Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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3
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50 views

Hyperbolic Systems ODE

Let $M_n$ the set of matrices of order $n \times n$ identified with $\mathbb{R^{n^2}}$ e $S=\{A \in M_n ; x'=Ax$ is hyperbolic$\}$. Show that $S$ is open and dense $M_n$.
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0answers
64 views

When to Interchange Limit & Integral

I got stuck in the proof of Cauchy's Integral Formula for higher derivatives: Under what Conditions over a function $f$, we can infer that : $\displaystyle\lim_{h\rightarrow ...
0
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3answers
35 views

sequence is bounded below

I have to show that the sequence $$c_n=1+\frac{1}{2}+\frac{1}{3}+ \cdots +\frac{1}{n-1}+\frac{1}{n}-\int_1^n \frac{1}{x}dx$$ is bounded below. I have thought the following: ...
3
votes
1answer
72 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
1
vote
4answers
43 views

verifying log rules via elementary properties of an integral

my attempt at this question so far, $$\int_1^x \frac{1}{t}dt +\int_1^y \frac{1}{t}dt= 2\int_1^x \frac{1}{t}dt+ \int_x^y \frac{1}{t}dt$$ But I am not sure hwo to prove it from elementary ...
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vote
4answers
51 views

Show limit of $n\log(1+ \frac{x}{n})$ exists

How would I do this question? The fact that $y \rightarrow \log(1+y)$ tells me that: $$\lim_{h \to 0} \frac{\log(1+0+h)-\log(1)}{h}$$ tends to a existing limit. How do I use this for my answer??
4
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1answer
70 views

Showing that $|f^{(n)}| \le n!n^n$ and then making this result sharper

Ahlfors: Show that the successive derivatives of an analytic function at a point can never satisfy $|f^{(n)}(z)| > n!n^n$. Formulate a sharper theorem of the same kind. Attempt for Part ...
1
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1answer
26 views

Steps in the solution of Korteweg-deVries PDE

In the following solution of the Korteweg-deVries PDE $$ u_t + 6uu_x + u_{xxx} = 0 \qquad (3.1) $$ I do not understand the second integration step and how they arrive at the expression for the ...
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2answers
63 views

Show that $x_n \rightarrow 0$

Let $f:[0,1] \rightarrow \mathbb{R}$ continuous, such that $f(0)=0$ We set $x_n=\int_0^1{f(x^n)}dx$ Show that $x_n \rightarrow 0$ $$$$ The function $f$ is continuous at a closed interval ...
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0answers
12 views

Rearranging 2 discriminant function to solve for 1 parameter (to derive a decision boundary)

I have a task where I want to classify patterns from 2 classes where the samples are drawn from a bivariate Gaussian distribution. I use the 2 discriminant functions ($g_1$ and $g_2$) to classify the ...
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0answers
73 views

Using Cauchy's integral formula to find the best estimate for $|f^{(n)}(0)|$ under a condition

Question: Is the following proof valid? Ahlfors: If $f(z)$ is analytic for $|z| < 1$ and $|f(z)| \le 1/(1-|z|)$, find the best estimate of $|f^{(n)}(0)|$ that Cauchy's inequality will yield. ...
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3answers
273 views

Why is f bounded?

I am looking at the exercise: Let $f:[0,1] \to \mathbb{R}$ integrable such that $|f(x)| \leq \int_0^x |f(t)|dt, \forall x \in [0,1]$. Show that $f=0$. At the solution,it is taken that $f$ is ...
6
votes
0answers
91 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
0
votes
3answers
49 views

Proving Differentiability rigorously

Assume that real function f is differentiable at $x_0$ with $f'(x_0)$ >0. How would one show that there exists a $\delta$>0 such that $$ f(x)>f(x_0) $$ for all x in between $x_0$ and $ x_0 + ...
4
votes
1answer
75 views

Differentiability implies continuity — possibly pedantic question about the common proof

The common proof that differentiability implies continuity arrives at this limit: $$\lim_{x\to a} [f(x) - f(a)] = 0$$ I'm failing to see the simple justification for moving to the next step, which ...
2
votes
1answer
34 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
1
vote
1answer
32 views

Question involving intermediate value theorem

The function $g: (0, \infty) \to \mathbb{R}$, is continuous, $g(1)>0$ and $$\lim_{x \to \infty} g(x) = 0$$ It is a fact that for every $y$ between $0$ and $g(1)$ the function takes on a value in ...
2
votes
1answer
45 views

sum of an arctan series using mathematical induction

How to solve this problem using mathematical induction: $$\arctan (1) + \arctan \Big(\frac13\Big) + ... + \arctan \bigg(\frac{1}{n^2+n+1}\bigg)=\arctan (n+1)$$
3
votes
4answers
167 views

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent?

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ? I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other ...
2
votes
1answer
30 views

Non-dimensionalise

A body of constant mass is thrown vertically upwards from the ground. It can be shown that the appropriate non-dimensional differential equation for the height $x(t;u)$, reached at time $t\geq0$ is ...
2
votes
2answers
38 views

How can I prove this function is not continuous for every point other than 0?

Define $g:[0,1]\rightarrow\mathbb R$ by $g(x)=\sqrt{x}$ if $x$ is rational and $g(x)=0$ if x is irrational. Prove that $g$ is continuous at $x=0$, but is not continuous at any other value of $x$. I ...
0
votes
1answer
51 views

Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
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1answer
21 views

Radius of convergence | ratio test

I need to find the radius of convergence of $\Sigma n^3z^n$ I want to use the ratio test because it would be simpler than the root test. If $C_n=n^3$ then $| \dfrac {C_{n+1}}{C_n}| > 1$ because ...
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0answers
83 views

Bounded linear functionals on $L^\infty$.

I am looking at a practice final and I am a bit confused by this statement I am trying to prove: "There is a nonzero bounded linear functional on $L^\infty[0,1]$ which vanishes on the subspace ...
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votes
2answers
424 views

Intuition Behind Maximum Principle (Complex Analysis)

Let $D$ be an open set in the complex plane and $f(z)$ be a non-constant holomorphic function on D. Then $|f(z)|$ has no local maximum on D. I can follow the proof fine - usually if I don't ...
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1answer
43 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
4
votes
1answer
84 views

Calculus and infinitesimals

In the definition of reimann integral, why do we put a 'dx' inside the integral sign when practically it serves no purpose except maybe telling what variable you are talking about. Then in some ...
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1answer
28 views

Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
1
vote
1answer
13 views

Defining a region as a data structure

Is there a way for one to define a curve or region (such as a closed, 2-d disk) as a data structure into the computer, and make an algorithm which detects if a point is a boundary point, limit points, ...
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vote
1answer
100 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
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0answers
29 views

Intermediate value of the derivative.

Hi all what would the best way be to approach this question? I tried using the hint but I can't seem to formulate an answer for the fist part. Any help for the first and second parts of the question ...
0
votes
1answer
29 views

Minimisation of Finite sum of a decreasing sequence

If $a_{1}<a_{2}<a_{3}<...<a_{n}$, find the minimum value of $$\sum_{i=1}^{n} (x-a_i)^{2}$$ Then find the value of $$f(x)=\sum_{i=1}^{n} |x-a_i|$$ Hi all, what would the best way be ...
0
votes
1answer
28 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
2
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0answers
42 views

Estimate $\displaystyle\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|$

I have to estimate the following integral $$\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|,\quad \forall k,n\geq 2 $$ According to Sogge (Oscillatory ...
2
votes
1answer
46 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
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vote
1answer
38 views

differential equation as taylor series

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a ...
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4answers
94 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
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2answers
38 views

Prove that the convergence of the sequence (s3n) implies the convergence of (sn).

I write $s_n-s$, as $(s_n^3-s^3)/(s_n^2+s_n*s+s^2)$, true for all $n>N$. I'm trying to show that the denominator is convergent. But I don't know how to do this. Need help! Thanks. (Sorry about ...
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vote
1answer
30 views

Does weak convergence of $\nu_{n}$ imply convergence of $\int{f_{n}(x)d\nu_{n}(x)}$?

Suppose that we know that $ \int{ |f_{n}(x) - f(x)| d\mu(x)} \longrightarrow 0 \qquad (1) $ for every probability measure $\mu \in \mathcal{A}$ in a certain class. Also, suppose that $\{\nu_{n}\}$ ...
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1answer
49 views

Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
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1answer
32 views

To show closedness of a subset in a metric spaces

Let $(X, d)$ be a metric space and $p\in X$, $\delta>0$ be fixed. Let $$A=\{q \in X : p \in X, d(p,q)>\delta\}$$ How to show that $A$ is closed? I tried to show that directly by taking $A$'s ...
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2answers
38 views

Why is $\int_C {dz \over z - a} = 2 \pi i$ not a counter-example to Cauchy's theorem in a disk?

Cauchy's theorem in a disk states that if $\Delta$ is an open disk and $f$ is analytic on $\Delta$, then if $\gamma$ is a closed curve inside $\Delta$ we have that $$ \int_\gamma f(z)\ dz = 0 $$ ...
0
votes
1answer
57 views

Cauchy's theorem in a disk (Proof Verification)

Consider the following proof of Cauchy's theorem in a disk. My question is pasted at the bottom of the picture. (Note that in the proof below, a reference is made to "Theorem 2". In my textbook ...
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2answers
66 views

Computing $\int_\gamma { |dz| \over |z-a|^2}$

Goal: Compute $$ \int_{|z|= \rho} {|\mathrm{d}z| \over |z-a|^2} $$ under the condition $|a| \ne \rho$. Ahlfors' Hint: make use of the equations $z \bar{z} = \rho^2$ and $$ |\mathrm{d}z| = -i ...
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0answers
22 views

Set of limit points of Riemann Integrable functions

I've looked around for answers to this question. It seems like perhaps I don't have enough knowledge of functional analysis to figure out the answer (or even understand the answer), but I'm intrigued. ...
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2answers
76 views

Approaches to teaching and learning analysis

I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take. IMHO, the ...
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0answers
29 views

With $\omega(\delta)$ being the modulus of continuity, prove $\omega( \delta_1 + \delta_2) \leq \omega(\delta_1) + \omega(\delta_2)$

If the modulus of continuity for the function $f: E \to \Bbb R$ is the function $\omega(\delta)$ defined for $\delta > 0$ by $$\omega(\delta) = \sup_{|x_1 - x_2| < \delta}_{x_1, x_2 \in \Bbb E} ...
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1answer
70 views

Show that if E is measurable set and f is continuous on E, then f(E) is measurable set

Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set
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0answers
39 views

show that f is integrable at $[a,c]$ and $[c,b]$

Let $f:[a,b] \to \mathbb{R}$ bounded and $c \in (a,b)$.Then $f$ is integrable at $[a,b]$ iff $f$ is integrable at $[a,c]$ and $[c,b]$.In this case,we have $\int_a^b f = \int_a^c f + \int_c^b f$. The ...
3
votes
0answers
46 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...