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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85 views

On Cauchy-Riemann equations

Given $f:\mathbb C\to \mathbb C$ is a non-constant entire function. Then which of the following is possible? Re $f(z)=$ Im $ f(z)$, Im$\,f(z)<0$, Re$\,f(z)$ is bounded, $f(z)\neq 0,$ for all ...
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1answer
69 views

How is an open set defined without referring to any distance function in a topology?

I am currently studying general topology. The definition given by Royden looks very confusing to me. It says that the elements of a topology is called open sets without actually defining what exactly ...
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4answers
61 views

Find the limit of this sequence $\lim_{n\to \infty}\frac{n}{1 + \frac{1}{n}} - n$

Find the limit of this sequence $$\lim_{n\to \infty}\frac{n}{1 + \frac{1}{n}} - n$$ First I tried dividing everything by $n$ but that would leave me with $$\lim_{n\to \infty}\frac{1}{\frac{1}{n} + ...
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0answers
42 views

How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
3
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1answer
41 views

Superior limit of integrals of entire functions

Let $f$ be an entire function on $\mathbb{C}$. If $f$ is not constant, then I want to prove \begin{equation} \limsup_{R\to\infty}\int_{\lvert z\rvert=R}\lvert f(z)\rvert\,\lvert dz\rvert=\infty. ...
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3answers
138 views

Problem in Chapter 2 (Walter Rudin).

"Principles of Mathematical Analysis" by Walter Rudin has the following question: Show that the following statement is false in $\Bbb{R}$: If $\{K_\alpha\}$ is a collection of closed subsets ...
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4answers
166 views

$f(f(x))=f(x)$ question

I am wondering what is the class of functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(f(x))=f(x)$? I think it should be: Constant Value functions the identity function absolute value ...
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2answers
75 views

Why is one set countable while the other uncountable?

This is an example from Chapter 1 of David Williams book - Probability with Martingales. Let $(S,\Sigma,\mu) = ([0,1],\mathcal{B}([0,1]),Leb)$. Let $V=\mathbb{Q}\cap [0,1]=\{v_n, n\in \mathbb{N}\}$. ...
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4answers
139 views

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to? As far as I can tell, this has no closed-form solution (not saying much, I don't know much math), but a friend of mine swears he saw a ...
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1answer
316 views

Proof of convergence of an infinite product

a) Show that $\Pi_{n=1}^\infty x_n$ converges if and only if for all $\varepsilon>0$ there exists an $N$ such that for all $m\ge n\ge N$, $\left|x_nx_{n+1}\cdots ...
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2answers
1k views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
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1answer
494 views

Inverse of the Joukowski map $\phi(z) = z + \frac{1}{z}$

We know the Joukowski map $$\phi(z) = z + \frac{1}{z}$$ which maps the upper semidisc of radius $1$ in the lower half plane, and the lower semidisc of radius $1$ in the upper half plane. What is the ...
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2answers
55 views

$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$

I'm reading the proof that $$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$$ There is a function $$h(z) =\pi \cot (\pi z) -[ \frac{1}{z} + \sum_{n ...
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2answers
245 views

show that $f$ is not integrable on $[0,1]$

show that $f$ is not integrable on $[0,1]$. hint: $M$ where $M = \sup f(x)$ on each subinterval $[X_{i-1},X_i]$ $M \geq \cos(x)-\sin(x)$. Then I'm not very sure about how to prove it. I would ...
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1answer
60 views

Oscilation Property of absolutely continuous functions

I have a question about absolutely continuous function $f:[0,T]\rightarrow \mathbb{R}$. First, as we know, as a function with finite variation, $f$ is almost everywhere differentiable. However, I ...
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2answers
1k views

Proving a complex function is continuous.

I've recently started complex analysis but I have very little background in complex numbers and to make sure I don't fall behind I'm doing some extra exercises one of which is Show $f$ is continuous ...
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2answers
148 views

Prove this integral, the Dirichlet's formula

Show that $$\int\int_R x^{p-1}y^{q-1}dxdy = \frac{\Gamma(\frac{p}{2})\Gamma(\frac{q}{2})}{\Gamma(\frac{p}{2}+\frac{q}{2}+1)},$$ where R is the region bounded by the first quadrant of the circle ...
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1answer
52 views

To express a integrable function as difference.

Assume $f$ is an integrable function on $[0,1]$. I want to find functions $g$ and $h$, so that $f=g-h$ almost everywhere. The functions $g$ and $h$ should be pointwise limits of continuous functions ...
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1answer
86 views

Showing that $\sum_{n=1}^\infty \frac{x}{n(1+nx^2)}$ Converges Uniformly via the Weierstrass M-test

Setting: Let $f_n(x):\mathbb{R} \rightarrow \mathbb{R}$ s.t. $$ f_n(x) = \frac{x}{n(1+nx^2)} $$ and let $f$ denote the series of the $\{f_n\}$: $$ f(x) = \sum_{n=1}^\infty f_n = \sum_{n=1}^\infty ...
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1answer
90 views

Smooth function with all derivatives zero

Assume that $f\colon\mathbb{R\to R}$ is an infinitely often differentiable function, and that a point $a\in\mathbb R$ satisfies $f^{(k)}(a) = 0$ for alle $k\ge 1$. Does there exist an $\epsilon > ...
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1answer
42 views

A characterization of differentiability of a convex function

Let $\phi : \mathbb R^n \to \mathbb R$ be a convex function. For all point $x\in \mathbb R^n$, define the subdifferential as $$\partial \phi(x) = \{ y\in \mathbb R^n | \ \phi(z) \geq \phi(x) + ...
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1answer
76 views

Converse of Barbalat's lemma?

We have a continuously differentiable function $x(t) : \mathbb{R} \to \mathbb{R}$ such that $\dfrac{dx}{dt} \to 0$ as $t \to \infty$. Under which assumption can we argue that $\lim_{t \to \infty} ...
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0answers
43 views

Example of a sequence function which is unbounded

Let $(X,d)$ be a metric space and $T:X\to X$ be self map on $X$. A map $\phi :X\to[0,\infty)$ is said to be sequence function with respect to $T$ if $$\lim_{n\to\infty}\phi(x_n)<\infty$$ whenever ...
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2answers
144 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
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1answer
125 views

Question about injective holomorphic functions on $\mathbb{D}$ and Koebe's quarter theorem

Let $f$ be an injective holomorphic function on $\mathbb{D}$ such that $f(0) = 0$, $f'(0) = 1$. The open mapping theorem implies that $f(|z| < 1)$ contains an open neighborhood of the origin. Then ...
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2answers
107 views

Is this set a manifold?

For which $ ( \alpha , \beta ) \in \Bbb R^2$ set: $\{ (x_1,x_2,x_3,x_4) \in \Bbb R^4 | x_1+x_4= \alpha, x_1 x_4 - x_2x_3 = \beta \}$ is a manifold? I made a Jacobian matrix: $ \begin{bmatrix} ...
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1answer
146 views

Tangent Cone is a cone?

I first give two definitions. Def1: A set $S$ is a cone if $x \in S, \lambda \geq 0 \implies \lambda x \in S$. Def2: Let $S$ be any set (we may assume $\mathbb{R}^n$ with the usual Euclidean ...
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0answers
126 views

Nice convergent subsequence of $\cos(n)$.

This question is related to a few questions which have been posted on the website : Is there a limit of $\cos(n!)$ Converging subsequence on a circle The limit of $\sin(n!)$ Because of the ...
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2answers
4k views

The difference between pointwise convergence and uniform convergence of functional sequences

$f_n$ converges pointwise to $f$ on E if $∀ x ∈ E$ and $∀ \epsilon > 0$, $∃ N ∈ N$ so that $∀ n ≥ N$ we have $|fn(x) − f(x)| < \epsilon$. $f_n$ converges uniformly to $f$ on E if $∀ \epsilon ...
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3answers
37 views

Analysis (Absolute Value )

The question is let $a \in \mathbb{R} $ does not contain 0. Prove that $|a+\frac{1}{a}| \ge 2$. I have no idea how to start this problem and any help on it would be greatly appreciated.
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2answers
43 views

Extreme Value Theorem over the Real Numbers

I'm stuck on where to start with this. I can tell it is to do with the extreme value theorem, but past that point I'm stuck. Any help would be appreciated. If $f(x)$ is continuous on $\mathbb{R}$, ...
3
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4answers
410 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
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1answer
18 views

Need help proving a statement about non-negative functions and integration

![a] I've proved question 2 and I'm on question 3, using the hint, I'm having troubles deducing why $f(x) > \alpha /2$ - I think from there the answer is given (you get a contradiction with $d ...
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1answer
22 views

$f$ extends continuously to the completion $(\bar{X},\bar{d})$

I have to prove of disprove the following fact: Let $(X,d)$ be a metric space and $f: X \rightarrow \mathbb{R}$ be a continuous function. Then $f$ extends continuously to the completion ...
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1answer
210 views

Stirling approximation / Gamma function

Is it possible to obtain the Stirling approximation for the factorial by using the gamma function ? $$\Gamma(z) = \lim_{n \to +\infty} \frac{n! n^{z}}{z(z+1) \dots (z+n)}$$ Any hint ?
3
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2answers
259 views

Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
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0answers
40 views

The space of bounded mean oscillation $BMO(B_R)$, live in the Campanato space $\mathcal{L}^{1, n}(B_R)$

Let $B_R$ be an open bounded ball in $\mathbb{R}^n$. I am trying to show that if $u\in BMO(B_R)$ then $u\in \mathcal{L}^{1, n}(B_R)$ and that \begin{equation} \|u\|_{\mathcal{L}^{1, n}(B_R)}\leq ...
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1answer
136 views

SOR and Gauss-Seidel Method - Confusion

Can anyone explain to me the SOR Method for finding the root(s) of a function? Its supposedly very similar to the Gauss-Seidel method. The Gauss-Seidel method, from my understanding, is similar to ...
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2answers
169 views

Why is the set of all $\infty$-tuples with finitely many non-zero rational terms dense in $\ell_2$?

This statement has been given as an example in the book "Introductory real analysis" written by Kolmogorov and Fomin: The set of all points $x=(x_1,x_2,\cdots,x_n,\cdots)$ with only finitely ...
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3answers
216 views

Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
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9answers
418 views

If $\,\,x+\dfrac{1}{x}=5,\,\,$ find $\,\,x^5+\dfrac{1}{x^5}$.

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
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0answers
63 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
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2answers
451 views

Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}}$ for all $n\in\mathbb N$?

Is there a function that is analytic in an open neighbourhood of $z=0$ and satisfies $$ f\left(\!\dfrac 1 n\!\right)=\dfrac 1 {\sqrt{n}}, $$ for all natural numbers $n$? I guess this problem ...
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3answers
134 views

Showing that $\sinh(\mathrm{e}^z)$ is entire

I am attempting to show that $\sinh(\mathrm{e}^z)$, where $z$ is a complex number, is entire. The instructions of the problem tell me to write the real component of this function as a function of $x$ ...
3
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1answer
193 views

Prove that $\int_{E}f =\lim \int_{E}f_{n}$

I'm doing exercise in Real Analysis of Folland, and got stuck on this problem. I try to use Fatou lemma but can't come to the conclusion. Can anyone help me. I really appreciate. Consider a ...
5
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1answer
261 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
4
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2answers
99 views

How to show $\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$ for continuous $f$?

Let $f$ be a continuous function. I wish to show $$\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$$ I can try to split up the integral over intervals $[0, 1-\delta]$, $[1-\delta, 1]$. ...
0
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1answer
166 views

Special Counterexample to Kakutani's Fixed-Point Theorem

For reference, here is the statement of Kakutani's fixed point theorem. Let $X$ be a compact, convex subset of $\mathbb{R}^n$ and let $f:X\to \mathcal{P}(X)$ be a set-valued function such that $f(x)$ ...
0
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1answer
35 views

Is this correct? $f(z) = z ^2\bar{z}^3$, $f_z(z) = \ldots$

I would like to know if this derivative is right. Let $z = x + iy \in \mathbb{C}$ and $f(z) = z^2\bar{z}^3 = (x+iy)^2(x-iy)^3$. Then $$ \frac{df}{dx} = 2 (x + iy) (x - iy)^3 + 3(x + iy)^2 (x - iy)^2, ...
0
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1answer
51 views

Proof of Projection property of convex set

suppose a set S$\subseteq$ $R^{m*n}$ is convex.Prove that T={$x_1$ $\in$$R^{m}$ :($x_1$,$x_2$) $\in$ S } is convex. This is projection property of convex.Can somebody tell how to prove it.