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Questions tagged [complex-numbers]

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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For a matrix $A(z)$ that represents the operation of multiplication with a complex number $z$, what does it mean for $e^{A(z)t} = A(e^{zt})$?

We can have a complex number $z = a + bi$ that determines a matrix $A(z)$ in the following way: $$ A(a + bi) = \begin{bmatrix} a & -b \\ b & a \end{bmatrix} $$ This matrix represents the ...
Rishav Dhariwal's user avatar
0 votes
0 answers
44 views

Polar form of the sum of two complex numbers.

I was trying to express the addition of two complex numbers only in function of its absolute values and arguments while preserving the result in polar form, with the fewest terms possible, for ...
g. bressan's user avatar
1 vote
2 answers
79 views

Explanation for why $(-1)^{-i} = e^\pi$?

I can't find any explanations online as to why the following holds: $$(-1)^{-i} = e^\pi$$ I assume there's a simple explanation pursuant to the rules of complex number manipulation, but is there any ...
Svenn's user avatar
  • 77
2 votes
1 answer
85 views
+50

Introduction to the Binary Tetrahedral group and the 24-cell

Context and introduction I was playing with complex number sequences $Z_n=r_n\omega^n=u_n+iv_n$ represented in space and realized that it's always possible to associate up to 48 naturally symmetric ...
phionez's user avatar
  • 320
0 votes
0 answers
52 views

factoring complex conjugation

Let $e_1$ and $e_2$ be variables such that $$e_1e_1 = 1,\quad e_2e_2 = -1,\quad e_1e_2 = i,\quad e_2e_1 = -i.$$ Denote by $(a, b) := ae_1 + be_2$ and $[a, b] := a + bi$. As usual, $$[a, b][c, d] = [ac ...
node196884's user avatar
-5 votes
0 answers
53 views

A simple yet complex proof that I am unable to solve. [closed]

prove that: ((e^(ax ))cos(bx))^n=((sqrt(a^2+b^2))^n)(e^(ax))cos(bx + n.arctan(b/a))
Rajrup Chattopadhyay's user avatar
-1 votes
1 answer
58 views

What are the solutions of $z^2=-1/\overline{z}$ [closed]

I was only able to find the solution $z=-1$, but according to the fundamental theorem of algebra, shouldn't it have two roots?
Mogipit243's user avatar
4 votes
0 answers
59 views

Approximating powers of elements on the unit circle

Let $R \subseteq \mathbb{N}$. We say that $R$ is adequate if: $$\forall n \in \mathbb{N} \; \; \forall \varepsilon > 0 \; \; \forall w \in S^1 \; \; \exists r\in R \; \; |w^n - w^r| < \...
J. S.'s user avatar
  • 412
3 votes
1 answer
98 views

How to find principal value of the cubic root?

I tried to find principal value for $\sqrt[3]{z}$ , I started from $$ z=w^3 $$ So $$ w_1=\sqrt[3]{r} \exp\left(\frac{Arg(z)}{3} i\right)$$ $$ w_2=\sqrt[3]{r} \exp\left(\frac{Arg(z)+2\pi}{3} i\right)$$ ...
Faoler's user avatar
  • 1,637
4 votes
2 answers
286 views

Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$

While exploring possible applications for exponential substitution, I stumbled upon the following integral identity: $$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
Emmanuel José García's user avatar
0 votes
1 answer
113 views

Trying to solve a complex number question

Currently, I have been preparing for ioqm and this question came in a mock of it. Cant figure out whether the question gives sufficient information to solve it.... Here is the statement: Suppose a, b, ...
Dhaval Bothra's user avatar
1 vote
1 answer
101 views

Using the residue theorem to compute two integrals [closed]

Classify the singular points for the function under the integral and using the residue theorem, compute: (a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$ (b) $$ \int_{|z|=2} \sin\left(\frac{...
GENERAL123's user avatar
-2 votes
0 answers
33 views

Replacing z with its conjugate to solve questions in complex numbers [closed]

I have noticed that in many questions of Complex numbers , specially those involving a polynomial in z , we replace z with zbar and solve the question, what's the rationale behind doing this. And does ...
satyam sharma's user avatar
-1 votes
0 answers
39 views

Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]

$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
Mark Ren's user avatar
12 votes
7 answers
5k views

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

I am studying the history of complex numbers, and I don't understand the part on the screenshots. In particular, I don't understand why a cubic always has at least one real root. I don't see why the ...
Tereza Tizkova's user avatar
3 votes
2 answers
113 views

Precisely sketch the mapped region and write the equations of its boundaries.

Find the formula for the fractional linear transformation $f$ that maps the points $0$, $-i$, and $2i$ respectively to the points $-1$, $0$, and $3$. Determine and write where the mapping $f$ maps the ...
math123's user avatar
  • 21
14 votes
2 answers
638 views

Can the definition $i=\sqrt{-1}$ be made sense of rigorously without using $\mathbb R^2$ or similar construction of complex numbers?

For me, the natural way to define complex numbers seems to be to take $\mathbb R^2$ and then define addition and multiplication on top of that an boom, you have complex numbers. You then pretty ...
user1747134's user avatar
5 votes
1 answer
70 views

Calculate the integral using direct parameterization

Calculate the integral using direct parameterization: (a) $\int_K \overline{z} \, dz$, where $K$ is the line segment from the point $2i$ to the point $2 - 4i$. (b) $\int_K \sqrt{z} \, dz$, where $K = \...
general123's user avatar
2 votes
1 answer
96 views

The function $ f $ is given by the formula $ f(z) = \frac{1 - \cos z}{z^5 + z^7}$

The function $ f $ is given by the formula $$ f(z) = \frac{1 - \cos z}{z^5 + z^7}. $$ (a) Classify the singularity at the point $ z = 0 $ and write down the principal part of the Laurent series ...
user1718's user avatar
3 votes
1 answer
36 views

Map the region $D$ using the function $g$

The functions $f: \mathbb{C} \to \{x + iy \mid (x > 0) \vee (x = 0 \wedge y \ge 0)\}$ and $g: \mathbb{C} \setminus \{0\} \to \mathbb{R} \times (-\pi, \pi]$ are given by the equations $f(z) = \sqrt{...
math123's user avatar
  • 21
2 votes
1 answer
95 views

$ |\sinh(\Im z)| \leq |\cos z| \leq \cosh(\Im z),$ what estimate do we get if $z \in \mathbb{R}$?

Prove that for every $z \in \mathbb{C}$ the following holds: $$ |\sinh(\Im z)| \leq |\cos z| \leq \cosh(\Im z). $$ What estimate do we get if $z \in \mathbb{R}$? Attempt: To prove the given ...
lolip123's user avatar
4 votes
1 answer
60 views

Compute the integral $\int_{|z|=3} \frac{f(z)}{z^2} \, dz$ for $f(z) = \sin(iz) - z^2$

Let $f(z) = \sin(iz) - z^2$. (a) Write the real and imaginary parts of the function $f$ and check if $f$ is holomorphic. (b) Compute the integral $\int_{|z|=3} \frac{f(z)}{z^2} \, dz$. Attempt: (a) ...
lolip123's user avatar
1 vote
1 answer
118 views

Write a Möbius transformation that maps the region

Write a Möbius transformation that maps the region $D = \{ z \in \mathbb{C} \mid |z| \geq 1 \land |z + i| \leq \sqrt{2} \}$ to the region $D' = \{ z \in \mathbb{C} \mid \frac{\pi}{4} \leq \arg(z) \leq ...
user1718's user avatar
1 vote
0 answers
62 views

determine $\int_{-\infty}^{\infty} \frac{\cos(ax)}{\pi (1 + x^2)} \, dx$ and $\int_{-\infty}^{\infty} \frac{\sin(ax)}{\pi (1 + x^2)} \, dx$ [duplicate]

Let $a \in (0, \infty)$. Using integration of the function $f$ with the rule $f(z) = \frac{e^{iaz}}{\pi (1 + z^2)}$ along the positively oriented boundary of the region $D = \{ z \in \mathbb{C} \mid |...
user1718's user avatar
2 votes
1 answer
71 views

Determine and justify where the function $ f $ maps the region $ D = \{ z \in \mathbb{C} \mid 0 \leq \operatorname{Re} z \leq \pi \} $.

Given the function $ f : \mathbb{C} \to \mathbb{C} $ with the rule $ f(z) = \cos z $. (a) Find all $ z \in \mathbb{C} $ for which $ f(z) = \frac{5}{4} $. (b) Determine and justify where the function $ ...
Markus's user avatar
  • 45
4 votes
1 answer
159 views

Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.

Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$. (a) Determine and classify the singular points of the function $f$ and calculate the residues at these ...
lolip123's user avatar
-3 votes
1 answer
63 views

The imaginary part of $\frac{ax + iay + b}{cx + icy + d}.$

What is the imaginary part of $$\frac{ax + iay + b}{cx + icy + d}$$? I know that I should multiply by the conjugate of the denominator, but what is the conjugate of the denominator? Is it $c (x - iy) -...
Emptymind's user avatar
  • 2,087
3 votes
2 answers
88 views

$\int_{|z-2|=1} \frac{g(z) \, dz}{(z-2)^2} = 12 \int_{|z-2|=1} \frac{f(z) \, dz}{(z-2)} + 8 \int_{|z-2|=1} \frac{f(z) \, dz}{(z-2)^2}$

Let $f: \mathbb{C} \to \mathbb{C}$ be a holomorphic function and let $g(z) = z^3 f(z)$. Prove that the following holds: $$\int_{|z-2|=1} \frac{g(z) \, dz}{(z-2)^2} = 12 \int_{|z-2|=1} \frac{f(z) \, dz}...
math123's user avatar
  • 21
1 vote
0 answers
63 views

$\int_K |z| dz$ and $\int_K \text{Im}(z) dz$ for different $K$-s

Calculate the integral: (a) $\int_K |z| dz$, if $K$ is the line segment from the point $0$ to the point $2 - i$. (b) $\int_K \text{Im}(z) dz$, if $K = \{ z \in \mathbb{C} \mid |z| = 1 \land \text{Im}...
Markus's user avatar
  • 45
6 votes
2 answers
137 views

Find all holomorphic functions $f(x + iy) = u(x, y) + iv(x, y)$, for which it holds everywhere that $ u(x, y) + v(x, y) = x^2 - y^2. $

Find all holomorphic functions $f(x + iy) = u(x, y) + iv(x, y)$, for which it holds everywhere that $$ u(x, y) + v(x, y) = x^2 - y^2. $$ Express the solution as a function of the variable $z = x + iy$....
lolip123's user avatar
1 vote
1 answer
115 views

Two questions about the function $f : \mathbb{C} \to \mathbb{C}$ defined by the expression $f(z) = \sinh z$

Given is the function $f : \mathbb{C} \to \mathbb{C}$ defined by the expression $f(z) = \sinh z$. (a) Prove that for all $z, w \in \mathbb{C}$, the following holds: $\sinh(z + w) = \sinh z \cosh w + \...
GENERAL123's user avatar
0 votes
0 answers
84 views

Given the function $f: \mathbb{C} \to \{ x + iy \mid (x > 0) \vee (x = 0 \land y \geq 0) \}$ with the rule $f(z) = \sqrt{z}$ [closed]

Given the function $f: \mathbb{C} \to \{ x + iy \mid (x > 0) \vee (x = 0 \land y \geq 0) \}$ with the rule $f(z) = \sqrt{z}$ and the region $D = \{ z \in \mathbb{C} \mid |z| > 1 \land \left(\...
user1718's user avatar
3 votes
2 answers
191 views

Möbius transformation mapping the upper half of the unit disc to the first quadrant

Find the Möbius transformation that maps the region $D_1 = \{ z \in \mathbb{C} \mid |z| < 1 \land \operatorname{Im} z > 0 \}$ to $D_2 = \{ z \in \mathbb{C} \mid \operatorname{Re} z > 0 \land \...
user1718's user avatar
0 votes
1 answer
27 views

How is this calculation of solution to damped oscillator done in MIT OCW "Vibrations and Waves" course?

I'm taking a course from MIT OCW called "8.03 - Vibrations and Waves". The first two lectures are reviewing simple harmonic motion and damped free oscillators. In lecture 2 there is a ...
xoux's user avatar
  • 5,021
2 votes
1 answer
77 views

Puzzled by asymmetry of cosine integral

I used Mathematica to calculate the antiderivative of $\cos (\pi x)/x$. I obtained the cosine integral $$ \int \frac {\cos (\pi x)}{x} dx = Ci(x) $$ where $$ \begin{aligned} Ci(x) &:= - \int_x^\...
Richard Burke-Ward's user avatar
0 votes
2 answers
93 views

How do I find the angle of the second right triangle?

Additional context (not necessary to read) (This is actually an electrical engineering problem. I just simplified it to a math problem. The three triangles actually represent three complex numbers, ...
tryingtobeastoic's user avatar
1 vote
1 answer
27 views

Show that if $I_C =C \frac{dV_C}{dt}$ and $V_C=\cos(\omega t)$ then $\frac{V_C}{I_C} = \frac{1}{i\omega C}$

A capacitor is a component that fulfills the relationship $I_C(t) = C \frac{\text{d}V_C(t)}{\text{d}t}$. Show that if $V_C(t) = \cos(\omega t)$ then the capacitor's impedance is $Z_C = \frac{V_C}{I_C}=...
Carl's user avatar
  • 539
2 votes
3 answers
127 views

Minimum value of $|z^4+z+\frac{1}{2}|$ on the unit circle

Let $z$ be a complex number. What is the minimum value of the expression $|z^4+z+\frac{1}{2}|$ for $|z|=1$? I wanted to explore the long process of considering $z=x+iy$, and substituting to get the ...
whatamidoing's user avatar
  • 2,879
0 votes
1 answer
71 views

What's the history of defining the amplitude of a complex number as its argument(phase)?

I see some confusing definitions saying the amplitude of a complex number is the argument or phase of the complex number, as in the following examples. What is the amplitude of a complex number? ...
relent95's user avatar
  • 111
-1 votes
2 answers
75 views

Can i express $\arg(z_3)$ as a combination of $\arg(z_1)$ and $\arg(z_2)$? [closed]

if $z_1$ and $z_2$ are two complex numbers and $z_3=z_1+z_2$, can I express $\arg(z_3)$ in terms of $\arg(z_1)$ and $\arg(z_2)$. I want to do this so that I can see the individual contributions of $\...
jones871's user avatar
1 vote
1 answer
89 views

Prove $p \colon \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} \colon z \mapsto \bar{z}^{2}$ is a covering map

I'm stuck on this problem and generally struggle to show that some maps cover maps. Consider $p \colon \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} \colon z \mapsto \bar{z}^{2}$. Prove that $p$ is a ...
JLGL's user avatar
  • 795
-1 votes
1 answer
43 views

Linear Independence without Vandermonde Determinant [closed]

Let $n > 2$ be an integer, $X_1, \ldots,X_n$ be vectors in a vector space, and $\lambda_1, \ldots, \lambda_n$ are nonzero, mutually different scalars. I want to prove the following implication: $$ ...
yassine's user avatar
  • 35
0 votes
1 answer
43 views

Help with Analytic Function Zero in Annulus

I'm working on a problem involving an analytic function in an annulus, and I need some help with the second part of the question. Here is the problem statement: Suppose that $ f(z) $ is analytic in ...
tree tree juice's user avatar
-2 votes
0 answers
141 views

Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]

So the question was $$\sqrt{x+1}=-2$$ And obviously there is no value for it, However, If you do the thing with $e$ and $\ln{}$ $$e^{\ln{\sqrt{x+1}}}$$ and $$e^{\frac{1}{2}\cdot (\ln{x+1})}$$ Then ...
Jkt's user avatar
  • 19
0 votes
2 answers
58 views

How is it that Archimedes's spiral is the derivative of the unit circle w.r.t. $i$?

The unit circle in the complex plane is given by $z=e^{i\theta}$. Therefore, $$ \frac{\partial z}{\partial i}=\theta e^{i\theta} $$ which is precisely Archimedes's spiral. This may be a fluke, but ...
Cye Waldman's user avatar
  • 7,778
1 vote
0 answers
24 views

Introductory questions to fourier transform: help with the visualization of integration (with respect to one axis) of a 2D function

I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn ...
S.C.'s user avatar
  • 5,064
2 votes
3 answers
102 views

Complex Numbers - Find the minimum value of ${|{z^2-1}|+|{z^2+1}|}$

Question - If ${|z|=1}$ and ${z\neq \pm1}$, then the minimum value of ${|z|+|\frac{z^2-1}{z}|+|\frac{z^2+1}{z}|}$ _ Tried a couple ways to solve this question. The essence of the question is basically ...
Ekarshi's user avatar
  • 63
2 votes
3 answers
58 views

Using Euler's Identity to simplify $\cos(2t + \pi/4) \cdot \cos(2t)$

To verify if $\cos(2t + \pi/4) \cdot \cos(2t)$ simplifies to $\cos(4t + \pi/4)$ using Euler's identity, I converted both cosine functions into their exponential forms respectively, multiplied the two ...
Rishav Dhariwal's user avatar
1 vote
0 answers
43 views

Real part of an eigenvalue lies in numerical range

I‘m struggling to prove this Lemma and will be glad to get hints: If $A\in\mathbb{R}^{n*n}$, $\lambda$ is an eigenvalue of $A$, then $Re(\lambda)\in\{\frac{x^TAx}{x^Tx}, x\in\mathbb{R}^n$\ $\{0\}\}$ ...
veirab's user avatar
  • 61
3 votes
3 answers
161 views

If $z_1+z_2+z_3=z_1z_2z_3$, then $z_1$, $z_2$ and $z_3$ cannot lie all above the real axis.

I am trying to solve the following problem: If $z_1$, $z_2$ and $z_3$ are complex numbers such that $z_1+z_2+z_3=z_1z_2z_3$, then $z_1$, $z_2$ and $z_3$ cannot lie all above the real axis. It is a ...
Taladris's user avatar
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