots at $\mathbb{C}$. I saw that the roots are $$\Large{e^{(\frac{\pi}{6}+\frac{2k\pi}{6})i}}\;\small{k=0,1,2,3,4,5}$$ this what I understand. ...
4
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3answers
111 views

Does $\operatorname{Log}(1+i)^2 =2\operatorname{Log}(1+i)$

And similarly, does $\operatorname{Log}(1-i)^2=2\operatorname{Log}(1-i)$? If we were dealing with real numbers, it would hold. But I'm guessing that the fact that there are imaginary numbers involved ...
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1answer
111 views

What is the principal argument of $-5-5i$?

When i calculated it by $\tan^{-1}\dfrac{y}{x}$, I got $\dfrac{\pi}{4}$, then i added $\pi$ to make it in the right quadrant, so my final answer is $\dfrac{5\pi}{4}$ However, the correct answer is ...
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0answers
94 views

$\operatorname{arg}$ and $\operatorname{Arg}$, what's the difference?

So to get the argument of a complex number sometimes we use $$\operatorname{arg}(z)$$ and sometimes we use $$\operatorname{Arg}(z).$$ I didn't know the difference between them so I searched the web. I ...
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1answer
39 views

Show that for any complex number $z$, $|r^z| = r^x$, where $r$ in the real numbers, $r>0$.

The question asks for the magnitude of $r^z$, and I take it that the $x$ will be from the fact that $z$ can be written as $z=x+iy$. Been stuck on this for a while now, it's only worth 2 marks, I ...
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2answers
64 views

One to the power of i with two different results.

We can write $1$ in polar form as $1=e^{ik2\pi}$ with $k\in\mathbb{Z}$. If we then take both sides to the power $i$ we arrive at $$1^i=e^{-k2\pi},\quad k\in\mathbb{Z}.$$ This looks indeed strange ...
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1answer
35 views

Complex sum question

If $\omega = \cos(\frac{2\pi}{n})+i\sin(\frac{2\pi}{n})$ show that $1+\omega^h + {\omega}^{2h}+ \ldots + \omega^{(n-1)h}=0$ if $h$ is not multiple of $n$
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0answers
43 views

Rules for $(x^a)^b$ - complex number. [duplicate]

How potentiate $(x^a)^b$ for complex numbers?
4
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2answers
287 views

Prove that: $\sin{\frac{\pi}{n}} \sin{\frac{2\pi}{n}} …\sin{\frac{(n-1)\pi}{n}} =\frac{n}{2^{n-1}}$

Using that: $$ x^{n - 1} + x^{n - 2} + \cdots + x + 1 = \left(x - w\right)\left(x - w^{2}\right)\ldots\left(x - w^{n - 1}\right) $$ Prove that: $$ \sin\left(\pi \over n\right)\sin\left(2\pi \over ...
1
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2answers
102 views

Prove $p_0-p_2+p_4-\cdots=2^{n/2}\cos{\dfrac{n\pi}{4}}$ and $p_1-p_3+p_5-\dots=2^{n/2}\sin{\dfrac{n\pi}{4}}$

Consider: $$(1+x)^n= p_0 + p_1 x + p_2 x^2+\cdots$$ From where $$p_0=1,\quad p_1=\dfrac{n}{1},\quad p_2=\dfrac{n(n-1)}{2!},\ldots$$ Are the coefficients of the Newton´s Binomial expansion, using $x=i$ ...
2
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4answers
203 views

Where is the world of Imaginary numbers?

Complex numbers have two parts, Real and Imaginary parts. Real world is base of Real numbers. but where is (or what is) the world of Imaginary numbers?
3
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3answers
140 views

Complex numbers - system of equations

Let $z$ and $w$ be complex numbers such that $|2z - w| = 25$, $|z + 2w| = 5$, and $|z + w| = 2$. Find $|z|$. Any tips where to start? Is there a better way then just squaring both sides and solving ...
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2answers
78 views

complex numbers - Proving

Prove algebraically that $|w+z|\le|w|+|z|$ for any complex numbers w and z. This is what I got so far: Since $|z|^2 = z \overline{z}$, we square both sides: $ |w+z|^2 = (w+z) \overline{(w+z)} \le ...
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2answers
148 views

Does anything interesting come out of this identity?

Students often make the mistake of writing the following: $$\frac{1}{a+b} = \frac{1}{a}+\frac{1}{b}$$ However, after doing a bit of algebra, it turns out that the above has solutions defined by: ...
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2answers
135 views

Complex numbers - Quadratic formula?

Let a and b be real numbers. The complex number 4 - 5i is a root of the quadratic $z^2 + (a + 8i) z + (-39 + bi) = 0$. What is the other root? I did a lot of work on hand and plugging this into the ...
2
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6answers
141 views

Complex numbers - proof

Let z and w be complex numbers such that $|z| = |w| = 1$ and $zw \neq -1$. Prove that $\frac{z + w}{zw + 1}$ is a real number. I let z = a + bi and w = c+ di so we have that $\sqrt{a^2+b^2} = ...
1
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1answer
60 views

Finding order of pole

I have some problems with the following excersice: Find the order of the pole of: $$\frac{1}{(2\cos z -2 + z^2)^2}$$ at $z=0$. I thought it is maybe better to work here with $1/f$ and find the order ...
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4answers
2k views

find two numbers that add up to 8 and multiply to 20

Find two numbers that add up to 8 and multiply to 20. Only the complex number "i" (imaginary number) is allowed other than the real numbers. But you do not necessarily have to use "i" if it is ...
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1answer
22 views

regarding a proof of $\|\theta(e^{i\lambda})\|^2$

When studying the spectral representation of time series, I read the following formula, I am not clear how to prove the second equation. I expand the left side of the second equation with the ...
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2answers
79 views

Do complex nos makes sense only if they exist in pairs (as conjugates)?

I am not sure if this is correct question but please reply so i can make sense out of complex numbers. Thanks. . It just occured to me. In case of x^2+1=0. We have i and -i roots and if multiplied ...
2
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2answers
342 views

Find all the solutions of $e^z=i$

Find all the solutions of $e^z=i$ I don't know where to start. I want to do $z=\ln(i)$, but have no idea where that would lead me. Thanks in advance.
1
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2answers
66 views

Finding power series

I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$. I know that $1 + a + a² = 0$. I have tried to differentiate the expression and give values ...
0
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1answer
143 views

Product of all complex roots of z^n=a+bi?

How can one prove that the product of all the roots of a complex equation is the same as one root to the power of equation? e.x. $z^n=a+bi$ has $n$ roots (from de Moivre's formula), prove that their ...
0
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1answer
111 views

In which book will I find these types of problems and theory too.

In which book I will find problems of the similar kind? were you have been given these complex half planes and then asked to find their convex polytope.
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1answer
72 views

Locus in the complex plane given an equation

I have the question Let $a$ and $e$ be two positive real numbers, with $0 < e < 1$. Describe the locus of the points $z$ in the complex plane which satisfy $|z - ae| + |z + ae| = 2a$. I ...
2
votes
2answers
134 views

Understanding Euler's Identity

I would like to understand one specific moment in Euler's Identity, namely $$e^{j\theta}=\cos(\theta)+j\sin(\theta)$$ where $j=\sqrt{-1}$. We also know that $$e^{j2(\pi)}=\cos(2\pi)+j\sin(2\pi)$$ ...