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

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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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122 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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118 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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174 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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161 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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298 views

prove the general arithmetic-geometric mean inequality

$(a_{1}a_{2}...a_{n})^\frac{1}{n}\leq\frac{a_{1}+a_{2}+...+a_{n}}{n}$ for all $a_{i}$ positive real numbers. I keep getting stuck half way. This is review material for me (which I feel like I should ...
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98 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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124 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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293 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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93 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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148 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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459 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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275 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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214 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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436 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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266 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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140 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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47 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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23 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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85 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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69 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 ...
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80 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 ...
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54 views

What is the “$\cdot$” in this: $\left\langle \nabla f,\:\cdot\right\rangle$

I've read the Wikipedia page on exterior derivatives, and it states the following: \begin{align} df&=\sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i=\left\langle \nabla ...
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52 views

Proving a strictly decreasing sequence which tends to zero is positive

Suppose $(a_n)$ is a strictly decreasing sequence such that $a_n\underset{n\to\infty}{\rightarrow}0$. I'm asked to prove that $(a_n)$ is positive. My approach: suppose there is a negative element ...
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63 views

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
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63 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
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803 views

A difficult integral (expectation of the function of a random variable)

For $H>L$ , $p,q,\alpha,\beta>0$, and B(.,.) the beta functon, trying to solve this integral: $$\mathbb{E}(X)=\frac{\alpha H }{\beta B(p,q)}\int_0^H \frac{x \left(\frac{-H \log ...
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27 views

Convergence range of $\sum_{n=0}^{\infty}\frac {(-1)^n3n}{(x-4)^n}$

Find the range of convergence of $\displaystyle\sum_{n=0}^{\infty}\frac {(-1)^n3n}{(x-4)^n}$ Finding the radius first, setting $t=x-4$, $R=1/|((-1)^n3n)^{1/n}|=1$ So $\mid \frac 1 ...
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19 views

Spivak Calculus Chapter II Exercise 22 :: Arithmetic mean and geometric mean

the exercise says: If $a_1,\ldots,a_n\ge0$ then the A-M is $A_n=\frac{a_1+\cdots+a_n}{n}$ and G-M is $G_n=\sqrt[n]{a_1a_2\cdots a_n}$ we would like to show that $G_n\le A_n$ (1) suppose that ...
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111 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
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1k views

A difficult integral

For $\gamma>0,\delta>0$, trying to evaluate this integral: $$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log ...
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101 views

Root finding and Bolzano Theorem

Suppose that $f:\mathbb{R} \to \mathbb{R}$ is continuous on the closed interval $[a,b]$, and that $f(a)<0$ and $f(b)>0$. Then it must be $0$ at some point. How to Mathematically proof this ...
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48 views

How can we show that the functions are differentiable?

Show that the following functions $$f(x, y)=\frac{xy}{\sqrt{x^2+y^2}} \\ f(x, y)=\frac{x^2y}{x^4+y^2}$$ are differentiable at each point of the domain. Determine which of them is $C^1$. $$$$ The ...
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77 views

Prove a simple relation involving the function $f(x)=\sum_{n=1}^{\infty}\frac{\sin(n x)}{n^2}$

Prove that the following identity holds: $$ 16f\left(\frac{\pi}{4}\right) + 16f\left(\frac{3\pi}{4}\right)= 27 f(\alpha) -9 f(2\alpha) + f(3\alpha)$$ where $\alpha = 2\arctan\left(\sqrt{2}\right)$
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46 views

Chain rule for the curl of a vector-valued function

I am looking for a vector expression for the curl of a composite vector-valued function. In other words $$ \nabla\times\mathbf{A}(\mathbf{B}) = ? $$ In indicial notation, this can be ...
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21 views

Equivalent Definition Using Limit of Continuity

Definition : Note that $f: {\bf R}\rightarrow {\bf R}$ is continuous at $x_0$ if $$ \lim_{x\rightarrow x_0} f(x)= f(x_0)$$ And $f$ is continuous on $(a,b)$ if $f$ is continuous at any $x_0\in ...
3
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52 views

Multivariable Calculus - Stokes' theorem and conservative fields - showing that a vector field does not have a potential on a domain.

I understand that the curl of a vector field $\textbf{F}$ being zero means that the field is conservative if its domain is simply connected. This was demonstrated in part a where I showed that the ...
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95 views

Graphs of interesting integrals of the form: $\int \sin^a(x^a)\cos^a(x^a)$

Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer. $a = 2$ $a = 4$ $a = 6$ Now, a few graphs of the form:- $$\int ...
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50 views

Analytical solution for volume when a plane cuts a hemi-sphere

I need to find the analytical solution when the plane $ P: z = grad\cdot y + z_{cut} $ cuts the hemi-sphere $ S: x^2 + y^2 + z^2 = r^2;\:y \leq 0 $. I constructed two 3D images in MatLab of the ...
3
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47 views

Prove that there is an interval $[a, b] ⊆ I$ such that $f$ coincides with a polynomial there.

Let $f ∈ C^{\infty}$ satisfy the following property: To each $x ∈ I=[0,1]$ there corresponds a finite integer $N_x$ such that $f^n(x) = 0 $for all $n ≥ N_x.$ Prove that there is an interval $[a, b] ⊆ ...
3
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60 views

Help me to figure out this integral

$$\int_{0}^{\infty} \frac{x^3\sin\left(\frac{1}{2}\pi x\right)}{e^{2\pi\sqrt{x}} - 1}~dx$$ I've been thinking a long time ,but I have no idea how to do it.