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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3
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1answer
94 views

Curves Bounding Complex Function

So I have the following complex function: $$f(z) = z^2$$. This function maps the following region: $$ D = \{z=x+iy : 1\leq x \leq 3 \hspace{3mm} \& \hspace{3mm} 1 \leq y \leq 2\}$$ onto $G ...
0
votes
2answers
62 views

Is there any flaw in the proof of $-1 = 1$? [duplicate]

We have $i^2 = -1$ where $i = \sqrt{-1}$. Now consider $$\sqrt{-1}\cdot \sqrt{-1} = \sqrt{(-1)\cdot(-1)} = \sqrt1 = 1$$ Which proves $-1 = 1$. Is there anything wrong with ...
0
votes
1answer
32 views

Does the commutative property not apply when multiplying Quaternions

I've been looking up Quaternion multiplication and many resources have stated that $j*k=i$ and also in other sources I've found $k*j=-i$ But I have not found any sources stating $j*k=-i$ and I ...
0
votes
0answers
28 views

Prove that $f(z(t)), f(w(t))$ are perpendicular at $t=0$

I have the following problem but I'm not sure if my proof is correct: Let $f(z)$ be a holomorphic function. Let $z(t)=a(t)+ib(t)$ and $w(t)=c(t)+id(t)$ be perpendicular at $t=0$. We have shown in ...
0
votes
0answers
17 views

Find $f'(z)$ of the following $f(z)$

There are 2 problems that I solved simply, but was wondering if there is a deeper concept that I may be missing. Whenever something seems too simple, I worry that I missed something. We are learning ...
0
votes
0answers
9 views

Solving a transformation problem in complex (Argand) planes

Below is an example in a text-book section on using complex-number arithmetic to represent transformations in Argand planes Example A transformation $T$ of the $z$-plane to the $w$-plane is ...
1
vote
2answers
26 views

Inverse tangent of a complex variable

Show that $$\tan^{-1}(z) = \frac{i}{2}\ln\left(\frac{i + z}{1 - z}\right)$$ I tried this approach: $$\tan(w) = z$$ $$\tan(w) = \frac{\sin(w)}{\cos(w)}$$ $$\tan(w) = ...
0
votes
1answer
16 views

Find the image of $f(S)$ and draw what's happening

$$w=f(z)=z^2$$ $$S=\{z\mid Re(z)=a\}$$ I don't get what this is asking and what that second part means. What is the concept here? If I let $z=x+iy$, then I can square it. The real term is then ...
0
votes
0answers
47 views

Solving complex quadratic with $x x^{*}$ and not $ |x|^{2}$

Iā€™m trying to solve a fairly simple AC circuit analytically for $V_2$. After a little bit of work, I end up with the following expression: $$V_2\cdot V_2 āˆ’ V_1\cdot V_2^* + S\cdot Z = 0$$ or ...
3
votes
2answers
24 views

Limits with complex numbers

$$\lim_{z\to 0} \frac{z^*}{z}$$ The way I see it is that it's asking what happens when $z$ approaches $0$. However, I can't just say undefined because $z$ is actually $z=x+iy$. So if I take the ...
1
vote
2answers
28 views

When should I add $\pi i$ to the exponent when computing the polar form of complex nubmers?

This is maybe math $101$ question: Let $z_1=1+i$. I know that $r=\sqrt 2$ and $\theta=\arctan(1/1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}} .$$ But now if I take a look at $z_2=-1-i$, I ...
1
vote
1answer
53 views

If $f(z) = e^x \cos{y} + i e^x \sin{y}$ , then what does $f(Ļ€/4)$ equal?

My attempt: $$e^x(\cos(y)+i\sin(y))$$ $$e^x\cdot e^{iy}$$ Let $$z=x+iy$$ Then, $$e^{x+iy}=e^z$$ Thus: $$e^{\frac{\pi}{4}}$$ This makes sense to me in some way, but we are learning about the ...
0
votes
1answer
43 views

Is $2\pi /2\pi$ considered an indeterminate ratio for imaginary exponentiation?

It seems that I can force any expression of the form $a^{bi}$ to become 1 by changing its base to the natural number and raising it to the power of one in the form of $\frac{2\pi}{2\pi}$ so that it ...
1
vote
4answers
118 views

Proving Trig Identities (Complex Numbers)

Question: Prove that if $z = \cos (\theta) + i\sin(\theta)$, then $$ z^n + {1\over z^n} = 2\cos(n\theta) $$ Hence prove that $\cos^6(\theta)$ $=$ $\frac ...
4
votes
0answers
85 views

Trigonometric Expression for $1 + \cos \alpha + \cos 2\alpha + \cdots + \cos n \alpha$ using complex numbers

This question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2} + \cdots + z^{n})$, where the question this ...
4
votes
2answers
50 views

Proving trig identity using De Moivre's Theorem

Question: Prove $$\cos(3x) = \cos^3(x) - 3\cos(x)\sin^2(x) $$ by using De'Moivres Theorem So far (learning complex numbers at the moment) that De Moivre's theorem states that if $z$ $=$ ...
2
votes
1answer
23 views

Value of $z$ in $\sin z=\frac{1}{2}\sqrt{2}(1-i)$?

I'm trying to find the value of $z$ in $\sin z=\frac{1}{2}\sqrt{2}(1-i)$. I have tried with the equation $\sin z=\frac{e^{iz}-e^{-iz}}{2i}=\frac{1}{2}\sqrt{2}(1-i)$ but I get stuck and can't make it ...
0
votes
1answer
29 views

Express in terms of $u+iv$

$$z^{-2}$$ I guess the tricky part of this one is whether or not I am allowed to multiply by the complex conjugate raised to the power of 2. Is my attempt valid? ...
0
votes
1answer
15 views

Proving Equations (Complex Conjugates)

The question is: $z$ is a complex number given by $z$ $=$ $sin$$(\theta)$ $+$ $i(1-cos(\theta))$, $-\pi < \theta < \pi $ Show that if $w$ $=$ $\frac 1{z-i}$ then $w$ $=$ $z^* + i $ where ...
1
vote
1answer
42 views

Sketch the region described by $\text{Im}\left[\frac{z-z_{1}}{z-z_{2}}\right] =0$

This question is similar, but please do not mark this as a duplicate, because it is not exactly the same as what is being asked in that question, and besides, the answers given to it do not answer my ...
3
votes
2answers
55 views

Why does it seem to be that I can raise negative numbers to the power i?

I recently encountered the ided of raising a number to the imaginary unit, and I've been trying to figure out what that means and haven't really found any useful resources. So, I came across this ...
1
vote
1answer
30 views

Sketch the region defined by $Re z + Im z < 1$

I have to sketch the region given by $Re z + Im z < 1$, and I'm stuck. For any complex number $z$, $Re z = x$ and $Im z = y$. Is this as simple as graphing the inequality $x + y < 1$, then? If ...
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vote
5answers
38 views

Region on the complex plane: $|z-z_{1}| = |z-z_{2}|$. Intersection of two unit circles?

I have to draw the region on the complex plane defined by the following relation: $|z-z_{1}|=|z-z_{2}|$. After squaring both sides, we obtain the equality $(z-z_{1})\overline{(z-z_{1})} = ...
1
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0answers
50 views

Why can't the exponent laws be extended to complex numbers?

$(x^a)^b=(x^b)^a=x^{ab}$ only when real numbers are involved. This implies that the base must be a positive number. For example, with $x=-1$, $a=2$, and $b=\frac{1}{2}$, ...
0
votes
2answers
76 views

1 = -1 Clearest way to explain why this proof is wrong. [duplicate]

Say you are a high school student or a young undergrad. You are being taught about complex numbers and you are told that $i = \sqrt{-1}$. You go home and you write this: \begin{equation} ...
1
vote
3answers
54 views

What is the argument of $0$? [duplicate]

Very easy but bit more thinking is making it complicated. What is the argument of $0$? The number $0$ can be written as $0+0i$ thus its argument is $\tan^{-1}(0/0)$ which is undefined; so, is the ...
0
votes
1answer
28 views

Is the set satisfying $-1 \leq Re(z) \leq 5$ closed?

I want to prove if such set is closed. We define for $z = x+iy$ in $\mathbb{C},$ $$\mathcal{O} := \{z \in \mathbb{C}: -1 \leq Re(z) \leq 5\} = \{(x,y) : x \in [-1,5], y \in (-\infty,\infty)\}.$$ ...
0
votes
0answers
28 views

Derive the formula: $f(z)=2u(\frac{1}{2}z,\frac{1}{2i}z)-2u(0,0)$

Let $š‘¢(š‘„, š‘¦)$ be a harmonic function which is the real part of a holomorphic function $š‘“ (š‘§)$, so that $$š‘¢(š‘„, š‘¦)=\frac{1}{2}(f(z)+\overline {f(z)})$$ Argue that $\overline {š‘“(š‘§)} = ...
3
votes
1answer
59 views

Summation of the given series

Is there anyway to find the sum of: $\cos(A)+\cos^2(2A)+\cos^3(3A)+....$ upto 'n' terms. Actually original question was to find sum of : $\cos(A)+\cos(2A)+\cos(3A)+...$ upto 'n' terms and I found it ...
2
votes
2answers
52 views

When $\cosh (z)=0$?

I'm studying complex analysis and I'm wondering about all complex values of $z$ that satisfy the equation: $$ \cosh(z)=0 \,\, . $$ Is there a smart way to show all values that vanish with the ...
1
vote
1answer
42 views

$(1+i)(e^{(1+i)\phi})$ expressed in polar and rectangular form

$$(1+i)(e^{(1+i)\phi})$$ I need to express this in both polar and rectangular form, but the difficult part is that extra $i$ above the $e$. Also, what am I supposed to make of $\phi$? We normally use ...
1
vote
1answer
36 views

Write $e^{\ln(5)}i$ in polar and rectangular form

Is there something I'm missing? Below is my attempt, but I feel as though I might have missed something to learn about complex numbers when $r=e^n$. $$|ie^{\ln(5)}|=e^{\ln(5)}$$ ...
2
votes
0answers
29 views

Limit of $\exp(z^2)$ as $|z|$ tends to infinity

Let $g(z) = \exp(z^2)$ and $L$ a ray starting at the origin. Determine those $L$ along which $g$ has a limit (finite or infinite) as $|z|$ tends to infinity and $z āˆˆ L$. Find the value of the limit ...
0
votes
1answer
24 views

Complex number algebra, help

I have $ \lambda^{2} = +/- i \frac{\omega}{\nu} $ when I try and solve for lambda i dont get the same answer as my text book. the book $ \lambda =+/- \frac{1}{\sqrt{2}}(1 + ...
0
votes
2answers
40 views

Find all $z\in\mathbb{C}$ such that $e^z = 1$.

We write $z=a+ib$. Now, $$1 = e^z = e^{a+ib} = e^a e^{ib} = e^a(\cos b + i\sin b) = e^a\cos b + ie^a\sin b$$ We have $$1 = e^a \cos b \\ 0 = e^a\sin b$$ Now, I don't understand why it has to be ...
0
votes
1answer
50 views

$z^{19}=(-1+i)$ find the value of $z$. complex number

I have to find the value of $z$ which satifsfies this equation and which has the second smallest positive argument $\theta$, $0<\theta<2\pi$. I have to find $r$ and $\theta$. The answer I got ...
0
votes
2answers
27 views

Find the real and imaginary parts of the following.

$$\frac{z-a}{z+a}; a \in \Re$$ The part I'm confused about is the $a \in \Re$. I know that this means that $a$ is a real number (not imaginary), but then how do I interpret the addition/subtraction ...
1
vote
1answer
25 views

Give the Taylor series for the following $f(z)$; also, find $f^{(100)}(0)$

$$e^{3z}$$ I'm not sure how to approach this complex number problem. I know that $$1+3 x+\frac{9 x^2}{2}+\frac{9 x^3}{2}+\frac{27 x^4}{8}+\cdots$$ is true for $e^{3x}$, but how does this apply to ...
0
votes
2answers
44 views

Complex Numbers: How do I know which plane to use?

I'm new to complex numbers and I want to know precisely when I need to use which plane for graphing and a general idea of what the plot would look like. $$z=a+bi$$ When it's in the form above, I ...
2
votes
0answers
45 views

What is the motivation behind the solution of this problem involving complex numbers?

The problem is Suppose for three distinct complex numbers $a, b, c$ such that $|a|=|b|=|c|>0$ all of the three numbers $a+bc, b+ac, c+ab$ are purely real. Prove that $abc=1$ By playing with ...
1
vote
1answer
18 views

Complex Numbers - Quartic

Find two distinct real roots of the equation $z^4-3z^3+5z^2-z-10$, and hence solve this equation completely. The problem is how do you find the two distinct real roots?
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votes
0answers
12 views

Coordinates of symmetry of point with respect to line?

A, B and Z are three complex numbers, Pa, Pb and Pz their representation in the complex plane. what is the complex number that corresponts to the symmetry of Pz with respect to the line PaPb ? I ...
0
votes
1answer
23 views

Convert to Cartesian (rectangular) form

Convert the following to Cartesian (rectangular) form and provide a graph. $$e^{i7\pi /2}$$ The problem comes after a long series of similar problems. However, the noticeable difference with this ...
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vote
0answers
24 views

Draw Regions On the Complex Plane that Satisfy this Relation

I'm looking to draw a region that satisfies the following: $$ Im\left(\frac{z-z_1}{z-z_2}\right)=0 $$ What I know so far is this: the expression as it's given is not of the form $ a + bi $, as ...
2
votes
1answer
29 views

Plotting the graph of $\operatorname{Re}(z)<2$

$$\operatorname{Re}(z)<2$$ My idea was that, in the complex plane, the graph looks like a ray starting at the origin and extending up to $2$, but not including that point. Then there would have to ...
4
votes
3answers
45 views

Compute all roots of $(-8)^{\frac{1}{3}}$

$$(-8)^{\frac{1}{3}}$$ The problem states to compute all roots of the complex number above. Below is my attempt, but my inquiries are if I did it right and why it doesn't match Wolfram. Wolfram only ...
1
vote
2answers
37 views

Is this f(z) function analytic?

Is $f(z) = z^2\sin z$ an analytic function for $z \in \mathbb{C}$ ? $z = x + iy $ I really don't know how to split this at this format $f(x,y) = P(x,y) + iQ(x,y)$, so I can prove that if this ...
0
votes
0answers
18 views

Axis of a glide reflection

I am currently taking a gap year before starting university and am trying to get a head start by teaching myself some of the course content. As a result I have no one to ask and no solutions to check ...
0
votes
2answers
70 views

Compute all the roots (complex number problem)

$$ (-1+i)^{\frac{1}{3}} $$ Below is what I've attempted, but I'm not 100% positive if it's right. Also, and more importantly, how do I know if I've computed ALL of the roots of a complex number? ...
2
votes
4answers
131 views

Solve for $z$ (complex numbers)

$$ z^3=i $$ The problem simply states to solve for $z$, but I know that there is some concept to be practiced here about the nth roots of unity. I'm just beginning to learn this concept so I didn't ...