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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3answers
339 views

A function which is R-integrable but does not have an antiderivative

How is it possible that this function: $$ f(x)=\left\{\begin{array}{ll} 0, & -1\le x < 0 \\ 1, & 0\le x \le 1\end{array}\right. $$ is R-integrable in $[-1,1]$ , but does not ...
2
votes
1answer
54 views

sum of an arctan series using mathematical induction

How to solve this problem using mathematical induction: $$\arctan (1) + \arctan \Big(\frac13\Big) + ... + \arctan \bigg(\frac{1}{n^2+n+1}\bigg)=\arctan (n+1)$$
2
votes
2answers
41 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 ...
2
votes
1answer
33 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 ...
1
vote
1answer
25 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 ...
1
vote
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 ...
8
votes
2answers
567 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 ...
4
votes
1answer
98 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 ...
1
vote
1answer
34 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, ...
1
vote
1answer
332 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
48
votes
5answers
3k views

Why are gauge integrals not more popular?

A recent answer reminded me of the gauge integral, which you can read about here. It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue ...
1
vote
1answer
43 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 ...
0
votes
2answers
75 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 ...
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 ...
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 ...
0
votes
0answers
30 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
74 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
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)= ...
3
votes
0answers
43 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 ...
0
votes
1answer
100 views

Weierstrass Caratheodory on open interval

I have been working on this question for a while now, and if I have understood it correctly shouldn't the answer be that $\phi_{c}=f'(x)$ for all $x \in (a,b)$ as the function f , is now said to be ...
6
votes
4answers
104 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?
0
votes
2answers
42 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 ...
1
vote
1answer
37 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 $\{\nu_{n}\}$ ...
4
votes
1answer
69 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 ...
0
votes
1answer
93 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 ...
0
votes
2answers
42 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 $$ \int_\gamma f(z)\ dz = 0 $$ ...
2
votes
2answers
94 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 ...
0
votes
0answers
31 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. ...
1
vote
1answer
62 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
0
votes
1answer
51 views

How to interchange sum and integral when measure is in terms of Dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit Dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
2
votes
1answer
30 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
0
votes
1answer
63 views

Prove that $F$ is dense in $C(X\times Y,\mathbb{R})$?

Let $X$ and $Y$ be compact metric spaces. Let $$ F= \Bigl\{\sum_{i=1}^n A_i f_i(x) g_i(y): f_i\in C(X,\mathbb{R}),g_i\in C(Y,\mathbb{R}), 1\le i\le n \Bigr\}. $$ Prove that $F$ is dense in $C(X\times ...
2
votes
0answers
43 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 ...
0
votes
1answer
42 views

How to calculate the Laplace transform?

we know that ( http://en.wikipedia.org/wiki/Laplace_transform )\begin{equation} \mathcal{L} \{t^{5}f(t)\} = (-1)^5\frac{d^5}{ds^5} F(s)\end{equation} My question is \begin{equation} \mathcal{L} ...
3
votes
0answers
50 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 ...
3
votes
1answer
315 views

Example of a false proof when a Fourier series is not unique?

I am attempting to come up with an example to illustrate why one should care that a function has a unique Fourier series expansion. Inspired by the fact that one can rearrange terms in a ...
1
vote
1answer
192 views

Definition of upper hemicontinuity of a correspondence.

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...
1
vote
1answer
100 views

munkres analysis integration question

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2 \to \mathbb{R}$ be defined by setting $f(x,y)=0$ if $y \neq x$, and $f(x,y) = 1$ if $y=x$. Show that $f$ is integrable over $[0,1]^2$.
1
vote
1answer
64 views

What is the limit of this function as $(x,y)$ approaches $(0,0)$?

Let the function $f \colon (\mathbf{R}^2 \setminus \{(x,y) \in \mathbf{R}^2 \colon x+y = 0 \}) \to \mathbf{R}$ be defined as follows: $$ f(x,y) \colon= \frac{xy}{x+y}$$ if $(x,y) \in \mathbf{R}^2$ ...
1
vote
3answers
58 views

Computing $\int_{|z|=2} {dz \over z^2 + 1}$

Goal: To compute $$ \int_{|z|=2} {dz \over z^2 + 1} $$ by decomposition of the integrand in partial fractions. Attempt: Let $\gamma$ be the circle around the origin of radius $2$. Let us ...
2
votes
1answer
557 views

Prove that set has zero Jordan content iff its closure has measure 0

Prove that set has zero Jordan content iff its closure has measure 0. I am having trouble with both directions , any tips would be great. THanks!
7
votes
1answer
148 views

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer.

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $$a_{n+2} = a_{n+1} - a_na_{n+1}/2$$ for $n$ a positive integer. Find $$\lim_{n\to\infty}na_n$$ if it exists. Well, we can deduce that $\lim a_n=0$ by ...
0
votes
2answers
63 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
2
votes
1answer
82 views

Showing the winding number of the unit circle is $1$

Let $\gamma$ denote the unit circle parameterized on the domain $[0,2\pi]$. I'm trying to compute $n(\gamma, 0)$ as follows: $$ n(\gamma,0) = {1 \over 2\pi i}\int_\gamma {dz \over z} = {1 \over 2 ...
0
votes
1answer
63 views

Show that the iteration $x_{n+1} = x_n - 2\frac{f(x_n)}{f'(x_n)}$ converges quadratically to $x_*$ provided $x_0$ is sufficiently close to $x_*$

We have the following conditions for the above slightly-modified Newton's method iteration: $f$ is a real function of one real variable $f''$ is Lipschitz continuous $f(x_*) = f'(x_*) = 0$ I also ...
6
votes
1answer
265 views

Smooth map with surjective Jacobian is open

I'd like to show that if $U\subset R^n$ is open, $f:U\to R^m$ is smooth, and $J_f(x)$ is surjective (full rank) for every $x\in U$, then $f(U)$ is open. My thoughts so far: For any $f(x)\in f(U)$, ...
1
vote
3answers
31 views

Negation of continuity applied to a sequence

Show that if it is not true that $\lim_{x \to a} f(x)=l$ then $\exists$ $\epsilon$>0 and a sequence $(x_{n}) \rightarrow a$ as $n \rightarrow \infty$ such that $|f(x_{n})-l| \geq \epsilon$. Now ...
4
votes
2answers
43 views

Is there a word for describing “smoothness” quantitatively?

I've long wondered how to "quantitatively" describe how smooth a function is. For instance, a 1000 term Fourier series for a the Heaviside step function is technically smooth, as it has infinite ...
1
vote
1answer
50 views

Graphs with the property that $f=f^{-1}$

Ive been working on this question and can't seem to progress, I know that for a function to have the property $f=f'$ it must be symmetrical about the line $y=x$,I can't find reasoning behind in the ...