Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
21,073
questions with no upvoted or accepted answers
26
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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^...
19
votes
0
answers
1k
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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,...
18
votes
0
answers
530
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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 ...
17
votes
0
answers
244
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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 ...
16
votes
0
answers
257
views
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: &...
16
votes
0
answers
754
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 ...
16
votes
0
answers
285
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 ...
15
votes
0
answers
523
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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 ...
15
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0
answers
271
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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
$$
f(x)=\sum_{n\geq 0}...
15
votes
0
answers
350
views
Question on the paper Donal F. Connon, "Some integrals involving the Stieltjes constants"
I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants.
It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
14
votes
2
answers
465
views
Is there an analytic solution for such problem?
Given function $$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,$$
where $n \in \mathbb{N}$ and $\...
14
votes
0
answers
565
views
prove or disprove an inequality on bounds of derivatives for radial functions
Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$,
and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$.
Prove or disprove the following.
Given any ...
12
votes
0
answers
347
views
Approximate the sum of a non $C^1(0,1)$ function by its integral
Consider the function $f: [0,1] \to \mathbb{C}$ defined by
$$
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|.
$$
I'm interested in approximating $...
12
votes
0
answers
367
views
Integral In Ramanujan's Letter To G.H. Hardy
In Ramanujan's first letter to G.H. Hardy he defines a function $\phi(n)$ as such
$$
\phi(n) = \int_{0}^{\infty} \frac{\cos n x}{e^{2 \pi \sqrt{x}}-1} d x
$$
And then he gives a functional equation
$$
...
12
votes
0
answers
301
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.
$$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, \...
11
votes
0
answers
366
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Integrating $\int_0^{\infty} \frac{e^{\cos(ax)}\cdot \sin(\sin(ax))}{x^2+1} dx$
I am trying to compute the family of integrals:
$$\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$$
for ...
11
votes
0
answers
253
views
L'Hôpital's rule in topological vector spaces
Let $E$ be a (separated) topological vector space over $\mathbb{R}$, $f\colon [0,1]\to E$ continuous. Assume that for every $t \in (0,1)$ we have a derivative
$$f'(t) = \lim_{h\to 0} \frac{f(t+h)- f(h)...
11
votes
0
answers
2k
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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 ...
11
votes
0
answers
315
views
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 ...
11
votes
1
answer
613
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 \...
11
votes
0
answers
223
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 ...
11
votes
0
answers
741
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 $k = \sqrt{2}/2$, $E(k)$...
11
votes
0
answers
410
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 ...
11
votes
0
answers
346
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:
$$\prod_a^b f(x)^...
11
votes
0
answers
270
views
Separable non-linear ODE (with radicals)
I am trying to solve the equation
$$
\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}
$$
$y(0) = 0$; $t_{0}=0$; ...
10
votes
1
answer
834
views
An intriguing integral representation of $\zeta^2(3)$
In a short presentation on RG, that is A Mesmerizing Integral Representation of $ζ^2(3)$ by C. I. Valean, we have, if I'm allowed, the following intriguing integral representation of $\zeta^2(3)$,
$$\...
10
votes
0
answers
208
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 ...
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,
$$I_1^{(p)}(N)\equiv\frac{1}{2^{N+p}}\int_0^1(1+t)^{N-1}(1-t)^pB\left(\frac{1}{t+1};N+p+1,N\right)\text{d}t$$
$$I_2^{(p)}(...
10
votes
0
answers
286
views
Integral of $\int_0^{\infty} \ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm 1}dx$
so I have this integral to try and evaluate
$$(*)=\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
votes
0
answers
665
views
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 ...
10
votes
0
answers
269
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, ...
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
$$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$
...
10
votes
0
answers
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. ...
9
votes
1
answer
296
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, ...
9
votes
0
answers
111
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 ...
9
votes
0
answers
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:
$$\frac{k^2}{h} \mathrm{sech}^2(k) \sqrt{1 + \left(\frac{k}{h} \tanh(k)\right)^2} = \ln\left( \frac{k}{h} \tanh(k) ...
9
votes
0
answers
308
views
Calculate in closed form: $\int_{0}^{\pi/2}\arctan\left(\frac{1}{\sqrt{2\sin x}}\right)dx$
I did the replacement, $u = \sqrt{2\sin x}$, but I did not succeed. $u = \sqrt{2\sin x}$. I found,
$$
\int_{0}^{\pi/2}\arctan(x)\cot(x)\,dx, \quad \int_{0}^{\pi/2}\arctan(\sin x)\,dx
$$
But the ...
9
votes
1
answer
176
views
What kind of "geometric" regularity $f'^2$ gives on $f$
When solving real-analysis' problems I like to represent the functions involved and think geometrically what is going on.
Today I got the following exercise :
Let $f \in \mathcal{C}^1(\mathbb{R},\...
9
votes
0
answers
440
views
Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$
This is my first question here, so I hope I'm not giving too little/too much information. I need some help calculating (or even approximating) an integral which I've been wrestling with for a while.
...
9
votes
0
answers
441
views
Why do these two integrals use roots of reciprocal polynomials?
There is the nice integral by V. Reshetnikov,
$$\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$$
also discussed in this post. By ...
9
votes
0
answers
385
views
Closed form of $\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}$
Is it possible to calculate the sum
$$
\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} \...
9
votes
0
answers
217
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
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 ...
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 ...
9
votes
1
answer
634
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.
$$ \lim_{n\to\infty}\frac{n}{\ln{n}}\...
9
votes
1
answer
238
views
How find this value of $A$?
Question:
Let $z\in C$
Find this value $A$,such
$$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$
where $i^2=-1$,and
$w_{k}(z)$ is Lambert $W$ function:see http://en....
9
votes
0
answers
894
views
Nasty Integral - Closed form solution?
Any suggestions on how to integrate this beast?:
$$\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}$$
where:
...
8
votes
0
answers
430
views
Improvement of Simpson's rule
Let $I_n$ denote the approximation of $$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}.$$ ...
8
votes
0
answers
226
views
An idea for this difficult integral
I am being stuck in caculating this integral: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$$ I tried to change to another variable: $x = - t$ then $dx = - dt$, ...
8
votes
0
answers
234
views
How to find $ \int_0^1 \frac{x^n}{(1-x) \ln ^n(1-x)} d x? $
Latest news
Thanks to Gary Liang and metamorphy who had given me links of relevant materials so that the closed form of our integral can be found as
$$
\boxed{\int_0^1 \frac{x^{n+1}}{(1-x) \ln ^{n+1}(...