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

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2answers
81 views

How to reciprocal this imaginary exponent?

Assume $x\gt 0$, how does one simplify $$e^{(-x^2t)/i}\ ?$$ I don't understand how we could change the i under to the top so I could use Euler's formula
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1answer
75 views

Recurrence sequence over the complex field

Consider the following recurrence relation $$z_{n} = c^2 + 2cz_{n-1}^2 + z_{n-1}^4 - (c+c^2)z_{n-1} - 2cz_{n-1}^3 - z_{n-1}^5$$ where $z_{n}, c \in \mathbb{C}$. I google a while but the formula for ...
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1answer
72 views

Why is it safe to assume point-wise voltage here is real-valued?

Please see my EMFT online notes. Where it says I assumed $\hat{V}(x)$ to be real-valued out of laziness. But down lower where a general solution for $\hat{V}(x)$ is given, it appears that $V(x)$ ...
5
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1answer
164 views

Show $|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$ for complex $a$, $b$, $c$

How to prove for any complex numbers $a$, $b$, $c$, the inequality $$|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$$ is correct?
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4answers
2k views

Does the inequality $|\sin(x)|\leq |x|$ extend to the complex numbers?

As it is well known: $$|\sin(x)|\leq |x| \forall x \in \mathbb{R}.$$ Now, if we have a complex number $z$; can I preserve the same inequality $$|\sin(z)|\leq |z|\quad \forall z \in \mathbb{C}?$$
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1answer
610 views

Deriving Taylor series for function from geometric series

Given the geometric series $\frac{1}{1-z} = \sum_{n=0}^n = 1 + z + z^2 + ...$ If there is a function $f(z)=\frac{1}{z+j}$ how would you get it's Taylor series about center z = 1? I have tried the ...
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1answer
100 views

Plotting a complex argument arc

I am having trouble sketching a complex argument arc $$ \text{Sketch the following on an arcand diagram:}\\ \arg\left(\frac{w+1}{w}\right)=\frac{\pi}{6}$$ I've tried to devise a method on my own ...
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2answers
114 views

Complex numbers lead a trigonometric equality

Times ago, I used to think about some trigonometric equalities. Now, I have faced a new one with different one: Show that if $z^7+1=0$ then ...
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2answers
236 views

Apply Cauchy-Riemann equations on $f(z)=z+|z|$?

I am trying to check if the function $f(z)=z+|z|$ is analytic by using the Cauchy-Riemann equation. I made $z = x +jy$ and therefore $$f(z)= (x + jy) + \sqrt{x^2 + y^2}$$ put into $f(z) = u+ ...
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1answer
244 views

Help with complex number phasor notation

I am having trouble understanding how $10jy$ is converted to $10 e^{j\pi/2}$. Here $x$ and $y$ are unit vectors: (original image) $$\large=\operatorname{Re}\left[(10\hat{x}-10j\hat{y})e^{-j10\pi ...
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1answer
55 views

Hectic absolute values? (where $a=ix$ and $b=-ix$)

Where $a=ix$ and $b=-ix$ then what is: $$|a+b|^2$$ $$|b-a|^2$$ And then is this equality true? $$|a+b|^2=|a|^2+|b|^2$$ because it seems $a+b=0$!
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3answers
102 views

Is there any difference between the absolute values operators $|z|$ and $\|z\|$?

Is there any difference between the absolute values operators $|z|$ and $\|z\|$ where $z=a+ib$?
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2answers
215 views

Find the imaginary part of this sum

Let $$S = e^{i\alpha} + \frac{e^{i3\alpha}}{3} + \frac{e^{i5\alpha}}{3^2} + \cdots$$ Find Im$(S)$ and show that it is equal to the sum $$I = \sin(\alpha) + \frac{\sin(3\alpha)}{3} + ...
3
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1answer
116 views

How do I derive these roots

Let $z = \cos(\frac{\pi k}{5}) + i\sin(\frac{\pi k}{5})$ Consider the imaginary part of $z^5$, and deduce that $x^4 - 3x^2 + 1 = 0$ has solutions: $$2\cos(\frac{\pi}{5}), ...
8
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3answers
226 views

Putting ${n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ in a closed form

As the title says, I'm trying to transform $\displaystyle{n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ into a closed form. My work: $\displaystyle\left(1 + ...
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3answers
659 views

Find a maximum of complex function

I am trying to find a simple method that does not use the tools of advanced differential calculus to find following maximum, whose existence is justified by the compactness of the close ball ...
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1answer
1k views

Cauchy–Schwarz inequality for complex numbers

How can I prove the Cauchy– Schwarz inequality for two complex numbers? $$z_1=x_1+iy_1$$ $$z_2=x_2+iy_2$$ I can prove the triangle inequality for two complex numbers: $$|z_1+z_2|\le |z_1|+|z_2|.$$ ...
3
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2answers
114 views

Put a transformation under the form of a rotation in the complex plane

On the complex plane, I have a transformation "T" such that : $z' = (m+i)z + m - 1 - i$ ($z'$ is the image and $z$ the preimage, $z$ and $z'$ are both complex number) and $m$ is a real number. ...
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5answers
135 views

simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $?

How can I simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $ to rectangular form $z = a+bi$? (Note: Wolfram Alpha says the answer is $z=-8$. My professor says the answer is $z=\pm8$.) I've tried to figure ...
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1answer
104 views

Show that the function $g: S \rightarrow \mathbb{C}$, given by $z\mapsto z^3$, is surjective but not injective.

Let $S$ denote the closed sector $0 \leq \arg (z) \leq 2\pi/3$, in the complex plane, including the vertex at $z = 0$. Show that the function, $g: S \rightarrow \mathbb{C}$ , given by $z\mapsto z^3$, ...
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3answers
638 views

Factorize the polynomial $P(z) = z^4 - 2z^3-z^2+2z+10$, into linear and/or quadratic factors with real coefficients

$2+i$ is given to be one of the roots of the polynomial. I am doing this as a practice for exam prep. Since $2+i$, is a root, then $(z-2-i)$ is a factor? So I have: $(z-2-i)(z^3-Az^2-Bz+C) = ...
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1answer
78 views

Classifying continuous characters $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$.

I recently saw the following claim: Let $\mathbf{C}$ denote the field of complex numbers together with its usual topology. If $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$ is a continuous ...
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3answers
228 views

sketch set satisfying $|z-2|+|z+2|\le5$

In the complex $z$ plane, $z = x+iy$, sketch the set satisfying the inequality: $|z-2|+|z+2|\le5$ I know from experience that this is an ellipse, but if I just wanted to find the $x$ and $y$ ...
3
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1answer
173 views

Topological shape of sphere-like equations with complex radius

For fixed complex number $s≠0$, what 4-dim shape is given by complex solutions $z,w$ of $z^2+w^2=s^2$ and higher dimensional version 2N-dim shapes of $z_1^2+z_2^2+...+z_N^2=s^2$ ?
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0answers
19 views

Optimal bounding constant for partial sums of a signed sum of numbers in the unit disk.

This recent question received several answers, and GenericHuman's answer and the comments below provide a good synthesis of all the other answers in my opinion. In this synthesis, only one related ...
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1answer
363 views

Complex Numbers in Standard Form and other assorted problems

Some help on these practice questions and how they are solved would be much appreciated. Ran into some problems in Precalc. 1) Write the complex number in standard form. $6 − \sqrt{-50}$ 2) Perform ...
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2answers
9k views

Writing Complex Numbers in Standard Form

Can someone show me how to write complex numbers in standard form? I missed a few days of class and do not have the text book. Answering a simple question like the one below would help Write the ...
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2answers
521 views

modulus of $\sin(z)$ where $z$ is a complex number

I'm asked to show if there exists $z$ in $\mathbb{C}$ such that, the two following conditions are simultaneously satisfied $$|\sin(z)|>1, |\cos(z)|>1$$ For $|\sin(z)|^2$ I find ...
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0answers
241 views

Complex numbers and Fourier transform

Here I am stuck in solving Fourier transform and the funny part is that I am stuck in the basics, in the complex part. I hope someone can help me solve this part. $$ 3 + 3 ( \cos \frac{4\pi}3 + j ...
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2answers
229 views

Euler's formula and $i^x = \cos(x \cdot \frac{\pi}{2})$

While playing around with a plotting software, i just found out that $$f(x) = i^x = \cos(x·\frac{\pi}{2})$$ How does this connect to Euler's formula? Obviously, here, the alternating sign change ...
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1answer
2k views

Locus of Z as cartesian equation

Could you please help with this locus problem? I think I am aiming for a cartesian equation in terms of $x$ and $y$ that may look like a circle equation e.g. $(x+a)^2 + (y+b)^2$ but I'm not sure. ...
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3answers
203 views

Factorization of polynomials over $\mathbb{C}$

I'm stuck I don't know how to write this complex number equation as two factors although I know one of those factors is $z - 3$. Any ideas/advice appreciated. $$ f(z) = z^3 + (-6+2j)z^2 + ...
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1answer
343 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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3answers
2k views

How to show $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$

How would I show that $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$? Im not sure how to begin, does it involve using $\sinh z=\frac{e^{z}-e^{-z}}{2}$ and $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$?
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5answers
2k views

Complex number: calculate $(1 + i)^n$.

I have to solve the following complex number exercise: calculate $(1 + i)^n\forall n\in\mathbb{N}$ giving the result in $a + ib$ notation. Basically what I have done is calculate $(1 + i)^n$ for some ...
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1answer
374 views

How to calculate partial derivatives of $f(x+iy)=x^2-y^2 + 5xi$ using limits

Let $f(x+iy)=x^2-y^2 + 5xi$. So hence $u(x,y)=x^2-y^2$ and $v(x,y)=5x$ In my notes it calculated $\frac{\partial u}{\partial x}$ at $0$ as follows: $\frac{\partial u}{\partial ...
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3answers
302 views

Explaining why sin and cos are *not* at right angles

Disclaimer: I'm an engineer, not a mathematician I recently had a fierce discussion (lots of blood) on electronics.stackexchange about phase shifts. The impedance of a resistor is real, that of a ...
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1answer
237 views

Radius of convergence of $\displaystyle\sum\limits_{n=0}^\infty2^{-n^2}z^n$

I was reading examples to find the radius of convergence for power series. The power series is defined as $\displaystyle\sum\limits_{n=0}^\infty c_n(z-z_0)^n$. And to find the radius of convergence ...
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3answers
204 views

How to plot $\{z \in \mathbb{C} : |z-i|>|z+i|\}$

How would I draw the set $\{z \in \mathbb{C} : |z-i|>|z+i|\}$ and $\{z \in \mathbb{C} : |z-i|\not=|z+i|\}$? Im not sure how to solve the second one, and for the first one, I tried squaring both ...
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4answers
813 views

Finding square roots of $\sqrt 3 +3i$

I was reading an example, where it is calculating the square roots of $\sqrt 3 +3i$. $w=\sqrt 3 +3i=2\sqrt 3\left(\frac{1}{2}+\frac{1}{2}\sqrt3i\right)\\=2\sqrt ...
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3answers
1k views

Calculate square root of $i \Leftrightarrow z^2=i$

Let $z = r(\cos\theta+i\sin\theta)$. In my notes there was this example to calculate the square roots of $i$. What was done was: $z = r(\cos\theta+i\sin\theta)\\z^2 = ...
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2answers
202 views

Complex numbers argument

Question: $$z=\frac{a+3i}{2+ai}$$ Show that there is only one value of $a$ for which $\operatorname{arg} z= \frac{\pi}{4}$, and find this value. My attempt: ...
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3answers
299 views

Does my definition of double complex noncommutative numbers make any sense?

I wanted to factorize $a^2+b^2+c^2$ into two factors in a similar way to $$a^2+b^2 = (a+ib)(a-ib)$$ This doesn't seem to be possible using real or complex numbers. However I came up with the following ...
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2answers
188 views

Solving for an exponent?

$1^x = i$ I can't solve it through logs, because $\log 1 = 0$. Does this mean $x$ is undefined?
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2answers
321 views

How do I prove that a complex number equals infinity?

I need to prove that $z=x+iy$ equals infinity is equivalent to $x = \infty$ and $y=\infty$. I also have to give an example of a complex number $z$ so that $\sin(z)=\infty$.
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1answer
881 views

Finding roots of unity?

The $n$th roots of unity are the complex numbers: $1, w,w^2,...,w^{n-1}$, where $w=e^{\frac{2\pi i}{n}}$. Why is this true? I understand why $w$ is 1 root of unity, but why are $w^0,..., w^{n-1}$ ...
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1answer
63 views

Which region is defined by $\frac{z}{i}-\bar{z}=0$

Which region on complex plane is defined by the geometric images of $z$ that satisfied this condition: $\frac{z}{i}-\bar{z}=0$ One of my trials was: $\frac{z}{i}-\bar{z}=0\Leftrightarrow z- ...
3
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2answers
780 views

Cubic with complex roots

I have a problem figuring out how exactly I find the cube roots of a cubic with complex numbers. I need solve the cubic equation $z^3 − 3z − 1 = 0$. I've come so far as to calculate the two complex ...
2
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3answers
2k views

Square root of negative numbers

If: $$a = \sqrt{ b^2 - b }$$ The problem I have is that for values of: $0 < b < 1$ the result of: $b^2 - b$ Is a negative number which gives rise to an error on Excel and my calculator. ...
3
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2answers
479 views

Roots of unity?

The $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even: The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = ...