0
votes
1answer
34 views

What is missing? (Rudin's Principles of Mathematical Analysis - Theorem 2.30)

Let us first give a definition: Definition Given a metric space $X$, and a subset $Y\subseteq X$, we say a subset $E$ of $Y$ is open relative to $Y$ if for each $p\in E$ there is an associated ...
2
votes
1answer
48 views

How to arrive at desired equality?

Why is the following second equality true? $$e^{1+1/2+...+1/(n+1)} - e^{1+1/2+...+1/n} \\= ...
0
votes
5answers
81 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
1
vote
1answer
117 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
0
votes
0answers
13 views

Coordinate Change Operator

Let $ f: \mathbb{R} \rightarrow \mathbb{R} $ be analytic. Recall that for $ h \in \mathbb{R} $, the translated function $ \tilde{f} (x) = f(x+h) $ can be formally written as $ \tilde{f} = e^{ h ...
0
votes
1answer
36 views

Prove limit of function

Let $f:(a,+\infty) \to \mathbb{R}$ and on every finite $(a,b)$ interval function $f$ is bounded. Then $$\lim_{x \to \infty}\frac{f(x)}{x}=\lim_{x \to \infty}f(x+1)-f(x)$$ How can we prove or ...
2
votes
2answers
100 views

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
0
votes
0answers
67 views
+50

Understanding last step of a proof that $\text{two trajectories cannot cross at a finite value of } t$ (Phase trajectories/nodes)

I have read that two trajectories cannot cross (Phase plane and nodes). The following proof was attached: Proof: Suppose $\vec y_A(t_A) = \vec y_{P_0} = \vec y_B (t_B)$ Set $\vec y(t) = \vec ...
0
votes
1answer
62 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
0
votes
1answer
33 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
3
votes
1answer
81 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
1
vote
2answers
102 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
0
votes
1answer
50 views

Which technique of integration should use to solve the question?

Quote from the paper I read: Given $F=(1-\lambda)f$ + $\lambda zf'$, we find that $f(z)= \tfrac{1}{\lambda} z^{1- \tfrac{1}{\lambda}}\int_0^zF(t)t^{\tfrac{1}{\lambda}-2}dt.$ My questions is Which ...
3
votes
3answers
85 views

What is $\frac{d^n}{dx^n} \frac{e^{\lambda x}}{x}$?

I was wondering whether there is an explicit way to say what the derivative of $\dfrac{d^n}{dx^n} \dfrac{e^{\lambda x}}{x}$ for $n \in \mathbb{N}_0$is, where we assume that $\lambda \neq 0$.
5
votes
1answer
48 views

Fubini's theorem applied to heaviside step functions

First of all, I should probably mention that I am a physicist, not a mathematician so I sincerely apologize for any lack of rigour in my explanation of my problem. Recently, I have been trying to ...
1
vote
1answer
17 views

Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
2
votes
2answers
67 views

Can we expect to find some constant $C$; so that, $\sum_{n\in \mathbb Z} \frac{1}{1+(n-y)^{2}} <C$ for all $y\in \mathbb R;$?

Fix $y\in \mathbb R;$ and consider the series: $$\sum_{n\in \mathbb Z}\frac{1}{1+(n-y)^{2}}.$$ My Question is: Can we expect to find some constant $C$; so that, $$\sum_{n\in \mathbb Z} ...
1
vote
1answer
41 views

$\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C$ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we choose $r$ large enough so that $I(y)< C$ for ...
1
vote
1answer
45 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
5
votes
1answer
39 views

how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$ If for all $y\in(y_0-\delta,y_0+\delta)$, ...
2
votes
1answer
38 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ ...
0
votes
1answer
18 views

Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...
1
vote
3answers
91 views

Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
0
votes
3answers
94 views

Assumptions in Word Problems.

My dilemma has been that I am confused on how we make mathematical assumptions in WORD problems. Suppose you are given a related-rates word problem. (Q#) Air is being pumped into a spherical balloon ...
1
vote
0answers
42 views

Are the two properties of a function equivalent?

$f(x)$ is a function defined on $\Bbb R^n$. $A$: $\forall x,y$ $$ |f(y)-f(x)-\nabla f(x)^T(y-x)| \le \frac{\beta}{2}\|y-x\|_2^2 $$ $B$: $\forall x,y$ $$ \| \nabla f(y)-\nabla f(x)\|_2 \le \beta ...
2
votes
2answers
49 views

Proving some statements only by the definition of Real numbers.

Let $f \colon \ [0.1] \to \mathbb R$ is monotonically increasing function and $f(0)>0$ and $f(x)\neq x $ for all $x\in [0,1]$. $$A=\{x\in [0,1] : f(x)>x \}$$ We know: every non-empty subset ...
8
votes
11answers
404 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
0
votes
1answer
97 views

Proof of $\displaystyle\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$

I want to prove $$\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$$ without useing L'Hôpital's rule.
9
votes
1answer
140 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
1
vote
1answer
38 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
1
vote
3answers
94 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
0
votes
0answers
26 views

Proving $\cos$ is Lipschitz continuous with $L=\frac{\sqrt3}2$ on $[-\frac12,1]$, using $\frac{\sqrt3}2=\cos\frac\pi6=\sin\frac\pi3$

I'm working my way through some analysis exercises to gain a better understanding and I stumbled upon an exercise where I could really use a hint. The task is to show that the inequality $|\cos ...
1
vote
1answer
26 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
2
votes
2answers
56 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
1
vote
0answers
15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
3
votes
1answer
22 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
58 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
4
votes
1answer
102 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
3
votes
2answers
80 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
4
votes
2answers
137 views

Solve nonlinear differential equation

Could you help me solve or give me some advice about following differential equation $$ 2(y')^2 + 3xy'y'' + 3yy'' = 0 $$
1
vote
2answers
52 views

Difference between the two definitions about the equality of two functions

From a long time I have found there are two definitions about the equality of two functions (or identity of two functions). I quoted the two definitions in the following: Zorich's definition ...
1
vote
1answer
85 views

Is there a differentiable function f which the differential function f' is bounded but has no maximum on a closed interval.

Is there a differentiable function $f$ in which the differential function $f'$ is bounded but has no maximum on one closed interval? Thanks
3
votes
2answers
243 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
62 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
0
votes
2answers
41 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) ...
3
votes
3answers
453 views

Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
3
votes
4answers
299 views

Analysis question limit sin function as n goes to infinity

can you help me with the following: $\lim_{n \rightarrow \infty} \sin^{2} \pi \sqrt{n^2 + n}$ Thanks a lot!
2
votes
2answers
136 views

Asymptotic Behaviour Of A Bizarre Function

It is relatively easy to show that the asymptotic behaviour of $f(x)$, where $$ f(x)= \left[\frac{x}{2}\right] + \left[\frac{x}{4}\right] + \left[\frac{x}{8}\right] + \left[\frac{x}{16}\right] + ...
1
vote
0answers
30 views

existence and uniqueness 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. ...
1
vote
1answer
25 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...