Questions tagged [calculus]

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

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60
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Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,...
33
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697 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider $f(x)=...
28
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0answers
590 views

The “natural” Sophomore's Dream integral: $\int_{0}^{\infty} x^{-x}\ dx$

I have been wondering about this for a while, but with no real luck in figuring it out. The famous "Sophomore's dream" identity refers to two similar integrals, one of which is $$\int_{0}^{1} x^{-x}\ ...
28
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1answer
905 views

The Modulus of all the roots of a Polynomial are equal to $1$

Suppose the real number $\lambda \in (0,1)$, and let $n$ be a positive integer. Prove that all roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{...
20
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502 views

What can be said about the level set of the real part of an analytic function?

I am working with a function $F(z;a)$, for $z\in \mathbb{C}$ and $a$ being a set of parameters, from which I need to analyze the level set $\text{Re}(F(z))=0$ (for a fixed set of parameters $a$, which ...
18
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316 views

$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^...
16
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716 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
16
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750 views

Convergence/Divergence of infinite series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{1+\left|{\cos n}\right|}}$

It is well known that $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent while $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon}}$ is convergent for a fixed positive value of $\...
13
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0answers
172 views

Show That A Particle In A Bounded Force Field Can Reach Any Point In Fixed Time Span

I tried to proof that for a smooth bounded force field $F$ and $x\in{\bf R}^n$ there exists some $v\in{\bf R}^n$ such that a particle starting in $0$ with mass $1$ and velocity $v$, obeying Newton's ...
13
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476 views

Computing the exact value of $\sum_{n=1}^\infty \left(\frac{2n+3}{3n+2}\right)^n$

I found this problem in my textbook, and I know that it converges, but I wanted to know if there was a way to find the exact value of the convergence (similar to what Euler did with the sum of ...
13
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989 views

Gauss-Lucas Theorem (roots of derivatives)

Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots $\alpha_1,\ldots,...
12
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160 views

Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Given the complete elliptic integral of the first kind $K(k_\color{blue}m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\...
12
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395 views

An integration to first order

I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me. I wish to evaluate $$\int_0^\pi {\cos\theta\cos \left[\omega t-{\...
12
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1answer
437 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
11
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167 views

Is there an analytic solution for such problem?

Given function $$f_n(x) = \cos x - (\cos \cos x) + (\cos \cos \cos x) - (\cos \cos \cos \cos x) + \dots + (-1)^{n-1} \underbrace{ \cos \cos \dots \cos }_n x,$$ where $n \in \mathbb{N}$ and $\...
11
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0answers
206 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq 0}...
11
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224 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
11
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586 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h \...
11
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1answer
164 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
11
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1answer
843 views

What is the best way to learn Differential forms?

I'm taking a Multivariable Calculus class and my teacher has just started Differential forms. It is not making a lot of sense, though. I have tried reading "Geometric Approach to Differential forms" ...
11
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1answer
671 views

Integral involving square root of sine and cosine

Is there any closed formula for $$ \int_{0}^{\pi/2} \dfrac{e^{-x}\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} $$ I know $$ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}...
10
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113 views

How to interpret Newton's 6th Lemma?

In Newton's "Principia Mathematica" Book 1, Section 1 ("Of the Motion of Bodies") there is the following Lemma 6: "LEMMA VI. If any arc ACB, given in position, is subtended by its chord AB, and ...
10
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106 views

Symbolic approximation through integration by parts

This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
10
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0answers
145 views

Integral of $\int_0^{\infty} \ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm 1}dx$

so I have this integral to try and evaluate $$(*)=\int_0^{\infty} ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm1}dx$$ So far, I have managed to evaluate a very similar integral $$\int_0^{\infty}...
10
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0answers
261 views

The diferential equation $y' = \frac{\ln(x^2+y^2)}{x^2 + y^2}$

In my University, the integral calculus teacher gave me this diferential equation to solve. $$ y' = \frac{\ln(x^2+y^2)}{x^2 + y^2} $$ I dont have any clue of what form has the solution of this ...
10
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0answers
224 views

Bear of an integral

I have a pretty ferocious integral to solve, and would be over the moon if I were able to get some sort of analytic expression / insight for it. $$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, \...
10
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0answers
843 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.
10
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775 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f{} = 2\pi+2\tan^{-1}(y,x)$ $y = -...
10
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1answer
297 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)} }{...
9
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1answer
155 views

What kind of “geometric” regularity $f'^2$ gives on $f$

When solving real-analysis' problems I like to represent the functions involved and think geometrically what is going on. Today I got the following exercise : Let $f \in \mathcal{C}^1(\mathbb{R},\...
9
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0answers
167 views

Strong Counterexample to MVT on Q

A well-known application of the MVT is to prove that any $f: \mathbb{R} \to \mathbb{R}$ with $f'= 0$ is constant. But of course, the MVT relies fundamentally on the properties of $\mathbb{R}$, and in ...
9
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0answers
287 views

Evaluate $\sum_{n=0}^{\infty} \frac{n!}{(n^2)!}$

I'm interested in a method of evaluating $\sum_{n=0}^{\infty} \frac{n!}{(n^2)!}$. If there was a linear equation with leading coefficient $1$ in the denominator or a quadratic with leading ...
9
votes
1answer
101 views

Prove a probability inequality

Let $X,Y\sim\phi(x)$ be i.i.d. ($\phi:\Bbb R\to\Bbb R_{\ge 0}$ denotes the PDF), show that $$P(|X+Y|\le1)\le2P(|X-Y|\le1).$$ My thoughts are standard, $$P(|X+Y|\le1)=\int_\Bbb R \rm dx\phi(x)\...
9
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0answers
402 views

Evaluating Elliptic Integrals in terms of Gamma Function

Some complete elliptic integral of first and second kind $E(k)$ and $K(k)$ can be evaluated for some particular values of $k$ in terms of Euler Gamma function. For example, for $k = \sqrt{2}/2$, $E(k)$...
9
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0answers
1k views

What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
9
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0answers
257 views

An Additional Rule for Calculus

Background The rules for differentiating elementary functions (arithmetic, exponential, trigonometric, etc.) together with the chain rule for differentiating compositions of functions are often ...
9
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1answer
389 views

Find the limit of $ \lim_{n\to\infty}\frac{n}{\ln{n}}\left(\frac{1}{p+1}-na_{n}^{p+1}\right) $

Problem:Let postive real sequence$\{a_{n}\}$ satisfy $\displaystyle\lim_{n\to\infty}a_{n}\left(\sum_{i=1}^{n}a_{i}^{p}\right)=1$,where $p>-1$,Find the limit. $$ \lim_{n\to\infty}\frac{n}{\ln{n}}\...
9
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1answer
251 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 $1+2+3+\cdots = -\frac{1}{12}$? The Euler-...
9
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1answer
207 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see http://en....
9
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0answers
170 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{\left(\gamma-1+\frac{2\alpha\beta}{2\alpha-1}\right)e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1}\tag{1} $$ $y(0) = 0$; $t_{0}=0$; ...
8
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0answers
199 views

Integrals involving powers and beta (or hypergeometric) function

I have the three following integrals, very similar the one to the others, $$I_1^{(p)}(N)\equiv\frac{1}{2^{N+p}}\int_0^1(1+t)^{N-1}(1-t)^pB\left(\frac{1}{t+1};N+p+1,N\right)\text{d}t$$ $$I_2^{(p)}(...
8
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0answers
46 views

Matrix performing local differintegral analysis being its own inverse. Coincidence?

I found a curious matrix $$T = \begin{bmatrix}1&2&1\\1&0&-1\\1&-2&1\end{bmatrix}$$ This matrix (or actually $\frac 1 2 T$) performs Local mean value (integral) estimation. ...
8
votes
0answers
293 views

An $\operatorname{erfi}(x)e^{-x^2}$ integral

I want to find an elementary evaluation of $$I=\int_0^\infty \left(\frac{\sqrt\pi}2\operatorname{erfi}(x)e^{-x^2}-\frac1{1+2x}\right)dx$$ where $\operatorname{erfi}(x)=\frac{2}{\sqrt\pi}\int_0^...
8
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0answers
174 views

A wrong proof for an (evident) lemma

(Eliashberg, Y.; Mishachev, N.M., Wrinkling of smooth mappings and its applications. I, Invent. Math. 130, No.2, 345-369 (1997). ZBL0896.58010. \cite{EM}) Let $ \alpha : [a, b] \to \mathbb{R}$ is ...
8
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0answers
249 views

Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$

This is my first question here, so I hope I'm not giving too little/too much information. I need some help calculating (or even approximating) an integral which I've been wrestling with for a while. ...
8
votes
1answer
316 views

An extremely mysterious integral: $\int_0^1 \frac{k \tan^{-1}(t)}{k^2 + t^2}\mathrm d t$

$$f(n) = \int_0^1 \frac{n \tan^{-1}(t)}{n^2 + t^2}\mathrm d t \tag{n > 2}$$ Introduction: This is one of the most beautiful and mysterious integrals I've every encountered. It's very simple, but ...
8
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0answers
131 views

Complete this table of general formulas for algebraic numbers $u,v$ and $_2F_1\big(a,b;c;u) =v $?

(This extends this post.) Given fixed rationals $a,b,c,$ the problem of determining, $$_2F_1\big(a,b;c;u) =v $$ such that both $u,v$ are algebraic numbers may be solved by appealing to modular ...
8
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0answers
426 views

Find $ ? = \sqrt[3] {1 + \sqrt[3] {1 + 2 \sqrt[3] {1 + 3 \sqrt[3] \cdots}}} $

I wonder about a closed form for $ ? = \sqrt[3] {1 + \sqrt[3] {1 + 2 \sqrt[3] {1 + 3 \sqrt[3] {1 + 4 \sqrt[3] {1 + 5 \sqrt[3] \cdots}}}}} $ To be clear $$? = \sqrt[3]{ 1 + \color{Red}{1}\sqrt[3]{ 1 ...
8
votes
0answers
372 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in his ...
8
votes
1answer
748 views

Integral (Tanh and Normal)

I am trying to evaluate the following: The expectation of the hyperbolic tangent of an arbitrary normal random variable. $\mathbb{E}[\mathrm{tanh}(\phi)]; \phi \sim N(\mu, \sigma^2)$ Equivalently: $...