How does Mathematica obtain this result?

FullSimplify[  Sqrt[2 \[Pi]] InverseFourierTransform[1/(x^2 - a^2), x, p],   Element[a, Reals]] 

Gives the output

-((\[Pi] Sign[p] Sin[a p])/a) 


$ \int_{-\infty }^{\infty } \frac{e^{-i k x}}{x^2-a^2} \, dx$ is not a defined integration. Mathematica also returns undefined as answer if you compute it.

So, I am trying to understand how does Mathematica calculates that Fourier transform of the non-integrable function.

How to obtain eigenvalues from an ODE’s general solution?

For this second-order linear ODE (eigenvalue e) $ $ y”(x) – 2g\, y'(x) + [-e + g^2 – (\frac{b^2x^2}{2}+a)^2 + b^2x]\, y(x)=0$ $ I know the correct analytical general solution form with two undetermined constants c1,c2 (verified in the code below where y1,y2 is given) $ $ y=c1\,y1(x)+c2\,y2(x)$ $ I want to impose the boundary condition $ y(\pm5)=0$ in order to get the eigenvalue $ \lambda$ . (Actually $ y(\pm\infty)=0$ , but not much numerical difference, I suppose.) So I put c1=1 and tried FindRoot e and c2 together, but it didn’t work well. So I was wondering if I need to tune any option or some other methods?
I also add a direct numerical solution as a reference.

F := (D[#, {x, 2}] -       2 g D[#, x] + (-e + g^2 - (b^2 x^2/2 + a)^2 + b^2 x) #) &; y1[x_] := E^(-(1/6) b^2 x^3 - x a + x g) HeunT[e, 0, -2 a, 0, -b^2, x]; y2[x_] := E^(-(1/6) b^2 x^3 - x a + x g) HeunT[e, 0, -2 a, 0, -b^2, x]; y[x_] := c1 y1[x] + c2 y2[x]; F[y[x]] == 0 // FullSimplify  g = 0.2; a = 0.3 - I 0.3; b = 1; FindRoot[0 ==     y[x] /. {{c1 -> 1, x -> 5}, {c1 -> 1, x -> -5}}, {{e, -0.3}, {c2,     1}}, WorkingPrecision -> 55]  FF = F /. e -> 0; {vals, funs} = NDEigensystem[FF[yy[x]], yy[x], {x, -5, 5}, 5]; vals Plot[Evaluate@Abs@funs, {x, -5, 5}, PlotRange -> All,   PlotLegends -> Automatic] 

What method can be used to obtain a fifth and sixth “bonus feat”?

The goal is to obtain a total of six feats that clearly have the label of "bonus feat" at first level. The character class is Erudite, which has two first level bonus feats. Taking two flaws grants two more bonus feats.

How can I obtain the last two bonus feats at first level? Is there a race or template which has two features or abilities using the words bonus feat?

Not interested in TOs (theoretical optimizations), or methods which require helper NPCs, or wealth beyond a starting Erudite’s Wealth by Level.

Does an Arcane Hierophant without animal companion obtain a companion familiar?

Upon entering the Arcane Hierophant class you obtain a “Companion Familiar”, the feature description says you have to dismiss your familiar but you may retain your animal companion if you have one.

Companion Familiar: Upon becoming an arcane hierophant, you must dismiss your familiar, if you have one (You do not risk losing XP for doing so.) You may retain any one animal companion you already possess. You add your arcane hierophant class level to your druid or ranger level for purposes of determining your animal companion’s bonus Hit Dice, natural armor adjustment, and Strength/Dexterity adjustment (see the sidebar The Druid’s Animal Companion, page 36 of the Player’s Handbook). For example, a character who is a 4th-level druid/3rd-level wizard/4th-level arcane hierophant has the animal companion of an 8th-level druid (+4 bonus HD, +4 natural armor, and +2 Strength/Dexterity adjustment, or an animal companion chosen from the 4th-level or 7th-level lists). In addition, your animal companion (if any) gains many of the abilities that a familiar would normally possess. You add your arcane hierophant class level to your arcane spellcasting class level, and determine the Intelligence bonus and special abilities of your animal companion accordingly (see the sidebar Familiars, page 53 of the Player’s Handbook). For example, a 4th-level druid/3rd-level wizard/4th-level arcane hierophant has a familiar companion equal to that of a 7th-level wizard and 8th-level druid (Intelligence 9, alertness, improved evasion, share spells, empathic link, deliver touch spells, speak with master, speak with animals of its kind, devotion). The Hit Dice, hit points, attack bonus, saving throws, feats, and skills of the familiar companion are determined as normal for an animal companion. Due to the familiar companion’s unusual Intelligence score, it may very well have more skill points than other animals of its kind. The familiar companion is a magical beast (augmented animal), but you can bestow harmless spells on your familiar companion as if it were an animal instead of a magical beast. If your familiar companion is killed or dismissed, you do not lose XP. You can summon a new familiar companion by performing a ceremony requiring 24 hours of uninterrupted prayer.

It is not clear to me what happens when you become an AH using classes that don’t have the Animal Companion feature (like a Cleric/Scout/Wizard, for example). Do you still get a companion familiar?

In Ghosts of Saltmarsh, how do the players obtain a ship?

In Ghosts of Saltmarsh, there is a lot of information about ships, naval combat, and the sea in general. However, I can’t find any information on how to get a ship in the first place. I want to run this adventure compilation, and I don’t want to have to sideline the game for 30 minutes to find out how to get a ship in the first place. Could someone please explain how my players can get a ship?

How does a web browser obtain the public key of a certificate author?

As far as I know, when a web browser verifies the signature in a SSL certificate, it uses the public key of the certificate author (CA) which has signed the certificate. Does the web browser contacts the CA to obtain its public key? If so, how does it happen? Does the IP address of the CA is contained in the certificate?

Are the following two ways to obtain server certificates for web servers to host web applications? says for running a flask web application with gunicorn and nginx with https:

Certbot provides a variety of ways to obtain SSL certificates through plugins. The Nginx plugin will take care of reconfiguring Nginx and reloading the config whenever necessary. To use this plugin, type the following:

sudo certbot --nginx -d your_domain -d www.your_domain 

This runs certbot with the –nginx plugin, using -d to specify the names we’d like the certificate to be valid for. says that for running an web application with https:

On Ubuntu the standard mechanism would be:**

  • dotnet dev-certs https -v to generate a self-signed cert
  • convert the generated cert in ~/.dotnet/corefx/cryptography/x509stores/my from pfx to pem using openssl pkcs12 -in <certname>.pfx -nokeys -out localhost.crt -nodes
  • copy localhost.crt to /usr/local/share/ca-certificates
  • trust the certificate using sudo update-ca-certificates
  • verify if the cert is copied to /etc/ssl/certs/localhost.pem (extension changes)
  • verify if it’s trusted using openssl verify localhost.crt

I was wondering if the above two ways are to achieve the same goal as obtaining a server certificate for a web server to host a web application?

Specifically, do the single certbot command and the dotnet dev-certs https and openssl commands do the same thing?

Are the two ways working directly on web servers, instead of web applications?

Can the two ways replace each other in their use case scenarios? (suppose dotnet dev-certs https would work on Ubuntu, for simplifying my questions.)

I am new to digital certificate, and have seen the above two approaches for different web application frameworks, and am trying to understand the commonality.


What happens if you obtain a cursed item while already attuned to your maximum number of attuned items (three)?

Reference: daze413’s answer to “As a monk, can I use Deflect Missiles to protect my ally?”

Scenario: Monk already has his three attuned magic items. Monk picks up a shield of missile attraction, which is a cursed item, and uses it for awhile. This should cause him to pick up the curse. Monk discards the shield of missile attraction.

If I read correctly, most magic items do not function unless attuned, and using a cursed item causes the attunement to pick up automatically (otherwise what’s the point)?

But now we’re trying to exceed the attunement limit by using a cursed item. Oops. Contradiction. Now what?