How to plot the solutions of $f(z,a)=0$ on the complex plain of $z$?

I have an equation $ f(z,a)=0$ where $ z$ is a complex variable $ \frac {56}{10}<z<2\pi$ and $ a$ is a real variable $ 0<a<\pi$ .

I want to plot solutions of this equation on the complex plain $ z$ as $ a$ increases from $ 0$ to $ \pi$ , and show the behavior of $ z$ by changing the color of the curve (as $ a$ increases, the curve changes from blue to red), something like this plot

enter image description here

f[z_,a_]:=9 + 4 Cos[a - (273 z)/50] - 4 Cos[a - 2 z] - 2 Cos[2 z] -   3 Cos[(173 z)/50] + Cos[4 z] - 2 Cos[(273 z)/50] -   3 Cos[(373 z)/50] - 4 Cos[a + 2 z] + 4 Cos[a + (273 z)/50] -   4 I Sin[a - (273 z)/50] + 4 I Sin[a - 2 z] - 2 I Sin[2 z] -   3 I Sin[(173 z)/50] + I Sin[4 z] - 2 I Sin[(273 z)/50] -   3 I Sin[(373 z)/50] - 4 I Sin[a + 2 z] + 4 I Sin[a + (273 z)/50];   f[z,a]==0  56/10 <z<\[Pi] 0<a<\[Pi]  

How to recreate this complex iteration

I came across this image in a book Im reading:

enter image description here

(taken from Rational Iteration Complex Analytic Dynamical Systems from Steinmetz. I hope it is okay to share it here)

and I try to recreate it in Mathematica. I tried

JuliaSetPlot[(9 + 2 z + 9 z^2)/20, z]

which gives a complete different picture.

enter image description here

I also tried to use

ColorFunction -> With[{cf = ColorData["M10DefaultFractalGradient"]}, cf@Sqrt[#3] &]

which doesn’t improve anything. So JuliaSetPlot might not be the right method. Do you have any suggestions?

EllepticPi argumet is complex. so can not plot it. How to handle this problem?

inttau[r_]=-(1/Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + r) (62.8822 +   r))/(3000 r)]) 

This is my function of r, now I integrated it w r t r

tauanalytical[r_] = Integrate[inttau[r], r] 

This is what I got after integration, I got this result.

-((2.3094 Sqrt[1. + 62.1447/r] (-31.0723 + r) EllipticPi[3.,ArcSin[0.57735 Sqrt[1. + 62.1447/r]], 1.])/Sqrt[-0.965489 + 20./r + 0.000333333 r^2]) 

later I tried to find the numerical value of;

tauanalytical[30] // N  

I got this, no the exact number , which I was expecting

-126.491 EllipticPi[3., -1.5708 + 0.153761 I, 1.] 

Here the second argument of Ellepticpi comes out to be imaginary , it should be real . Please provide necessary assistance.

Can a simulacrum be healed by any means other the ‘complex process’ outlined in the spell?

Inspired by another discussion of Simulacra – can a simulacrum actually heal via any method other than the ‘complex process’ outlined in the spell? Or is that process simply another option? What if the copied creature type has Fast Healing, or Regeneration? Do Cure Wounds spells function on the simulacra? Is the simulacrum actually even a creature?

How complex of an object can be created via the Creation spell?

Inspired by this question about creating gunpowder with the Creation spell.

My reading of the spell (and components) would indicate that the spell can only create an item (or multiple items1) but of a single "type". This comes from most of the description using singular form when referring to type.

From the spell description:

You pull wisps of shadow material from the Shadowfell to create a nonliving object of vegetable matter within range: soft goods, rope, wood, or something similar. You can also use this spell to create mineral objects such as stone, crystal, or metal. The object created must be no larger than a 5-foot cube, and the object must be of a form and material that you have seen before.

And the required components:

Components: V, S, M (a tiny piece of matter of the same type of the item you plan to create)

But the linked question tries to create gunpowder; a combination of "a fuel (charcoal), an oxidizer (saltpeter or niter), and a stabilizer (sulfur) to allow for a constant reaction". This is not only multiple types of matter, but different types; some vegetable, some mineral 2.

There is also no mention of the time being different; as in, using the shortest duration of all types used in the creation.

I would think that if the spell was to able to create mixed types it would have very different phrasing.

So is the Creation spell limited to only one type of matter at a time, or can it be a combination of matter/types?

And if it possible to create something with multiple/mixed types, how complex can the item be?

  • Can they mix different elements of the same type like an apple and an orange to create a fruit salad?
  • Can they create a leather coat with a fur lining and metal buttons? Or would they have to create each thing separately and sew them together by hand?
  • How about a metal crown, with cloth padding on the interior and encrusted with gems?
  • They can create iron, but can they also create it fused with carbon to create steel?
  • People could be considered a combination of vegetable matter and minerals. Could a character create a corpse (nonliving object)?

1 I would also think that this creates only a single item (a rope), except for minerals it states "You can also use this spell to create mineral objects…". Otherwise the description sticks to singular object.

2 Admittedly, something like this could be done with multiple castings, each creating a specific type of matter. But the point of the question was do create a complete pile of gunpowder.

LIMIT and OFFSET a complex query

I have a query generated by a ORM (Sequelize), i ran into an issue where sequelize fails, look at those issues. https://github.com/sequelize/sequelize/issues/7344

https://github.com/sequelize/sequelize/issues/12200

Postgres Query:

 SELECT "feeds"."id",     "feeds"."title",     "feeds"."likes",     "feeds"."description",     "feeds"."files",     "feeds"."allowComments",     "feeds"."readConfirmation",     "feeds"."isDraft",     "feeds"."createdAt",     "feeds"."updatedAt",     "feeds"."companyId",     "feeds"."createdById",     "reads"."id" AS "reads.id",     "reads->feeds_reads"."createdAt" AS "reads.feeds_reads.createdAt",     "reads->feeds_reads"."updatedAt" AS "reads.feeds_reads.updatedAt",     "reads->feeds_reads"."feedId" AS "reads.feeds_reads.feedId",     "reads->feeds_reads"."userId" AS "reads.feeds_reads.userId",     "createdBy"."id" AS "createdBy.id",     "createdBy"."firstName" AS "createdBy.firstName",     "createdBy"."jobTitle" AS "createdBy.jobTitle",     "createdBy"."lastName" AS "createdBy.lastName",     "createdBy"."profilePicture" AS "createdBy.profilePicture",     "bookmarks"."id" AS "bookmarks.id",     "bookmarks->feeds_bookmarks"."createdAt" AS "bookmarks.feeds_bookmarks.createdAt",     "bookmarks->feeds_bookmarks"."updatedAt" AS "bookmarks.feeds_bookmarks.updatedAt",     "bookmarks->feeds_bookmarks"."feedId" AS "bookmarks.feeds_bookmarks.feedId",     "bookmarks->feeds_bookmarks"."userId" AS "bookmarks.feeds_bookmarks.userId",     "units"."id" AS "units.id",     "units"."parentId" AS "units.parentId",     "units->feeds_units"."createdAt" AS "units.feeds_units.createdAt",     "units->feeds_units"."updatedAt" AS "units.feeds_units.updatedAt",     "units->feeds_units"."feedId" AS "units.feeds_units.feedId",     "units->feeds_units"."unitId" AS "units.feeds_units.unitId",     "units->users"."id" AS "units.users.id",     "units->users->users_units"."createdAt" AS "units.users.users_units.createdAt",     "units->users->users_units"."updatedAt" AS "units.users.users_units.updatedAt",     "units->users->users_units"."userId" AS "units.users.users_units.userId",     "units->users->users_units"."unitId" AS "units.users.users_units.unitId",     "units->descendents"."id" AS "units.descendents.id",     "units->descendents"."parentId" AS "units.descendents.parentId",     "units->descendents->unitsancestor"."unitsId" AS "units.descendents.unitsancestor.unitsId",     "units->descendents->unitsancestor"."ancestorId" AS "units.descendents.unitsancestor.ancestorId",     "units->descendents->users"."id" AS "units.descendents.users.id",     "units->descendents->users->users_units"."createdAt" AS "units.descendents.users.users_units.createdAt",     "units->descendents->users->users_units"."updatedAt" AS "units.descendents.users.users_units.updatedAt",     "units->descendents->users->users_units"."userId" AS "units.descendents.users.users_units.userId",     "units->descendents->users->users_units"."unitId" AS "units.descendents.users.users_units.unitId",     "teams"."id" AS "teams.id",     "teams->feeds_teams"."createdAt" AS "teams.feeds_teams.createdAt",     "teams->feeds_teams"."updatedAt" AS "teams.feeds_teams.updatedAt",     "teams->feeds_teams"."feedId" AS "teams.feeds_teams.feedId",     "teams->feeds_teams"."teamId" AS "teams.feeds_teams.teamId",     "teams->peoples->teams_users"."createdAt" AS "teams.peoples.teams_users.createdAt",     "teams->peoples->teams_users"."updatedAt" AS "teams.peoples.teams_users.updatedAt",     "teams->peoples->teams_users"."userId" AS "teams.peoples.teams_users.userId",     "teams->peoples->teams_users"."teamId" AS "teams.peoples.teams_users.teamId",     "comments"."text" AS "comments.text",     "comments"."id" AS "comments.id",     "comments"."likes" AS "comments.likes",     "comments"."parentId" AS "comments.parentId",     "comments"."createdById" AS "comments.createdById",     "comments"."createdAt" AS "comments.createdAt",     "comments"."updatedAt" AS "comments.updatedAt",     "comments->createdBy"."id" AS "comments.createdBy.id",     "comments->createdBy"."firstName" AS "comments.createdBy.firstName",     "comments->createdBy"."lastName" AS "comments.createdBy.lastName",     "comments->createdBy"."jobTitle" AS "comments.createdBy.jobTitle",     "comments->createdBy"."profilePicture" AS "comments.createdBy.profilePicture",     "peoples->feeds_peoples"."createdAt" AS "peoples.feeds_peoples.createdAt",     "peoples->feeds_peoples"."updatedAt" AS "peoples.feeds_peoples.updatedAt",     "peoples->feeds_peoples"."feedId" AS "peoples.feeds_peoples.feedId",     "peoples->feeds_peoples"."userId" AS "peoples.feeds_peoples.userId" FROM "feeds" AS "feeds"     LEFT OUTER JOIN (         "feeds_reads" AS "reads->feeds_reads"         INNER JOIN "users" AS "reads" ON "reads"."id" = "reads->feeds_reads"."userId"     ) ON "feeds"."id" = "reads->feeds_reads"."feedId"     LEFT OUTER JOIN "users" AS "createdBy" ON "feeds"."createdById" = "createdBy"."id"     LEFT OUTER JOIN (         "feeds_bookmarks" AS "bookmarks->feeds_bookmarks"         INNER JOIN "users" AS "bookmarks" ON "bookmarks"."id" = "bookmarks->feeds_bookmarks"."userId"     ) ON "feeds"."id" = "bookmarks->feeds_bookmarks"."feedId"     LEFT OUTER JOIN (         "feeds_units" AS "units->feeds_units"         INNER JOIN "units" AS "units" ON "units"."id" = "units->feeds_units"."unitId"     ) ON "feeds"."id" = "units->feeds_units"."feedId"     LEFT OUTER JOIN (         "users_units" AS "units->users->users_units"        LEFT OUTER JOIN "users" AS "units->users" ON "units->users"."id" = "units->users->users_units"."userId"     ) ON "units"."id" = "units->users->users_units"."unitId"     LEFT OUTER JOIN (         "unitsancestor" AS "units->descendents->unitsancestor"         LEFT OUTER JOIN "units" AS "units->descendents" ON "units->descendents"."id" = "units->descendents->unitsancestor"."unitsId"     ) ON "units"."id" = "units->descendents->unitsancestor"."ancestorId"     LEFT OUTER JOIN (         "users_units" AS "units->descendents->users->users_units"         LEFT OUTER JOIN "users" AS "units->descendents->users" ON "units->descendents->users"."id" = "units->descendents->users->users_units"."userId"     ) ON "units->descendents"."id" = "units->descendents->users->users_units"."unitId"     LEFT OUTER JOIN (         "feeds_teams" AS "teams->feeds_teams"         INNER JOIN "teams" AS "teams" ON "teams"."id" = "teams->feeds_teams"."teamId"     ) ON "feeds"."id" = "teams->feeds_teams"."feedId"     LEFT OUTER JOIN (         "teams_users" AS "teams->peoples->teams_users"         INNER JOIN "users" AS "teams->peoples" ON "teams->peoples"."id" = "teams->peoples->teams_users"."userId"     ) ON "teams"."id" = "teams->peoples->teams_users"."teamId"     LEFT OUTER JOIN "comments" AS "comments" ON "feeds"."id" = "comments"."feedId"     LEFT OUTER JOIN "users" AS "comments->createdBy" ON "comments"."createdById" = "comments->createdBy"."id"     LEFT OUTER JOIN (         "feeds_peoples" AS "peoples->feeds_peoples"         INNER JOIN "users" AS "peoples" ON "peoples"."id" = "peoples->feeds_peoples"."userId"     ) ON "feeds"."id" = "peoples->feeds_peoples"."feedId" WHERE (         "peoples"."id" = 11         OR "feeds"."createdById" = 11         OR "teams->peoples"."id" = 11         OR "units->users"."id" = 11         OR "units->descendents->users"."id" = 11     )     AND "feeds"."companyId" = 4     AND "feeds"."isDraft" = false     AND "feeds"."createdAt" < '2020-12-09 12:59:34.017 +00:00' LIMIT 20; 

Here the limit is not applying to the feeds, i want 20 feeds but it’s giving me same feed 20 times.

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!