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

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Is there any relation between summation and indefinite integration?

Indefinite integral is the family of all its primitives or antiderivatives. It represents geometrically a family of curves having parallel tangents at their points of intersection with the lines ...
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54 views

Proving another digammabinomial series result

This series is related to some extent to the previous question of mine, that is Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$, where an approach by series only ...
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119 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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41 views

Help with an ODE

I need some help, I have this ODE but can't solve it for $y(x)$, I try every method I know, but with no succes,please, somebody can help me? $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ Thanks.
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103 views

Closed form of arctanlog series

What tools would you recommend me for $$\operatorname{arctan}\left( \frac{1}{1}\right)\log\left(1+\frac{1}{1}\right)+\operatorname{arctan}\left( ...
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47 views

When do Leibniz-like rules lead to unique linear operators

Background Usually one defines differentiation in terms of limits, and then shows that differentiation satisfies the Leibniz (product) rule, $$\frac{d}{dx}(f \cdot g) = f\frac{dg}{dx} + g ...
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208 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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382 views

Absolute Value in Linear Differential Equation

I have a question about dropping the absolute value sign when solving a linear differential equation. If $y'-y/x=1$ Integrating Factor $=e^{\int{-1/xdx}}=e^{-lnx}=1/x$ $y/x-y/x^2=1/x$ $[y/x]'=1/x$ ...
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102 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
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176 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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224 views

Is Courant's Introduction to Calculus and Analysis still up-to-date?

I just found this marvelous book and I think that it's the best book in this category, but I'm worried that it is not up-to-date. I've heard that Hardy's A Course of Pure Mathematics has some switched ...
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321 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $\mathscr{C}^1$ on $(c,d)$. Then ...
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94 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
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88 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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113 views

A trigonometric integral with sin(cos(x)) in exponent

Evaluate: $$\int_0^{\pi} x\csc^{\sin(\cos x)}(x)\,dx$$ I honestly don't know how to deal with this case. If I apply the property $\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$, I get: $$\int_0^{\pi} ...
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132 views

Integral of product spherical harmonics

I'm trying to calculate this integral: $$\int_0 ^\pi\int_0^{2\pi} [\sin\theta(\cos\theta \sin \phi + \sin\theta \cos(2\phi))]^2 \sin\theta d\phi d\theta $$ We could compute this integral ...
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114 views

L'Hopital quicky

suppose L'Hopital applies and $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \lim_{x\to\infty}\frac{f'(x)}{g'(x)}$$ under what conditions is it true then that $$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} ...
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150 views

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old. I am also looking for literature on these ...
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61 views

Calculation of $\lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2}$

Calculation of $\displaystyle \lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2}$ $\bf{My\; Try::}$ Given $\displaystyle \lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2} = \lim_{x\rightarrow ...
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103 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
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98 views

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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107 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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128 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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180 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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174 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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105 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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132 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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45 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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318 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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162 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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151 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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1k views

Good introductory book for matrix calculus

Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
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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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172 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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497 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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278 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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217 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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473 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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167 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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273 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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105 views

Find the derivative of the following integral

Find the derivative of $f(x)= \int_x^0 \frac{\cos(xt)}{t} dt$. My first reaction was to apply the FTOC, but I don't believe I can do this because $\frac{\cos(xt)}{t}$ is not defined at $t=0$ and thus ...
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58 views

A integral inequality

Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\left( ...
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35 views

gateaux derivative and frechet derivative

In calculus, we have the following equation $DF(x,y)=\partial F_xdx+\partial F_ydy$ if $F$ is differentiable. I think such equation still holds for frechet derivative, but not for gateaux derivative. ...
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23 views

an integral equation in a function with two arguments

Say we are given $C(s,t)=\min(s,t)+\zeta st$. How can we solve $$g(s,t)=C(s,t)+\lambda \int_0^1g(s,u)C(u,t)du.$$ Looking into some text books on integral equations I see that most of the kernels, $C$, ...