# Questions tagged [calculus]

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

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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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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 ...
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### 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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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
While seeing this post, the following integral is just struck me $$\int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1$$ I have tried like what user @OlivierOloa did in his ...
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: \$...