Questions tagged [calculus]

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

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$f\colon I\rightarrow G$ 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^...
lion2011's user avatar
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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,...
wieschoo's user avatar
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Prove $ \int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n}$

Nice little generalization: $$\int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n},\quad a=0,1,2,...$$ The point of this post is to save us some calculations ...
Ali Shadhar's user avatar
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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 ...
msm's user avatar
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16 votes
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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 ...
Sophie Agnesi's user avatar
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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 ...
lsr314's user avatar
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15 votes
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Trying to prove a proposition about the nth order derivative of a polynomial by induction - is this correct?

Recently, I decided to try and create a formula for the $n$th order derivative of a polynomial, and I believe I succeeded! I tried to do a proof by induction to confirm this for myself, but since I ...
cdog's user avatar
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Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Given the Elliptic integral singular value $K(k_m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\tfrac12}$. Conjecture: &...
Tito Piezas III's user avatar
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271 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}...
Julio's user avatar
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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 ...
Vladimir Reshetnikov's user avatar
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2 answers
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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 $\...
Elman's user avatar
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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 ...
booksee's user avatar
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Approximate the sum of a non $C^1(0,1)$ function by its integral

Consider the function $f: [0,1] \to \mathbb{C}$ defined by $$ f(x)=\sum_{n=1}^{9} e^{2\pi n i x}, $$ so that $$ |f(x)|=\bigg|\frac{\sin(9\pi x)}{\sin(\pi x)}\bigg|. $$ I'm interested in approximating $...
Itachi's user avatar
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364 views

Integral In Ramanujan's Letter To G.H. Hardy

In Ramanujan's first letter to G.H. Hardy he defines a function $\phi(n)$ as such $$ \phi(n) = \int_{0}^{\infty} \frac{\cos n x}{e^{2 \pi \sqrt{x}}-1} d x $$ And then he gives a functional equation $$ ...
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12 votes
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300 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, \...
Schwinger's user avatar
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363 views

Integrating $\int_0^{\infty} \frac{e^{\cos(ax)}\cdot \sin(\sin(ax))}{x^2+1} dx$

I am trying to compute the family of integrals: $$\int_0^{\infty} \frac{e^{\cos(ax)}\cdot \cos(\sin(ax))}{x^2+1} dx$$ and $$\int_0^{\infty} \frac{e^{\cos(ax)}\cdot \sin(\sin(ax))}{x^2+1} dx$$ for ...
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250 views

L'Hôpital's rule in topological vector spaces

Let $E$ be a (separated) topological vector space over $\mathbb{R}$, $f\colon [0,1]\to E$ continuous. Assume that for every $t \in (0,1)$ we have a derivative $$f'(t) = \lim_{h\to 0} \frac{f(t+h)- f(h)...
orangeskid's user avatar
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Can we further refine $\int_{0}^{1}x^{x^x} \ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{\infty}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$

Question Can we further refine the integral $$\int_{0}^{1}x^{x^x}\ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{n}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$$ ? To compute the result, first ...
Paras's user avatar
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11 votes
1 answer
600 views

The gradient of a function has constant Euclidean length $1$

Consider a function $f : \mathbb R^{2} \to \mathbb R$ that is defined on every point and is differentiable. Then it has a gradient $\nabla f$. Now, suppose that $|\nabla f(x,y)| = 1$ for all $x,y \in \...
calcstudent's user avatar
11 votes
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220 views

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

(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 ...
Tito Piezas III's user avatar
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734 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 $k = \sqrt{2}/2$, $E(k)$...
Hmath's user avatar
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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 ...
Eugene Shvarts's user avatar
11 votes
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340 views

How to compute product integrals?

From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b f(x)^...
Ruslan's user avatar
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11 votes
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267 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{\left(\gamma-1+\frac{2\alpha\beta}{2\alpha-1}\right)e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1}\tag{1} $$ $y(0) = 0$; $t_{0}=0$; ...
JMK's user avatar
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10 votes
1 answer
825 views

An intriguing integral representation of $\zeta^2(3)$

In a short presentation on RG, that is A Mesmerizing Integral Representation of $ζ^2(3)$ by C. I. Valean, we have, if I'm allowed, the following intriguing integral representation of $\zeta^2(3)$, $$\...
user97357329's user avatar
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10 votes
0 answers
202 views

Non trivial $f$ such that $\int_0^1 x^n f(x) dx=a_n\pi+b_n$

I need a non trivial function $f(x)$ such that $$\int_0^1 x^n f(x) dx=a_n\pi+b_n$$ where $a_n,b_n\in\mathbb{Z}$ and $n\in\mathbb{N}$ We know that $$\pi=4\int_0^1 \sqrt{1-x^2}\ dx$$ By Binomial ...
Max's user avatar
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10 votes
0 answers
257 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)}(...
ARWarrior's user avatar
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10 votes
0 answers
284 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} \...
MKF's user avatar
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Is the Risch algorithm useful for calculating antiderivatives by hand?

In a German forum, a user asked how the "Feynman"-trick works. The example was $$f(x)=xe^x$$ Another user mentioned that the Risch algorithm should be taught. Therefore, I wonder whether the Risch ...
Peter's user avatar
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10 votes
0 answers
268 views

Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ completely monotone?

Recall the definitions of the sine and cosine integrals: $$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$ Both functions are oscillating, ...
Vladimir Reshetnikov's user avatar
10 votes
0 answers
2k 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 $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
user153012's user avatar
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10 votes
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2k views

Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the height. ...
JustinBlaber's user avatar
9 votes
1 answer
293 views

Bound a function with parameter involving logarithm

In my master project I encounter the following function $$f_\varepsilon(x) = \ln\left(\frac{x^{1 + a} + \varepsilon^\beta}{\lambda(x)^2 + \varepsilon^2}\right)$$ for $a$ close to zero, $\beta \in (1, ...
Falcon's user avatar
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9 votes
0 answers
109 views

Is it feasible to use Operator Calculus to solve for $f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots))$ over $\mathbb{R}$

Consider the function $f(t)=\exp^{(t)}(0)$ where $\exp^{(0)}(0)=0$ and $\exp^{(t+1)}(0)=\exp(\exp^{(t)}(0))$. That is, $$f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots)).$$ Such a function is not ...
Graviton's user avatar
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9 votes
0 answers
2k views

Why are these integrals equivalent to the Riemann Hypothesis?

Riemann Hypothesis is equivalent to the integral equation $$\int_{-\infty}^{\infty} \frac{\log \mid \zeta (1/2+it)\mid }{1+4t^2} \ dt =0$$ Many other integral equations exist that are equivalent. How ...
mick's user avatar
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9 votes
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115 views

This (rather long) implicit equation has a short explicit solution, but how can it be found?

I am curious if a method exists for solving for $k$ or $h$ in this implicit equation: $$\frac{k^2}{h} \mathrm{sech}^2(k) \sqrt{1 + \left(\frac{k}{h} \tanh(k)\right)^2} = \ln\left( \frac{k}{h} \tanh(k) ...
David Brock's user avatar
9 votes
0 answers
305 views

Calculate in closed form: $\int_{0}^{\pi/2}\arctan\left(\frac{1}{\sqrt{2\sin x}}\right)dx$

I did the replacement, $u = \sqrt{2\sin x}$, but I did not succeed. $u = \sqrt{2\sin x}$. I found, $$ \int_{0}^{\pi/2}\arctan(x)\cot(x)\,dx, \quad \int_{0}^{\pi/2}\arctan(\sin x)\,dx $$ But the ...
Mathsource's user avatar
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9 votes
1 answer
175 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},\...
auhasard's user avatar
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9 votes
0 answers
437 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. ...
Philip's user avatar
  • 492
9 votes
0 answers
438 views

Why do these two integrals use roots of reciprocal polynomials?

There is the nice integral by V. Reshetnikov, $$\int_0^1\frac{dx}{\sqrt[3]x\ \sqrt[6]{1-x}\ \sqrt{1-x\,\alpha^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{3}\;\alpha}\tag1$$ also discussed in this post. By ...
Tito Piezas III's user avatar
9 votes
0 answers
384 views

Closed form of $\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}$

Is it possible to calculate the sum $$ \sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!} $$ in closed form? Formal naive argument gives $$ \sum _{n=0}^{\infty} \...
Tyrell's user avatar
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9 votes
0 answers
216 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 ...
Asier Calbet's user avatar
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9 votes
0 answers
367 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 ...
Ahmed S. Attaalla's user avatar
9 votes
0 answers
2k views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
Elimination's user avatar
  • 3,150
9 votes
1 answer
633 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}}\...
pxchg1200's user avatar
  • 2,040
9 votes
1 answer
238 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....
user avatar
9 votes
0 answers
892 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2\left(\frac{\omega_{12}}{2}\right)\sin^2\left(\frac{\omega_{23}}{2}\right)d\omega_{23}d\omega_{12}$$ where: ...
okj's user avatar
  • 2,499
8 votes
0 answers
427 views

Improvement of Simpson's rule

Let $I_n$ denote the approximation of $$I = \int_a^b f(x)dx$$ obtained by applying Simpson's rule with $2n$ intervals of uniform length. Define a new approximation $$J_n=\frac{16I_{2n}-I_n}{15}.$$ ...
emil agazade's user avatar
8 votes
0 answers
215 views

An idea for this difficult integral

I am being stuck in caculating this integral: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$$ I tried to change to another variable: $x = - t$ then $dx = - dt$, ...
Lê Trung Kiên's user avatar
8 votes
0 answers
232 views

How to find $ \int_0^1 \frac{x^n}{(1-x) \ln ^n(1-x)} d x? $

Latest news Thanks to Gary Liang and metamorphy who had given me links of relevant materials so that the closed form of our integral can be found as $$ \boxed{\int_0^1 \frac{x^{n+1}}{(1-x) \ln ^{n+1}(...
Lai's user avatar
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