Questions tagged [derivation-of-formulae]

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Derivation of an epitrochoid

I was working on an assignment and had a good idea to have a water jet model an epitrochoid to create a fountain. While the overall idea was approved by my teachers, I was told that I need to show a ...
TheShadowSider101's user avatar
-1 votes
1 answer
63 views

What is the formula for converting an improper fraction to a mixed number [closed]

There are methods for converting improper fractions to mixed numbers, but I am interested on finding a formula to which I can input the numerator and denominator of an improper fraction and get an ...
Aceffad's user avatar
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1 answer
52 views

What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?

Let all untyped variables be natural numbers. Formula? Given $x \geq 1$, $0 \leq r \lt d$ there are two cases to handle: $x \lt r$ and $x \geq r$. What is the 2-case closed form for $\sum_{i = 1}^{x} \...
Daniel Donnelly's user avatar
1 vote
1 answer
95 views

What are the formulas for the circumradius, surface area and volume of each Kepler-Poinsot polyhderon based on the length of the entire edge?

Every formula I've found online is based on only a part of the total edge. If anyone knows the formulas based on each red edge below I would greatly appreciate it. A derivation of those formulas ...
John's user avatar
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56 views

Is there a formula for finding the area of a regular polygram based on its edge length? [duplicate]

i.e. the red edges in each polygram below. Bonus: If there is a formula, how is it derived?
John's user avatar
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40 views

show two surface area formulas are equivalent

According to Wikipedia, for a uniform n-gonal antiprism with edge length $a$, SA = $\frac{n}{2} \left( \cot\frac{\pi}{n} + \sqrt{3} \right) a^2$ . This formula seems unnecessarily complex to me, as I ...
John's user avatar
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1 answer
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Is it possible to define an implicit function for the Kth N such that N(N+1)/2 is a perfect square?

The questions asks: Define a formula to yield the Kth N for which there exists an integer X less than or equal to N for which the sum of the integers from 0 to X (inclusive) is equal to the sum of the ...
BlueInfinite1729's user avatar
1 vote
1 answer
35 views

Correct a definite integral combining two Hermite polynomials and a Gauss function in two reference books

In the celebrated 8th edition (the latest) of I.S. Gradshteyn and I.M. Ryzhik, revised by D. Zwilinger and D. Moll (2015), on page 811 one reads formula 7.374.4 reproduced below: $$\int_{-\infty}^\...
user12030145's user avatar
  • 1,041
1 vote
2 answers
150 views

deriving antiprism formulas

According to Wikipedia: Let $a$ be the edge-length of a uniform $n$-gonal antiprism; then the volume is: $$V = \frac{n \sqrt{4\cos^2\frac{\pi}{2n}-1}\sin \frac{3\pi}{2n} }{12\sin^2\frac{\pi}{n}}~a^3$$ ...
John's user avatar
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How do I arrive at the formula for total clearance? (pharmacokinetics)

In Rang & Dale (10 ed) on page 152, the formula for total clearance is given by: $$Cl_{tot}=\frac{Q}{AUC_{0-\infty}}$$ $CL_{tot}=$total clearance Q=initial dose given $AUC_{0-\infty}$=Area under ...
Magnus's user avatar
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Unsure how kullback-leibler was invoked in derivation.

I am trying to understand how authors of the DDPM paper made the leap from equation 21 to equation 22 in appendix A. Specifically, it is not clear to me how they managed to convert the first term of ...
Spacey's user avatar
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2 votes
1 answer
73 views

Showing $r^2\left(\frac\theta2-\sin\frac\theta2\cos\frac\theta2\right)=\frac{r^2}{2}(\theta-\sin\theta)$

I derived the formula for the area of a circular segment as follows: However, the formula I find online when using radians is: A = $\frac{r^2}{2}\bigl(\theta - \sin(\theta)\bigr)$. Is there a ...
John's user avatar
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1 vote
1 answer
64 views

In principle, can every second order ODE be solved?

In studying differential equations, I came across a formula to solve the equation: $$(A\cdot y)(x)+(B\cdot d_xy)(x)+C(x)=0$$ Where $\{A,B,C,y\}$ are all elements in an algebra of continuous functions, ...
Simón Flavio Ibañez's user avatar
4 votes
1 answer
220 views

What is this boolean pattern?

I am trying to come up with an invertable 1 or 2 dimensional transformation that is not unlike the fast fourier transform. Given a prior sequence $A_0 .. A_{n-1}$, we can generate a new sequence of ...
micsthepick's user avatar
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1 answer
35 views

Conceptual doubt regarding the derivation of the surface area of a sphere

I tried doing this and didnt account for the width of each strip so I ended up with the formula: I ended up with 4πR instead of 4πR^2 My thought was that since integrals are summing over all the 1 ...
Shiven Pradeep's user avatar
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0 answers
38 views

Deriving Newton’s gravitational potential through Poisson’s equation for gravitational field.

I am trying to derive Newton’s gravitational potential $\phi_N = -\frac{GM}{r}$ from Poisson’s equation $\Delta \phi_N = 4\pi G\rho$, where G is the gravitational constant, M is the mass of which the ...
Sunny's user avatar
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0 answers
48 views

Integral as a product of integrals

I have conjectured that for all functions $$ g(t) = \sum_{n=0}^{\infty}\frac{a_n}{n!}t^n, $$ it holds that $$ \int_0^{\infty}\frac{(g(tx)-g(0))^j}{j!}e^{-t} dt = \frac{1}{j!a_0a_1...a_j}\prod_{r=1}^j(\...
Artur Wiadrowski's user avatar
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0 answers
16 views

Problem with deriving a standard deviation function

I have a function which depicts errors for a specific parameter. The function for the standard deviation of this parameter is also given, but I am having problems deriving it from the error function. ...
j.hed's user avatar
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1 answer
29 views

Trig derivation of $\sin(\alpha_1\cos(\omega t)$ to $\alpha_1\cos(\omega t)$ for $a1\gg 1$?

Hi im trying to rewrite an equation but i can't see how the answer given is obtained which annoys me. The answer states that $\sin(\alpha_1 \cdot \cos(\omega \cdot t)=\alpha_1 \cdot \cos(\omega \cdot ...
Aleksander Lauridsen's user avatar
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0 answers
65 views

Fourier Transform In Music

I am a senior in highschool and am currently trying to conduct an exploration on Fourier Analysis, specifically using the Discrete Fourier Transform to analyze a chord played on my piano. Basically I'...
Ralph Khouri's user avatar
4 votes
1 answer
167 views

How to derive A001187 formula combinatorically

I’ve learned recently that the formula for A001187 is defined recursively. The formula is: $$n2^{\binom n2}=\sum_{k=0}^{n} \binom{n}{k}kd_k2^{\binom{n-k}{2}}$$ Where $d_k$ is the number of connected ...
badatmathman's user avatar
2 votes
0 answers
125 views

The ultimate polylogarithm ladder

As you can see, here I performed a derivation of a quite simple formula, not much differing from the standard integral representation of the Polylogarithm. Seeking to make it fancier, I arrived at ...
Artur Wiadrowski's user avatar
1 vote
1 answer
51 views

All twin prime averages in the range $[9, 119]$ are of the form $6(5[3(z-x)]_{\pmod 7} + x)$ for some $x \in \{0,2,3\}, z \in \{0,2,3,4,5\}$.

Question. Can we come up with a general formula $f(x_5, x_7, x_{11}, \dots, x_{p_n})$ such that each twin prime average $a \in [p_n + 2, p_{n+1}^2 - 2]$ is expressible as $f(x_5, \dots, x_{p_n})$ for ...
Daniel Donnelly's user avatar
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0 answers
65 views

Laplacian matrix and its eigenvectors and eigenvalues

I am reading paper written by Newman on finding community structure in a network. I came across with the Laplacian matrix. And there is one equation derived from some others. Before diving into that, ...
Ilhom Sadriddinov's user avatar
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0 answers
65 views

Why is the Discrete Fourier series a sum from $0$ to $N - 1$?

I want to derive the Discrete Fourier series: $$f(x) = \sum_{k=0}^{N-1} X_k e^{i2\pi xk}$$ from the Continuous Fourier Series formula: $$f(x) = \sum_{k=-\infty}^{\infty} X_k e^{i2\pi xk}$$ ...
Kian's user avatar
  • 11
0 votes
1 answer
128 views

How is the equation for this virtual site derived?

I am trying to understand virtual sites in MD simulations, and I came across this configuration: Here, coordinate $\mathbf{s}$ represents the virtual site, which is formed by three other atoms $\...
Adupa Vasista's user avatar
1 vote
0 answers
58 views

Polylogarithm further generalized

Here I proposed a generalized formula for the polylogarithm. However, because of a slight mistake towards the end, visible prior to the edit, I was unaware that it yields just a result of an integral ...
Artur Wiadrowski's user avatar
1 vote
0 answers
57 views

Problems when deriving Bernhard Riemann's equation for $\zeta(1-s)$

I am using this source: https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf It includes a translated version of Riemann's famous 1859 paper, and I am trying to understand all the steps ...
Noddy's user avatar
  • 21
-1 votes
1 answer
86 views

Prove That an Approximation of $\sin(x)$ via Euler's Formula Approaches $\sin(x)$

I'm trying to approximate the trigonometric functions for a code library, and I want to ask if this is a good way to go about it. I'm aware of the Taylor series approach, but I wanted to go with ...
Sig Moid's user avatar
1 vote
1 answer
194 views

Verification of the generalized polylogarithm formula

Here I posted a generalized formula for the polylogarithm I had discovered. However, for $x=\frac{1}{2}$, $z=\frac{1}{2}$, $p=1$ wolfram alpha yields a result different than what the double integral ...
Artur Wiadrowski's user avatar
-3 votes
2 answers
318 views

How to derive: $\sin(x-a)\sin(x+a) = \sin^2(x) - \sin^2(a)?$

I was looking at the solution of $$\int\frac{\sqrt{\sin(x−a)}}{\sqrt{\sin(x+a)}}dx.$$ Where the formula, $$\sin(x-a)\sin(x+a) = \sin^2x - \sin^2a = \cos^2 a - \cos^2 x$$ is used, I want to know how it ...
XZCY's user avatar
  • 371
2 votes
1 answer
94 views

How to derive zeroth order bessel function as solution to exponential of sin/cos

I've come across an integral which I'm stumped on how the solution was reached. I want to know how it was derived so I can understand if it's possible to vary the limits of integration in the ...
Ian's user avatar
  • 31
0 votes
1 answer
41 views

Derivation of F from ¬E V F and ¬E -> F

I want to derive F from ¬E V F and ¬E -> F Here is what I have done: ¬E -> F // hyp ¬E V F // hyp ¬E // AA F // E ->(2) F // AA ...
Lzkb's user avatar
  • 31
0 votes
0 answers
31 views

Simplifying the Formulas for Weighted Means

I am reading the following Wikipedia article (https://en.wikipedia.org/wiki/Pooled_variance) and came across the following formula: $$\mu_{x \cup Y} = \frac{N_x \mu_x + N_y \mu_y}{N_x + N_y}. \tag{...
Uk rain troll's user avatar
0 votes
0 answers
135 views

Deriving green function for Biharmonic equation and more in general polyharmonic.

As I am not an expert on biharmonic and more in general polyharmonic equation. Is there a procedure similar to the one highlighted here to derive the green function? Some people however don't seem to ...
user8469759's user avatar
  • 5,263
0 votes
2 answers
58 views

Trying to write an optimization problem

So here is the problem: A small machine tool manufacturing company entered into a contract to supply $80$ drilling machines at the end of the first month, $120$ at the end of the second month and $...
Scipio's user avatar
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4 votes
0 answers
75 views

How to derive this polylogarithm identity (involving Bernoulli polynomials)?

How can one derive the following identity, found here, relating the polylogarithm functions to Bernoulli polynomials? $$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!...
WillG's user avatar
  • 6,469
1 vote
1 answer
154 views

Ellipse in polar coordinate [with center of coordinates not center of ellipse]

On an old wikipedia page describing the ellipse, we have the following formula that parametrise an ellipse (semi-axis $a,b$, tilt $\phi$ and center $(r_0, \theta_0)$ inside the ellipse) : $$r(\theta) =...
edamondo's user avatar
  • 1,217
2 votes
1 answer
235 views

Generalized formula for the polylogarithm

Some time ago, I discovered the formula for repeated application of $z\frac{d}{dz}$ here. Recently, I thought about taking the function to which this would be applied to be the integral representation ...
Artur Wiadrowski's user avatar
0 votes
2 answers
122 views

surface area and volume of a spherical cap [duplicate]

While researching the formulas for these calculations, I am completely stumped by how the ones circled in red are derived. I think the surface area formula is wrong here since that's the formula for ...
John's user avatar
  • 161
0 votes
1 answer
46 views

The problem of deriving the formula, the left term of the equation is missing 1/2

The derivation in the paper: $$ V=-\frac{\left( kT/e \right) \left( x+ct \right)}{ct},\ E=\frac{kT/\left( ec \right)}{t} $$ $$ \left\{ \begin{array}{l} \varepsilon _0\dfrac{\mathrm{d}^2V}{\mathrm{d}x^...
Vancheers's user avatar
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2 votes
0 answers
116 views

Formula for Mertens Function $M\left(x\right)$ in terms of the Prime counting function $\pi\left(x\right)$ using inclusion-exclusion

In this post I was interested in studying the behaviour of Mertens function $M\left(n\right)$, which is defined for all positive integers as $$M\left(n\right)=\sum_{k=1}^{n}\mu\left(k\right)$$ Less ...
Juan Moreno's user avatar
  • 1,010
1 vote
1 answer
63 views

Signs in the Cardano formula

When deriving the Cardano formula from $x^{3}+px+q=0$ we let $x$ be a sum and compare coefficients. So $x=u+v$, then we get a system for $u$ and $v$. We get $(1) -q=u^{3}+v^{3}$ and $(2) u^{3}v^{3}=-(\...
thereisnoname's user avatar
2 votes
0 answers
26 views

calculating limit by second moment argument

Let $0<p<1$ be fixed. Let $X_n$, $n=1,2,...$, be a sequence of random variables with non-negative integer values. Suppose their expectations satisfy $E(X_n)=\frac{n(n-1)}{2} p^{n-2}$. Question : ...
Shiquan Ren's user avatar
0 votes
1 answer
84 views

Mathematics behind diatonic scales [closed]

I am looking for the mathematics that can be used to calculate a diatonic scale. It is my understanding that a number of musical scales can be represented in logarithmic form using mathematical ...
Michael Moriarty's user avatar
3 votes
1 answer
212 views

Verifying formulas and process for surface area and volume of a spindle torus

While working on this geometry problem I reasoned that the surface area of the spindle torus is the surface area of the apple (outer surface) plus the surface area of the lemon (inner surface) while ...
John's user avatar
  • 161
4 votes
1 answer
124 views

How to find an explicit formula for this function?

Let us take $$ \mathbb{N} := \{ 1, 2, 3, \ldots \}, $$ and let the function $f \colon \mathbb{N} \longrightarrow \mathbb{N} \times \mathbb{N}$ have the following values: $$ \begin{align} & f(1) :=...
Saaqib Mahmood's user avatar
1 vote
2 answers
79 views

Derive a formula to successively decrement a given value by $x\%$ $n$ times

I have a value where it needs to be decremented by $x\%$ successively $n$ times For example, for $$\begin{align*} \text{value} &= 100 \\ x &= 2\% \\ n &= 3 \end{align*}$$ we expect the ...
Code Guy's user avatar
  • 109
2 votes
1 answer
100 views

Algebraic manipulation in summations involving the floor function

https://www.ivl-projecteuler.com/overview-of-problems/25-difficulty/problem-401 takes the following derivation steps: It needs to calculate: $$\sum_{k=1}^{n}{k^2\left\lfloor{\frac{n}{k}}\right\rfloor}...
cladelpino's user avatar
0 votes
0 answers
56 views

Rigorously expressing time as an integral of velocity and position

I am studying the brachistochrone problem and I have found in many books the following chain of implications for calculating the time to travel from one point $A$ to another point $B$: $$v = \frac{dr}{...
David's user avatar
  • 95