Analysis and Calculus

Let's fistly consider what is sectorsgenta then the hypothesis

This is a proof of a statement in Stillwell's "Reverse Mathematics", p.77: My question is, how do we know that the sequence [math]r_1, r_2, ...[/math] converges?

Check out this fortran program that I wrote, lnln-pole-int, that computes ⌠ ln(x+a)ln(x+c) ⎮ ────────────── dx ⌡ x+f for complex-valued parameters a, c, and f. Prompted for an indefinite integral of this type, math software tools spit out an expression that will be invalid for some integral parameters. Prompted for a definite integral of this type, at least some math software tools fall back on numerical integration when a, c, or f are complex, probably because an algorithm for analytical evaluation is not widely available. I aim to contribute toward filling this void with the linked program. One limitation that I know about is that it fails to compute integrals of t…

a = (Pi - 2)/4 + sin(a) a is unknown angle(rad) in unit circle a/2 is area of sector of angle a (Pi-2)/8 is area of segment of angle a

Hi, I've seen a video that reminded me of the definition of continuity (that you can find here). Somehow it seemed it was not what my "humble intuition" (sorry for that) excepted (although I remember it from school).. (no I don't confuse continuity with derivability ok). My intuition is "a function that you can draw without lifting the pen".. So I thought maybe I can find a function that is obviously not drawable without lifting the pen, but still match the classic definition for continuity, and I found a very simple one : Take any real number x, write it in decimal, and then recompute it in base 8 So, in "math" (I try my best) term : x = sum(dn * 10^n] f(x) = sum(dn *…

The class of finite unions of arbitrary (also unbounded) intervals is an algebra on Ω=R (but is not a σ -algebra). How to prove it? I know an algebra is a class of sets A⊂2Ω which holds the following conditions 1)Ω∈A 2)A is closed under complements. 3)A is closed under unions. σ -algebra fulfills all these three conditions. But in addition, it is closed under countable unions. Here countable means finite or countably infinite. I also know that R=(−∞,+∞) which is uncountably infinite unions of arbitrary (also unbounded) intervals.(is that correct? 🤔) Now, with these information available to me,, how can I answer this …

The text I'm reading proves uncountability of the Cantor's set by showing that the Cantor's function is a surjective map from the Cantor's set onto [0, 1]. I think that it can be shown by direct use of the Cantor's diagonal argument for the Cantor's set, i.e., without use of the Cantor's function. Am I right or am I missing something?

I've started studying measure theory by this book: Measure, Integration & Real Analysis (Graduate Texts in Mathematics) (It is free on Kindle, btw.) It pretty much begins with proving that it is impossible to simply extend the concept of length from a real interval to an arbitrary set of real numbers. The proof is a bit formal, and I want to make it more intuitive while still rigorous. Would like to hear if the following description is good and if it can be improved. So, we want to have a real-valued, non-negative function 'length' defined for any set of real numbers with the following desired properties: a) it gives length of b-a for an interval [a, b]…

PROBLEM STATEMENT: What is the least number of smaller circles that can be fitted inside a mother circle under the following conditions: 1. The smaller circles cannot intersect or be contained inside any other circle besides the mother circle. 2. The areas of the smaller circles must be N (N<1) times any existing circle inside the mother circle. 3. The system must contain the maximum number of circles of the same area as possible. Spoiler Spoiler MUNIM'S PROBLEM.docx

Hello everyone, For several days I have been thinking about the fractions like 10/3 and 100/16. Calculations show that when you do these problems, usually you would think about the recurring numbers at the end. However if we do the reverse action, the answer won't really equal to the first number we took the result times from the Second number. This is due to remain from the recurring numbers being taken away from the first number. Let's take for example this problem: 10 / 3 (to 1 significant figure) From this question we see that this is going to be a recurring number as any number that isn't a multiple of 3 - will definitely have recurring end. However th…

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Hello everyone, I’m currently studying infinite series in my calculus course and have encountered some concepts that I find quite challenging, particularly the criteria for convergence. I’ve read about the Comparison Test, Ratio Test, and Root Test, but I’m struggling to apply these tests effectively in different scenarios. To give you some context, I’ve worked through a few examples, but I often find myself unsure about which test to use or how to justify my choice. For instance, when faced with a series like ∑n=1∞n22n\sum_{n=1}^{\infty} \frac{n^2}{2^n}∑n=1∞2nn2, I initially thought the Ratio Test would be appropriate, but I’m not confident in my analysis. …

{(1,1), (2,0.1), (3,0.1), (4,0.01), (5,0.01)} assuming that this pattern continues on forever?

I programmed a program to test the Collatz hypothesis. For testing, I plugged the first 17 million numbers into the program, which showed no unexpected results. The Collatz hypothesis worked in every case. Many matematicians are into Collatz Conjecture so I have to test program on much bigger numbers. I will continue my research on much larger numbers.

While attempting to solve the differential equation: [math]\dfrac{dr'}{dr} = \dfrac{1}{\sqrt{1 - \dfrac{2GM}{c^2 r}}}[/math] expressing [math]r[/math] in terms of [math]r'[/math], I encountered a novel family of transcendental functions called "Leal-functions". These functions are similar to the Lambert W function (the function [math]W(x)[/math] that solves [math]W(x)e^{W(x)} = x[/math]), but (apparently) can't be derived from it. The link to the full article about these functions: https://www.sciencedirect.com/science/article/pii/S2405844020322611 The link to the section that defines these functions: https://www.sciencedirect.com/science/…

If I do the equation 1*0 the answer is 1*0=0 but If I have £1 in my hand and do the equation £1*0 the answer is £1*0=£1 because I still have the £1 in my hand . ????????????????????????????????????????????

Hello! I'm building a rebase algorithm that I need some help with. These are the variables, constants, and constraints (the values themselves are just examples): Set of variables 1: a1 = 1934464428151493937044 b = 54802476401439357 c = 1e18 result1 = ? Set of variables 2: a2 = 1837741206733939183658 b = 54802476401439357 c = 1e18 result2 = ? Final invariant --> result1 == result2 As you can see b and c are constants and the only variables shared between sets, so you can't use a1 during the operations of Set 2 and vice versa (a1 and a2 might fluctuate in value, but not b and c). Also, r…

I am in the middle of constructing a theory that is specifically focused on iterations. The symbol for iterations is uppercase phi. It is like a sum but instead of adding them together, you take iterations of the numbers. So, you use n in the equation, and if n is equal to let's say, 5000 you replace n with 5000 in the next iteration. But the beauty of this theory is that you do not have to take an infinite number of iterations. you just have to reason your way out of an iteration. Let me give you an example. Say we are taking an iteration of x to the power of n over n to the power of n. We can reason that n to the power of n will reach infinity, so we have something over…

Hello community, I have a question regarding the calculation of Value Added Tax (VAT), and I'm reaching out to get some insights from fellow members. Suppose I've received a net payment of 200 euros, and I need to calculate the gross amount for VAT purposes with a VAT rate of 25% in my region. I stumbled upon a formula that suggests multiplying the net amount by 1 plus the VAT percentage, which in this case is 1.25. According to the formula, the gross amount should be 250 euros. However, during my attempt to verify this calculation: If 250 euros represents 100%, 1% would be 2.50 euros. 25% would be 62.50 euros. Subtracting 62.50 euros fr…

Is f(x) used only with functions?

What is 'my' in calculus?

What is phi used for more than the golden ratio? What is the golden ratio?

I do not understand difference between a derative and differentiation and what each means. I have searched the web but not got wiser.

It is often written dy/dx, what is it simply explained?