For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
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105 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
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123 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
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124 views

$\int e^{-x} \log \log x dx$ - another special integral?

I came across this integral in some old notes. After several unsuccessful attempts I ran it in WA and got an interesting result: the antiderivative (closed form) doesn't exist, but the bounded ...
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177 views

Measuring how change in input variables contributes to output in non linear equation.

How do we measure how a variable contributes to an output as its value increases, and how it relates to other input variables? Let's say we're playing a video game, where you can buy items to augment ...
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41 views

General and basic question about convergence of a series

Let $(a_{i,j})_{i,j=1}^n$ be a sequence of real numbers such that the following series converges $$ S = \lim_{n\to\infty}\sum_{i=1}^n\sum_{j=1}^na_{i,j} $$ It is known that for each $i$th the ...
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167 views

Difficult Sum: Any Tips?

I'm trying to integrate a very difficult expression, and I've arrived at a particular step involving this sum, which I want to turn into a closed expression so that I can raise it to a power, re-sum ...
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54 views

integration of simple function

So I have this question I saw this week which I do not understand the answer for it. Let $a,b \in \mathbb{R} a <b \ $ and let $f_1:[a,b] \rightarrow \mathbb{R}$ and I know that $\int_a^x ...
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99 views

Extracting Taylor coefficients of a quotient

I was wondering if anybody has come across functions of the form $$\Phi_n(z):=\frac{f(z)^{n+1}}{zf'(z)-f(z)}\quad (n\geq 1).$$ Here, $$f(z)=\sum_{k=0}^{\infty} a_kz^k$$ is holomorphic on the open unit ...
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128 views

Differential calculus on Banach space

I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ...
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43 views

Laplacian of a generalized Whitehead potential

I'm trying to calculate the Laplacian $\triangle\Phi_W^{IJ}$ of the following "generalized Whitehead potential": $$ \Phi_W^{IJ}(\vec{x}) = ...
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303 views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points ...
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94 views

The Limit of the Following Derivative

Suppose you have two functions $F$ and $G$ with the following properties. $G(0)=F(0)=0, G'>0, G''<0, F'>0, F''<0 $ and also $\lim_{x\to0} F'(x)=\infty, \lim_{x\to\infty} F'(x)=0, ...
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160 views

How many points does one need for an epsilon-net

Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
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149 views

When are sums and integrals “identical” in form?

In the answer to this question Eric Näslund showed that logarithms can be written as the following limit of a sum: $$\displaystyle \log(x) = \lim_{k\to \infty } \, \sum\limits_{n=k}^{x k} ...
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217 views

Re: Rain droplets falling on a table

This questions is almost exactly similiar to the the following question, with an extra condition : Rain droplets falling on a table Suppose you have a circular table of radius R. This table has ...
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171 views

Evaluating $\int \limits_{a}^{\infty} \frac{\exp\left(-ax\right)}{\log(x)\left(c+x\right)^2}dx$

I have the following integral $$\int \limits_{a}^{\infty} \frac{\exp\left(-ax\right)}{\log(x)\left(c+x\right)^2} dx$$ that I do not know how to evaluate. Could you please give me a hint? Thanks in ...
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476 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
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276 views

How can I prove this inequality?

I have a pretty nasty looking function $$\sigma (t,y) = \sqrt{\frac{\sum_{i=1}^N \lambda_i \sigma_i \exp \left (-\frac{1}{2 t \sigma_i^2}\left(\ln{\frac{y}{S_0}} - \left(r - \frac{\sigma_i^2}{2} ...
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215 views

Understanding surface area of a revolution/length of curve

I don't quite understand why the formula to find the surface area of a revolution is what it is: $$A = 2\pi \int_a^b x\ \sqrt{1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2}\ \text{d} x.$$ I ...
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457 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
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166 views

Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$

This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion ...
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116 views

Why is this change of variables true?

I have an equation $F_{xx}+yF_{yy}+{1\over 2}F_y=0$ defined on $y<0$. I found that the characteristics are $\alpha={2\over 3}(-y)^{3\over 2}-x,\,\,\,\,\,\beta={2\over 3}(-y)^{3\over 2}+x$ and that ...
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1k views

Integral of the derivative of a function of bounded variation

Let $f\colon [a,b] \to \mathbb R$ be of bounded variation. Must it be the case that $|\int_a ^b f' (x) |\leq |TV(f)|$, where $TV(f)$ is the total variation of $f$ over $[a,b]$? If so, how can one ...
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269 views

Computing complex principal value integral - sgn-function?

I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$) $$ PV ...
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143 views

Tricky integral $\int_{a}^{b}\frac{\gamma d \gamma}{\gamma + \phi_{1}(\mu)-e^{-\frac{\phi_{2}(\mu)}{\gamma}}}$

in this integral $a=\psi_{1}(\mu), \ b=\psi_{2} (\mu)$. I expanded the function in Taylor series (3 terms) around ($\gamma= \frac{b}{2}$), numerically (for varioud values of $\mu$, and other constants ...
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44 views

Deriving expression for an integral that arose in Fourier analysis.

Note : This question arose when i am trying to solve this question. I am making this question self contained, and not to depend on the MO question, but one can look at MO question for understanding ...
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38 views

Origin of the Integral (Theory Behind It - How it came about)?

How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does ...
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77 views

Calculate in closed form $\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$

Playing with Taylor series is not helpful enough. What else would you try out? $$\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$$ $$\approx 2.1496160413898356727147400526167103602143301206321$$ It's ...
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65 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
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19 views

Analysis, area of rotating graph around x axis.

The proof comes to the following part which doesn't make a lot of sense for me: $$S(P)= 2 \pi \sum_{i=1}^{n} f(\epsilon_i) \sqrt{1+f'^2(\epsilon_i)}\triangle x_i + 2 \pi \sum_{i=1}^{n} ...
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29 views

Related Rates - possible textbook error

recently I've been asked to do some exercises from a textbook but I cannot get where the author came up with the answer. The exercise and my solution follow. Can you please help me? 2- At 8h boat A ...
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67 views

How to solve the Few Scientists Problem (big word problem) in its general form?

I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve the general form. Let me start with the concrete ...
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65 views

Problem using the Fourier transform and convolution to compute an integral

I'm trying to write a subroutine (in Fortran) to compute integrals of the form $$I=\int_{-L}^{L} f(x)g(y-x) \:\mathrm{d}x, $$ using the convolution theorem and fast Fourier transforms. In my routine, ...
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43 views

Prove $\frac{1}{\pi^2}\int_0^x \left(\sin \pi t\right)^2\left[\frac{1}{(t-a)^2}+\frac{1}{(t+a)^2}\right]dt\geq \frac{x-a}{1+x-a}$

Prove that $$\frac{1}{\pi^2}\int_0^x \left(\sin \pi t\right)^2\left[\frac{1}{(t-a)^2}+\frac{1}{(t+a)^2}\right]dt\geq \frac{x-a}{1+x-a}$$ for every $x\geq a>0$. I do not know where to start! Any ...
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Need help with this proof, I don't understand it , could anyone clarify some of the details. System of linear Differential equations.

$$(*)X'=A(t)X - system$$ $$(*)PX(\alpha)+QX(\beta)=0.$$-border conditions, where P,Q constant square matrices $n \times n $. Let $Y(t)$ be the fundamental matrix for the system $(*)$ normed for$ t= ...
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51 views

Are there concepts in nonstandard analysis that is useful for a introductory calculus student to know?

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense Can someone elaborate on this ...
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26 views

First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that $$\int_a^b G(t) dt = G(x)(b-a)$$ Assume $G$ is ...
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40 views

L'Hospital's rule for analytic functions of complex variable

Is there L'Hospital's rule for analytic functions of complex variable? If yes where can I find it, in what books, and how to prove it?
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31 views

Shortest system of roads between 4 cities

You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another ...
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49 views

How to evalute: $\int_0^1 \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)) dx$ and $a, b >0$

How to evalute: $$\int_0^1 \left[ \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}\left((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)\right) \right] dx$$ and $a, b >0$
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Theoretical Question regarding Taylor Expansion

I received the following question during a Calculus $2.0$ course in my university. I am not a native speaker, so please excuse my English. The question is as follows: Let $f$ be a function with ...
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78 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x ...
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55 views

Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
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77 views

Fast Hankel Transform

Can someone please explain what would be the expression for weights(Ho) in a Fast Hankel Transform.I found this in a paper and could not find any satisfactory answers .
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31 views

Gradient vector interpretation

I have trouble understanding the gradient vector $\nabla$. For a surface, when we find it's $\nabla$, why is the resultant vector the normal vector when $\nabla$ means gradient vector?? Thanks
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93 views

How to evaluate $\int_1^\infty \frac{1}{z} e^{-\left(\frac{z-1}{b}\right)^{\frac{1}{a}}} dz$?

I am really stuck on this integral: \begin{equation} \int_1^\infty \frac{1}{z} e^{-\left(\frac{z-1}{b}\right)^{\frac{1}{a}}} dz \end{equation} where $a,b$ are real constants. Is it a special ...
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52 views

detemine a limit involving log function

I just came up with the following limit $$ \lim_{n\to \infty} \frac{\displaystyle\log_a\Bigg(\sum_{\substack{k\in \mathbb{N}\\k\leq n~(1-\frac{1}{a})}}\binom{n}{k} (a-1)^k\Bigg)}{n} \qquad \text{ ...
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73 views

I can't find the critical points for this function. I showed my work :)

So, I have to find Critical Points of $y=\frac{1}{(x^3-x)}$ I know the derivative. Derivative = $(3x^2-1)/(x^3-x)^2$ To find Critical Points I equal to $0$. $x=1/\sqrt3$ and $x=-1/\sqrt3 $ But ...
3
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82 views

solution uniqueness of an algebraic system

$A(v),B(v),C(v)$ are positive, convexly decreasing functions on $\mathbb{R_+}$; $x$ is a random variable that obeys distribution F; Function $v(a,b,x)$ is implicitly defined as ...