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

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4
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75 views

Why is the infinite product of this quotient of $\sin$'s equal to $\left(\frac{3}{\pi}\right)^{2}$[SOLVED]

I was intrigued by this answer the other day, but perhaps lack a little bit of the necessary background to understand a certain step. Namely, the fact that $$\prod_{n=1}^{+\infty} \left(\frac{\sin\...
4
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0answers
478 views

How to solve an integral with a Gaussian Mixture denominator?

I am trying to solve this integral: $$t(v)\equiv\sum_{k=1}^{n}\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{w_{k}N(x-x_{k},B_{k})N(x-x_{j},v)}{\sum_{m=1}^{n}w_{m}N(x-x_{m},B_{m}+v)}dx $$ where $\sum_{...
4
votes
0answers
56 views

Calculus/Optimization - Implicit fuction theorem with equality constraints

I have the following constrained maximization problem, written as a Lagrangian: $$ L(x,y,\lambda) = f(x,y) - \lambda(g(x,y)) $$ I can derive a set of implicit equations that characterize the solution, ...
4
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0answers
220 views

Normal() convolved to Exp(polynomial)?

Is there a general exact solution for a Normal distribution convolved to $e^{y(x)}$, where $y(x)$ is a polynomial? $$e^{y(x)}*N(x,v)=\int_{-\infty}^{\infty}e^{y(z)}N(x-z,v)dz$$ $$y(x)\equiv\sum_{n=0}^...
4
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0answers
75 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 $k = \sqrt{2}/2$, $E(k)$...
4
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0answers
66 views

Upper bounding a definite integral

So I have the following problem. Let $F$ be the set of functions for which $|f(x)| \le 2$ for all $x$ and $\int_{0}^{5} [f(x)]^2dx \le 16$. Over all the functions in $F$, compute the maximum ...
4
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97 views

Is there a name for this type of integral $\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$?

Given a polynomial of arbitrary degree, $P(x)$, on $[a,b]$ is there a name for this type of integral: $$\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$$
4
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31 views

Effect of adding a constant to the torsion of a 3D curve

Let $\gamma$ be an arc-length parametrized curve in $\mathbb{R}^3$. Let say I add a constant to the torsion of $\gamma$ and let $\widetilde{\gamma}$ be the curve associated to the curvature of $\gamma$...
4
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124 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
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40 views

Asymptotic behavior of many derivatives

To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute $\frac{d^M}{dz^M} g(z)$ Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
4
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69 views

Volume of intersection between two horn tori

While playing around in Blender, I recently stumbled across a certain shape. The shape is found by taking the volume shared between two identical horn tori rotated at right angles to each other: The ...
4
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0answers
41 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, $f(...
4
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50 views

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$?

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$ ? To give context, this comes from: Dirac's Theory of General Relativity p19: http://imgur.com/mrkT5C7 I'm not comfortable with proofs regarding ...
4
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0answers
68 views

Sequences of rapidly decaying analytic functions

Note that we have $$\frac{1}{x}\gg\frac{1}{x^2}\gg\frac{1}{x^3}\gg\cdots\gg e^{-x}.$$ I was wondering if a generalization is possible. Namely, if $f_1\gg f_2\gg f_3\gg\cdots$ are analytic from $\...
4
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57 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
4
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67 views

Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
4
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0answers
125 views

Polar representation of conic sections $r(\theta)=\frac1{1 + e \cos\theta}$

Consider a curve given in polar coordinates by $r(\theta) = \dfrac1{1 + e \cos\theta}$, where $e\ge0$. a) Show that the distance of each point on this curve to the line $x=\frac1e$ is a constant ...
4
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63 views

Why can $x^0$ sometimes be simplified to 1 even when x can equal 0?

For example, the Taylor series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. It seems like it should be indeterminate or undefined at $x=0$, since the first term would contain $0^0$, but it's not ...
4
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0answers
104 views

In the mean value theorem, we are guaranteed $c$ such that $f'(c) = (f(b)-f(a))/(b-a)$. Does $c$ have a name?

The Mean Value Theorem says approximately that for differentiable $f$, there is a $c \in (a,b)$ such that $$ f'(c) = \frac{f(b)-f(a)}{b - a}. $$ I presume that the number $f'(c)$ is the mean value. My ...
4
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0answers
63 views

Find the slope at $t=16$ for $s(t) = $arctan$(\sqrt{t})$

A particle moves along the x axis so that its position at any time when t is greater than or equals zero is $s(t) = $arctan$(\sqrt{t})$. Find the velocity of the particle at $t=16$. The point of ...
4
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0answers
103 views

What is a good conceptual interpretation of a differential?

I'm having trouble with understanding what exactly a differential really is. For example, if we have the following function, $f(x,y)=x^2+xy+\frac{37}{x} +5$, does this differential, $df=\frac{\...
4
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0answers
59 views

Area under a curve with polar coordinates. Seems to be too simple?

Curve is given by equation: $$r^2 = 2a^2|\cos \phi|$$ I would like to use the formula: $$A = \frac{1}{2}\int_a^b (f(\phi))^2 \, d\phi$$ So, since equation is already squared, i can put the right ...
4
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0answers
84 views

Non-analytic smooth function

The Wikipedia page (http://en.wikipedia.org/wiki/Non-analytic_smooth_function) proves that $$f(x) = \begin{cases} \exp(-1/x), & \mbox{if }x>0 \\ 0, & \mbox{if }x\le0 \end{cases}$$ is a ...
4
votes
0answers
91 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
4
votes
0answers
105 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
4
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51 views

Elliptic integration with exponential numerator.

I was wondering if someone could help me with the evaluation of an integral: \begin{equation} \int_{x_1}^{x_2} \frac{e^{ax}}{\sqrt{1-b\cos(x)}}dx \end{equation} I'm familiar with elliptic integrals ...
4
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0answers
120 views

Do mathematicians consider functional integration to be good mathematics? If so, what is it?

I am trying to learn about path integration in physics and, from what I understand, this means learning to do functional integrals. (Note: I know what a functional is and I also have done functional ...
4
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0answers
81 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
4
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0answers
163 views

Solving $\int\frac{x}{(x^2+x+1)^{\frac{1}{12}}}$

I'm trying to solve $$\int_0^1\frac{x}{(x^2+x+1)^{\frac{1}{12}}}\mathrm dx$$ To calculate it I first tried to calculate the primitive function. So let $$\int\frac{x}{(x^2+x+1)^{\frac{1}{12}}}\mathrm ...
4
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0answers
66 views

a definite integral related to mean Gamma function

I am looking for a closed form of the following integral $$\int_0^1\frac{x^{a-1}\sin \theta dx}{x^2 - 2x\cos\theta + 1}$$ where $a>0$ and $0<θ<π$. I believe the answer is similar to the ...
4
votes
0answers
131 views

Integrate $\int\tan (x)\sin (ax)dx$

$$\int\tan (x)\sin (ax)dx$$ If $a$ integer number I used the wolfram.Alpha.com site to give me the following How can I use this result if I need to compute some values by using a program in ...
4
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0answers
109 views

Limit based on bounds

Let $k\in(0,1)$ be fixed. If $\limsup_{x\to\infty}f(kx)\leq \epsilon$ and $\liminf_{x\to\infty}f(\frac{x}{k})\geq -\epsilon$ for all $\epsilon>0$ then is it possible to say $$\lim_{k\to\infty}f(x)=...
4
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0answers
45 views

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 ...
4
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0answers
43 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.
4
votes
0answers
104 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( \frac{1}{2}\right)\log\left(1+\frac{1}{2}\right)+\...
4
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0answers
70 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 \frac{df}{...
4
votes
0answers
105 views

If Graham's number used $4$s instead of $3$s, at which $G$ would that number be bigger than Graham's number?

If $3\uparrow \uparrow\uparrow\uparrow3=G_1$, $ \quad G_2=\underbrace{3 \uparrow \ldots\uparrow3}_{G_1 \ \text{times}}, \quad G_3=\underbrace{3 \uparrow \ldots\uparrow3}_{G_2 \ \text{times}}$ , $ \...
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0answers
99 views

If a Sequence of Polynomials Converge to Another Polynomial Then the Roots Also Converge.

Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial ...
4
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0answers
655 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 $=e^{\int{-1/xdx}}=e^{-lnx}=1/x$ $y/x-y/x^2=1/x$ $[y/x]'=1/x$ ...
4
votes
0answers
109 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 ...
4
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0answers
113 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
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0answers
88 views

How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...
4
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0answers
266 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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0answers
4k views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
4
votes
0answers
282 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 ...
4
votes
0answers
459 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 ...
4
votes
0answers
111 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} ...
4
votes
0answers
108 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 ...
4
votes
0answers
164 views

The second derivative as a limit

It is well-known that if $f$ is twice differentiable at $a$, then $$ f''(a) = \lim_{h\to 0} \frac{f(a+2h)-2f(a+h) + f(a)}{h^2}. $$ See e.g. this question or this question. On the other hand, the ...
4
votes
0answers
147 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let $\...