How to plot a set of complex numbers with given argument and absoulute value bounds

I want to plot the following complex numbers $ $ z \in \text{(complex numbers)}:\pi/4 < \arg (z) \leq 5 \pi/4,\ 1 \leq |z| < 2 $ $

I don’t know how to graph it so that it would look like 2D without any unecessary details. The closest I found how I want it to look is, when I looked at how graphing of parametric function looks. I tried to use contour plot to graph it, but I just can’t seem to do it…

ContourPlot[ Im[F[z[x, y]]], {3 pi/4 < arg (z) <= 5 pi/4, 1 <= abs (z) < 2}, {x, -.2, .2}, {y, -.2, .2}, PlotRange -> All, Contours -> Range[-5, 5, .5], ContourLabels -> True]

Does anybody know how to graph my set?

How to Convert the complex notation into x, y-coordinates?

Convert the complex notation into x, y-coordinates; see also Accumulate and ListLinePlot.
enter image description here

I try to use the table to list all the summation value.( I suppose upper bounds N is 5)

plot1 = Table[{N,Sum[e^(2*Pi*I*n*Sqrt[2]),{n,1,N}]},{N,1,5}]; ListLinePlot[plot1]; 

But I just got an empty axis.
I think I need to convert the complex notation, I don’t know how to convert it.
Could you give me some suggestions?
Thank you!

How can we find the powers of a complex number?

this is the complex number im dealing with zb=(1/7)* (Cos[pi/3]+iSin[pi/3])

I need to represent for the first 10 powers in the plane. I also need to label the points with "z to the power n(n is the power for which i elevate zb)

I need to label the axes, i need to do all this in mathematica

My questions 1.How can I represent a complex number in mathematica 2.How can I represent for the first 10 powers the plane. 3.How can I label each point on the graph to the corresponding power?

my professor gave me a hint to use the callout[] command in mathematica. thank you!

Error in nonlinearmodel fit for a function with Definite Integral and complex number

I am trying to fit a function fun and Here is the code which I am trying, Please find the data here dataset

Data = Import[    "E:\Shelender\codes\Mathematica\Aelastic \ relaxation\datat.asc"]; real = Data[[All, {1, 2}]]; imag = Data[[All, {1, 3}]]; w = 1.26*10^8; k = 1.38*10^-23;  f[H_, s_, d_] := ((1/(Sqrt[2*Pi]*s*H))*Exp[(-(Log[(H/d)])^2/(2*s^2))]) dynamic[x_?NumericQ, s_?NumericQ, d_?NumericQ, A_?NumericQ,    t_?NumericQ] :=   A*(1 - NIntegrate[      f[H, s, d]/((1 + (I*w*t*Exp[((H)/(k*x))]))), {H,        0, \[Infinity]}])  fit = ResourceFunction["MultiNonlinearModelFit"][    Rationalize[{real, imag}, 0],     ComplexExpand[ReIm@dynamic[x, s, d, A, t]],     Rationalize[{{A, 1.0*10^-4}, {t, 1.0*10^-12}, {d, 10}, {s, 0.25}},      0], {x}, PrecisionGoal -> 3, AccuracyGoal -> 3];  fit["ParameterTable"] Show[ListPlot[{real, imag}],   Plot[{fit[1, x], fit[2, x]}, {x, 0,     Max[real[[All, 1]], imag[[All, 1]]]}, PlotRange -> All],   PlotRange -> All] 

Although I am not getting any error but fit values are completely off

How to determine if given “complex” time complexity is $O(n^2)$?

If a given time complexity, such as these:

  1. $ (n + \log n) * \sqrt{n+\log n}$
  2. $ n * (200 + \log^2 n)$
  3. $ (7+n^3)\log(n^5)$

is not determinable by just looking at it whether is it in class $ O(n^2)$ or not, how do I decide? If a time complexity is given, and in it there are more types of expressions (exponential, logarithmic, polinomial, … ) how do I decide which one determines the $ O(n^2)$ or $ O(n\log n)$ or … complexity?

Get modulus and plot complex function

I have the following function:

freq[a_, b_, t0_, tr_, s_] := -((b E^(-s (b + t0)) (b E^(s (b + t0)) (-1 +             b s) UnitStep[-b] - b E^(s t0) UnitStep[b] +          E^(s (b - tr)) (E^(s (t0 + tr)) (-1 + b s) UnitStep[-t0] +             E^(s tr) (-1 + b s - s t0) UnitStep[t0] -             E^(s (t0 + tr)) (-1 + b s) UnitStep[-t0 - tr] + (1 +                s (-b + t0 + tr)) UnitStep[t0 + tr])))/(s^2 tr)) 

Now I want to plot the function as follows:

Plot[ComplexExpand@Abs@ExpToTrig@freq[0, 1, 0, 10^-6, Iw], {w,0,10^9}] 

However that doesn’t work. I couldn’t exact the absolute value of the complex function to plot it.
(w is a real positive number)

Does anyone know how to plot that?

Making complex boolean circuits that give true as output only for a specific combination of boolean inputs

This is my first question on a stack exchange website so please bear with me. I am making challenges for a jeopardy style capture the flag event in my college and I had come across the minetest challenge in the hardware section of google CTF qualifier conducted last year. A clean and organized solution to this problem has been provided by liveoverflow.

I would like to design a simpler version of this problem for my college’s CTF event but I am unable to design a complex circuit that gives true output only for a specific combination of inputs. I know that a circuit with this functionality is not very difficult to implement and just needs to represent the following logic:

trueinput1 AND trueinput2 AND ... NOT falseinput1 AND NOT falseinput2 ...  

However I want it to be vast and complicated so that participants cannot decode its functionality just by doing a visual analysis. Is there any technique to complicate the boolean logic above and to design a corresponding circuit that looks ugly even for a small number of inputs(32/64).

Simplifying complex expresion (without ComplexExpand or User defined rules)

Many times, I’ve struggled with simplifying complex expressions, such as this extraordinarily simple expression, and Mathematica wont do it:

$  Assumptions = {l>0,a>0,a \[Element]Reals, l \[Element]Reals} Conjugate[(Exp[I a])^(2 l)] //FullSimplify 

Where Mathematica returns the same thing with no simplification. I told it the assumptions which make it very obvious to just change the sign of the exponent, and I don’t understand why it doesn’t work.

I looked here, but it seems like they have to define their own rules to do this kind of thing. I’m also saw here that you can do //ComplexExpand //FullSimplify for this expression, but it doesn’t seem to work in all cases. Why should it be nessecary and why doesn’t FullSimplify do it already? Also, I think ComplexExpand assumes that all variables are real, which was the case in this expression, but isn’t always the case. So how would I do it then?

What happens if a wizard attempts to create a complex object using True Polymorph?

Let’s say that a wizard tries to create a vessel using True Polymorph. Now, a vessel is made of a main wooden body and many other parts, more or less “removable” (e.g. sails, wheel, furniture, crates, ropes). Are they created along with the main body? Is it necessary to roll for some knowledge check to determine the result?