Repository navigation
A fractional power is inverted where its values lie - #2183
Conversation
The principal power z^p is e^(p ln z), so its argument is p arg(z). For 0 < p < 1 its values
lie in the sector -p pi < arg <= p pi, with 0, and for -1 < p < 0 in p pi <= arg < -p pi,
without 0. x^p = a was solved to a^(1/p) with no condition, which is no root where a is
outside the sector: sqrt(x) = a was { a ^ 2 }, and at a = -1 that is 1, whose square root is
not -1. The root now carries the sector, written with arg, as abs's inversion carries its own.
For |p| > 1 the sector is the whole plane and nothing is stated; a symbolic exponent has no
sector to state. A number on the other side is decided as before.
Providedf conjoined a nested condition without looking, so a system's rows carried the same
condition once a substitution, five times over for sqrt(y) - x, y - a. A conjunct already
stated is dropped now, and the conjunction is rebuilt only where something repeats.
That system was declined since #2123, its -sqrt(a) row failing at a = 137/100; each row now
says where it holds, and the test checks the rows by value at four points.
Part of #212.
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Codecov Report❌ Patch coverage is Additional details and impacted files@@ Coverage Diff @@
## master #2183 +/- ##
=======================================
Coverage 86.45% 86.46%
=======================================
Files 297 297
Lines 65406 65448 +42
Branches 19198 19215 +17
=======================================
+ Hits 56547 56588 +41
+ Misses 4804 4803 -1
- Partials 4055 4057 +2
Flags with carried forward coverage won't be shown. Click here to find out more.
🚀 New features to boost your workflow:
|
|
What if there was a |
(sqrt(x) = a) provided a in RR was solved to a ^ 2 provided a in RR and the sector of the
complex plane, which for a real a is only the half-line a >= 0. A provided equation is now
solved under TreeAnalyzer.Assuming of its condition, as a provided expression is already
differentiated: its roots are filtered by the condition anyway, so what holds under it can be
stated by it. The power's inversion asks what is known. A value known to lie on the half-line
the power reaches, a >= 0 for p > 0 and a > 0 for p < 0, needs no condition. One known to lie
off it has no root, and one known real, by its codomain, by membership of RR or by being ordered
against something, has the half-line for its condition in place of the sector.
(sqrt(x) = a) provided a in RR a ^ 2 provided a in RR and a >= 0
(sqrt(x) = a) provided a > 0 a ^ 2 provided a > 0
(sqrt(x) = a) provided a < 0 { }
Part of #212.
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
|
It was conjoined but not used:
A real value has argument 🤖 Generated with Claude Code |
|
Some solutions like |
|
Yes: 🤖 Generated with Claude Code |
|
Merged in #2186: 🤖 Generated with Claude Code |
On #212,
sqrt(y) = xwas the open question: the system's family needed a condition, and so did the single equation.sqrt(x) = awas{ a ^ 2 }, which is no root ata = -1, sincesqrt((-1)^2)is1. Of the three conditions offered there, Happypig375's answer opened the third, anargnode, which #2181 adds. This uses it.The power's inversion states the power's range. The principal power
z^pise^(p ln z), so its argument isp arg(z):0 < p < 1its values lie in the sector-p pi < arg <= p pi, with0;-1 < p < 0they lie inp pi <= arg < -p pi, without0;|p| > 1the sector is the whole plane, andv^(1/p)is a preimage of everyv.So
x^p = ais nowa^(1/p) provided ais in the sector, asabs's inversion already carriesr in RR and a >= 0. A symbolic exponent has no sector to state and is left as it was. A number on the other side is decided as before:sqrt(x) = -1is{ },sqrt(x) = iis{ -1 }.A condition is not stated twice.
Providedfconjoined a nested condition without looking, so the system's rows came back with the same condition five times over, one for each substitution. A conjunct that is already there is now dropped, and the conjunction is rebuilt only where something repeats, so every other condition keeps its shape.sqrt(x) = a, solved forx{ a ^ 2 }{ a ^ 2 provided a = 0 or -1/2 * pi < arg(a) and arg(a) <= 1/2 * pi }x^(1/3) = a{ a ^ 3 }{ a ^ 3 provided a = 0 or -1/3 * pi < arg(a) and arg(a) <= 1/3 * pi }x^(-1/2) = a{ a ^ (-2) }{ a ^ (-2) provided not a = 0 and -1/2 * pi <= arg(a) and arg(a) < 1/2 * pi }sqrt(y) - x,y - a, solved forx, y[[sqrt(a), (sqrt(a)) ^ 2], [-sqrt(a), (-sqrt(a)) ^ 2]]provided C(row's x),Cthe sector aboveThe system was declined on master since #2123, because the
-sqrt(a)row fails ata = 137/100. Each row now says where it holds: the first whereverais, and the second only ata = 0.SpuriousSystemRowTestchecks that at four values ofaevery row whose condition holds solves both equations, and that one of them does.Still open on #212. The family of
sqrt(y) - xalone, inxandy, is still declined. Its check putst^2back and needssqrt(t^2) - tto be0, which it is only under the condition, andSimplifydoes not use a condition to simplify. Integer-parameter families also remain open.Measured (on the branch before its rebase onto #2181's merge, an identical tree)
Part of #212.
🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura