mathematical functions

[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.)

“ 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

8

Taylor Series : an infinite sum giving the value of a function f(x) in the neighbourhood of a point a in terms of the derivatives of the function evaluated at a.

Not all functions can be expressed as a taylor series expansion. The function must be infinitely differentiable at the point of consideration.

The ideal example would be the exponential function. How many ever times you differentiate it, you get the same function back.

The following is the exponential function ex (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).

Notice how the red closely resembles the exponential function as more terms are added.

A similar could be done for the sine,cosine,arcsine,logarithmic, etc.

The tale of how the coefficients came to be in the taylor series expansion was the topic of discussion of today, hope you guys enjoyed it.

Have a great day!

Mathematics

Approximating a square wave using the Fourier series

Cont’d from “Fourier series

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 wave:

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 coefficient.

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

therefore,

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 7th term.

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.

Twenty-Four -- Ignis Scientia x Prompto Argentum

WRITING SHUFFLE PROMPTS
Song Selected: Twenty-Four by Switchfoot

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 valuable number.

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, don’t ya?”

“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.”

okay what the fuck

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”.