[10:15PM] i’m finally on vacation but that only means i have more time available to focus on the topics i find more difficult, so i can master them while i can – for me, of course, it means functions! after three days i have finally figured out how to do the past papers’ questions, and, fortunately, how to work with my calculator. yay?
(my head hurts a bit now, though. i’m sure i’ll have dreams about functions tonight.)
I saw someone working out with Battle Ropes the other day and this wonderful pattern emerged was absolutely fascinating. Of course, the waveforms are not purely sinusoidal but it helps us to understand why you see such patterns
In mathematics, the mountain climbing problem is a problem of finding the conditions that two function forming profiles of a two-dimensional mountain must satisfy, so that two climbers can start on the bottom on the opposite sides of the mountain and coordinate their movements to reach to the top while always staying at the same height.
“ Via Wikipedia.
Finally! The long-time-coming Fourier approximation of a square wave, as promised in this post.
Let’s start by considering a rectangular square pulse, which we’ll call the function f(x), and define such that
i.e. a square pulse of width π centred at the origin. We’ll let the periodicity be such that f(x) = f(x + 4π), meaning the function has a period of 4π. Graphically this will be a square
To analyse this using the Fourier series we consider only one square pulse:
First, we’ll evaluate the a0 coefficient, the average of the function. We can find this either by inspecting the graph or mathematically. generally, for a periodic function f(x) centred at the origin with period P, a0 is expressed as
In this case, P = 4π, and so
Now, we shall evaluate the an term. There’s a trick we can use here regarding a function’s odd or even properties:
If the function is even, the an term will contain the integral of an even term so it will contribute the Fourier series. However, the bn term will contain an integral of an odd function, which is zero. Therefore, in this case, we can ignore the bn term:
Conversely, if the function were odd, the an term would contain the integral of an odd function and the bn term would contain the integral of an even function. Hence, only the bn term would contribute and we could ignore the an term.
Now, since the function is symmetric in the y-axis it is even and we should analyse the an
Using piecewise analysis we can represent the function as the sum of the individual integrals of each respective parts of the function.
Since sin(−x) = − sin(x),
In turn, it can also be inferred that the integral of an odd function centred at the origin between symmetric limits ±ℓ is twice the integral of the same function between half of those limits.
Generally, this is,
Now we can obtain a final value for an. We can express an in terms of the sinc function, defined such that
Thus, we can substitute these expressions into the Fourier series.
So what does this function look like? Let’s approximate the function by taking the summation to the
Clearly we can see that this is not a completely perfect approximation. As we take it to higher orders, we’d expect the function to look more like the original piece-wise function.
Throwback to when I was working on Riemann hypothesis. This week has been somewhat less productive than usual as I’ve been dealing with a lot. So shout out to all the studyblrs who struggle with mental illness, please know that you’re absolutely welcome to message/ask me about anything including things you’re struggling with, whether it be managing sleep, anxiety or depression. Sending love to you all, keep going, I’m proud of you.
Whoo! My first time writing some Promnis! c: It was fun to write, though it was a hard song to try pulling inspiration from. A good challenge to keep going at it though, and I’d definitely want to explore Promnis more in the future too. vuv Thanks for the request, Tea anon!
Word Count: 647 Character(s): Ignis Scientia/Prompto Argentum (Promnis) Warnings: None
Twenty-four. A number that Ignis could associate with
everything. Despite it seeming like an odd number, it truly was a pleasant one.
Twenty-four hours in a day, meaning twenty-four hours to plan things to do.
Twenty-four in two dozen, meaning he could make about twenty-four pastries for
six to go to each of his companions and himself. Twenty-four carats to make up
100% pure gold. Twenty-four is the atomic number for chromium. The number
itself was a feat, for there was plenty of reasons why twenty-four was such a
So when Prompto asked Ignis to pick a number, Ignis’s
instantaneous reaction was to say the number. “Twenty-four.”
“Really? Twenty-four again?” Prompto couldn’t help but
laugh as he made a mental note of the number. “You really like that number,
“Of course. Twenty-four is a very valuable number. Unlike
its opposite, the number forty-two.”
“Like 4/20, yeah~?”
Ignis scoffed and rolled his eyes as he ran the back of
his knife over the cutting board to slide the carrots and celery into the pot.
Noctis will eat those vegetables whether he’d like it or not. “Twenty-four is a
far classier number, and it happens to be a favorite of mine.”
“Didn’t know that you could have favorite numbers.”
Prompto chuckled and rested his arms behind his head as he glanced up at the
sky. “Why twenty-four though? Like, I guess it could be fun to write, but what
makes a number special anyways? Figured it’d just mean more math!”
Ignis’s smile spread over his face as he shook his head.
The number itself didn’t really have a reason to be valuable itself, but
Prompto did make a good point. What made a number so special? It wasn’t because
Ignis enjoyed writing it, or the number was convenient. Math was surely not a
reason either, despite Ignis being content in doing basic mathematic functions.
He merely hummed to himself as he gently tilted his head while he cleaned his
cooking knife and slide it into its sheath to put it away. “There are plenty of
reasons to enjoy the number twenty-four.”
“Plenty of reasons, huh? What’s your favorite reason
though?” Prompto leaned forward with his elbows propping his chin up on the
table, looking at Ignis with a genuinely curious expression.
Ignis’s expression only softened as his green gaze met
the blond’s curious blue eyes. “My favorite reason? I suppose my favorite
reason for liking the number is because you have approximately twenty-four
freckles on your face. And each one of them light up when you smile. But do you
know what makes each one of those freckles even more special?” Prompto’s
reddening expression merely stared at Ignis with a dumbfounded expression. “Your
freckles remind me of at least twenty-four reasons as to why I love you.”
The look over Prompto’s face began to shift from
dumbfounded to overwhelmed with embarrassment and happiness. It was hard to
process all Ignis’s words at once! The blond-haired photographer wasn’t
expecting such an answer from Ignis, immediately growing flustered as he
nervously chuckled and rubbed the back of his head. The toothy grin over his
face was hard to hide as he found it harder and harder to keep his composure
after Ignis had said such a charming thing.
“See? They’re illuminating your face right now.”
“God, Iggy. You’re so embarrassing…! Now I’m going to be
self-conscious about me and my twenty-four freckles!” Prompto had to cover his
face with his hands, only causing Ignis to chuckle in response and lean over to
kiss the top of the photographer’s forehead. “I have twenty-four reasons why I
love you too, you know.”
“Is that so? Perhaps after we eat, you can elaborate them
all to me?”
Prompto gave a small nod as he leaned up and placed a
small kiss on Ignis’s lips. “Yeah. I’d like that a lot.”
this is not gonna make much sense to a lot of you but i‘m just so fucking angry. differential equations, okay? it‘s equations involving derivatives and you‘re looking for functions that satisfy the equations.
wHo ThE fUcK thought it was a good idea to just leave out the fucking variables
dr/dφ=r(1/sinφ + cotφ)
looks fine right? WRONG BECAUSE IT‘S NOT WHAT IT SAYS INSTEAD IT IS
r‘(φ)=r(φ)*(1/sinφ + cotφ)
I WAS READY TO ABANDON THIS TOPIC ALTOGETHER BECAUSE I HAD NO IDEA HOW TO SOLVE THIS TURNS OUT THE FUCKING NOTATION IS JUST TOTAL BULLSHIT
just out of spite i‘m always gonna write down the variables fucking physicists can fight me
The term “function” first appeared in 1692 in a mathematical article in the “Acta Eruditorem” to denote various tasks that a straight line may accomplish with respect to a curve, such as forming a chord, tangent, or normal. The article was signed O.V.E. but is attributed to Gottfried von Leibniz. In another article from 1964, Leibniz gave the term “function” a more specific meaning by letting it denote the slope of a curve, a definition that has very little in common with the present day mathematical definition of function.
The Swiss mathematician Leonhard Euler in 1749 defined a function as a variable quantity that is dependent upon another quantity, thereby approaching today’s definition. Euler’s definition was challenged when the French physicist and mathematician Joseph Fourier in 1822 presented his work on heat flow. For his investigations, Fourier introduced series with sines and cosines as terms, which led to the concept that a given representation of a function may be valid for only a certain range of values.
Based on Fourier’s investigations, Lejeune Dirichlet in 1837 proposed that, from the mathematical point of view, a function is a correspondence that “assigns a unique value of the dependent variable to every permitted value of an independent value”.