#intersection of a step function and a ln based function
tacit yarrow
oh wow i have underestimated its complexity
dull marlin
also using approximations of the sum function can exponentiate the number of intercepts
tacit yarrow
how can you find the points where this function abruptly changes direction
like, all these spikes
dull marlin
this is one of the spikes with an accuracy of n→30
30 is one of the approximation points that doubles the number of intercepts
tacit yarrow
whats the second derivative at this point
dull marlin
you done derivatives with sums?
tacit yarrow
dont you just differentiate each term in the series and find the sum?
dull marlin
the derivative of a sum is the sum of the derivatives yes
tacit yarrow
brother im only finishing calc 2
dull marlin
so you have a progression of derivatives
it's a real fun time
I'll brb after taking the 2nd derivative of the sum
-4nsin(2nx)/π
not bad
bad
nvm it's worse
desmos took a while
the peaks are infinite at n→∞ as expected
dull marlin
how can you find the points where this function abruptly changes direction
without the slant added by the 1/6x the turning points are lie on the line y=0 and y=1 with the contact with the X axis being π/8+nπ n∈Z
then you just apply that to the 1/6x slope
bottom points lies on y=1/6x, top on y=1/6x+1
then you can redefine your bases if you want
so it remains π/8+nπ
or you can do trig to find those points
(π/8+nπ)(6/√37)
cos(arctan(1/6))→cos(arccos(6/(√(1+6²)))=6/√37
so now you have a nifty new formula for contact points with each of the lines
oh yeah peaks are 7π/8+nπ
then you can do recursion to find xs
@tacit yarrow so you see, not Bad (I would end my life if I actually had to do this)
proposal for a server challenge:
form a general formula for the intercepts of a step function with an arbitrary slope and a log based function, with workings, in the simplest possible way
I've realised that the 1/6x 'rotation' is actually a rotation and squish
yeah seems like a waste of time
dull marlin
no way in hell im solving it
I'm tired, how about calculating the intersection points of 2 lines
straight lines
tacit yarrow
keke
dull marlin
oh I've had an idea for the step function thing
yeah it would make it a whole lot easier
it's very beautiful
doesn't fit in this margin
only thing left is me dying and some bloke discovering it 400 years later
did I ever have a strategy? who knows
anyways my actual idea was to define it as a piecewise function, then you can solve for the points geometrically
seeing as that piecewise function is what the Fourier transform is tending to
why not skip to the chase and just work with the piecewise function
also works much better under the rotation that I'm actually trying to apply
rather than needing scaling on the X component
still needs W but a lot simpler
I should sleep more often, maybe then it wouldn't take me 30 minutes of waffle to come up with an obvious solution
verbal tartan
the derivative of a sum is the sum of the derivatives yes
That's not always the case
U need to be sure that the series is unif convergent
dull marlin
dull marlin
U need to be sure that the series is unif convergent
and Majorant, but that doesn't matter here as I wasn't taking the derivative of the infinite series
I was taking the derivative of an approximation
verbal tartan
I was taking the derivative of an approximation
Whatever u say