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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5
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2answers
1k views

Series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$

Is it true that for $x\in[0,2\pi]$ we have $$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$$ How can I prove it? For other intervals what is the value of above ...
2
votes
1answer
150 views

How to find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ satisfying $f(XY)\le\min{(f(X),f(Y))}$

Let $M_n(\Bbb R)$ be the set of all real $n\times n$ matrices. Find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ such that $$f(XY)\le\min{(f(X),f(Y))}$$ for all $X,Y\in M_n(\Bbb R)$. ...
0
votes
1answer
93 views

Let $\delta$ be a linear functional equipped with the sup-norm. Show that $\delta$ is bounded and compute its norm.

Let $\delta:C([0,1])\rightarrow\mathbb{R}$ be the linear functional at the origin: $\delta(f) = f(0)$. If $C([0,1])$ is equipped with the sup-norm $$\|f\|_{\infty} = \sup_{0\leq x\leq 1}|f(x)|.$$ Show ...
4
votes
0answers
60 views

How prove that $\sum_{n=1}^{\infty}\frac{\pi{(n)}}{n^2}<\infty$ for some bijective $\pi(n)$ [duplicate]

Let $\mathbb{N}=\{1,2,\cdots\}$. Does there exist a bijective function $\pi:\mathbb{N} \to \mathbb{N}$ such that $$\sum_{n=1}^{\infty}\dfrac{\pi{(n)}}{n^2}<\infty ?$$ My try: note ...
1
vote
1answer
37 views

What is the Density Theorem in this context?

I have this exercise: Define $K:C([0,1])\rightarrow C([0,1])$ by $$Kf(x) = \int_0^1 k(x,y)f(y)dy,$$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. Prove that $K$ is bounded and ...
1
vote
1answer
54 views

Prove that $f$ is bounded if it converges as $x \rightarrow \infty$ and $x \rightarrow -\infty$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x) \rightarrow 7 $ as $x \rightarrow \infty $ and $f(x) \rightarrow -25 $ as $x \rightarrow -\infty $ and Prove that ...
0
votes
0answers
102 views

invertible differentiable function proof

Suppose that $f$ is an invertible differentiable function, that the domain of $f^{-1}$ contains an interval around $a$, and that $f^{-1}$ is continuous at a. Prove that $f^{-1}$ is differentiable at ...
1
vote
1answer
33 views

If $E_i$ is compact, then $\cup_{i=1}^{n} E_{i}$ is compact? $\cap_{i=1}^{\infty} E_{i}$ is compact?

If $E_i$ is compact, then $\cap_{i=1}^{n} E_i$ is compact: Since $E_i$ is compact then it is closed and bounded Then there exists $a_i$ and $b_i$ such that for all $x \in E_i$, $a_i \leq x \leq ...
3
votes
3answers
179 views

What other definite integrals can be computed in a manner similar to $\int_{-\infty}^\infty e^{-x^2}dx$?

The technique for computing $\int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}$ by computing the integral squared using polar coordinates is well known. Are there any other integrals that can be computed ...
2
votes
1answer
33 views

Prove that $(|u-s|+|x-y|)^2\leq 2|u-s|^2+2|x-y|^2$.

Prove that $(|u-s|+|x-y|)^2\leq 2|x-y|^2+2|u-s|^2$. My professor used this inequality for a proof last week. How would one prove this? I thought about using the Cauchy-Swartz inequality. This is ...
0
votes
1answer
108 views

Why does the limit exist on this interval?

Does the $\lim_{x \to x_0} f(x)$ exist at every point $x_0$ in $(-1,1)?$ I answered False, but the correct answer is True. Why? My thoughts: $f(x)$ is not the same number as $x \rightarrow 1$ from ...
1
vote
0answers
45 views

$f(x) = \sum_{k=1}^{\infty} \dfrac{1}{k} \sin \left(\dfrac{x}{k+1} \right)$

Show that $$f(x) = \sum_{k=1}^{\infty} \dfrac{1}{k} \sin \left(\dfrac{x}{k+1}\right).$$ converges pointwise on $\mathbb{R}$ and uniformly on each bounded interval in $\mathbb{R}$, to a ...
3
votes
4answers
247 views

Integral from zero to infinity of $\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda$

I know that the value of the integral is as follows $$\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda =z^a \frac{\Gamma(1-a)}{a}$$ However, how exactly the integral is calculated? ...
2
votes
3answers
71 views

Could anybody check this integral?

in a lengthy calculation by hand I got that $$ \frac{2}{\pi \sigma_k} \int_{-\infty}^{\infty} \frac{sin^2(\frac{\sigma_k}{2}(v_gt-x))}{(v_gt-x)^2} dx =1$$ Now I was wondering whether there is ...
1
vote
1answer
49 views

Is such a function differentiable?

Let $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function whose both partial derivatives of the first order exist on a dense vsubset $D\subset \mathbb R^2$ and these partial derivatives ...
0
votes
1answer
312 views

Lebesgue Measure of Intersection of two sets

I have a one question relating to one property of Lebesgue Measures. If I have two sets, say $A \subset B $ and $B \subset C$ (closed or open) and Lebesgue measure is denoted by $\lambda$. Then my ...
2
votes
1answer
74 views

Some more parameter integrals

It seems that the following formulas hold : $$\int_{0}^{\infty} \frac{1}{\sqrt{x^{2n}+1}} dx = \frac{\Gamma(\frac{n-1}{2n})\Gamma(\frac{2n+1}{2n})}{\sqrt\pi}$$ for any integer $n > 1$ and ...
6
votes
2answers
625 views

complex conjugates of holomorphic functions

I came across this question whilst doing some research into complex analysis, and I just can't see what to do! Let $f(z)$ be a holomorphic function on $\mathbb{C}$. Show that ...
1
vote
1answer
2k views

Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
1
vote
2answers
45 views

How to prove (-b)(-d)=bd?

George Peacock proved this by the so-called "Principle of the Permanence of Equivalent Forms" Detailed as below: Because (a-b)(c-d)=ac-ad-bc+bd holds for positive integers (that is general form ...
1
vote
1answer
173 views

How prove this inequality $f(a)\le f(b)$

Suppose $f(x)$ is continous on $[a,b]$,and for any $x_{0}\in [a,b]$. the limit $$\varliminf_{x\to x^{-}_{0}}\dfrac{f(x)-f(x_{0})}{x-x_{0}}\ge 0$$ show that $$f(a)\le f(b)$$ My try: I found ...
0
votes
1answer
44 views

Finding extremal of function J

Find a curve passing through $\left(0,0\right)$ and $\left(1,1\right)$ that is an extremal for the functional $\displaystyle{{\rm J}\left(x,y,y'\right) = \int\left\{\left[y'(x)\right]^{2} + ...
1
vote
2answers
59 views

Prove that $\lim_{n \to \infty} \int_0^2 e^{ x^2 / n}\,{\rm d}x$ exists and evaluate it.

I need to show that this limit exists and then evaluate it. It is from a section on uniform convergence of sequences. I know that if $f_n \rightarrow f$ uniformly and each $f_n$ is integrable, then I ...
4
votes
0answers
281 views

A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261) *3. Using Ex. 2, show that $p + q$ maps $\Omega$ in a one-to-one manner onto a region bounded by convex contours. Comments: ...
3
votes
1answer
38 views

Intuition tells me this function doesn't converge uniformly but not sure how to put it formally?

$\mathbb{R}$ is the domain. Let $$f_n(x) = \frac{4n}{n+x^2}$$ As $n$ becomes large the $x^2$ term becomes insignificant and the function converges to $4$ pointwise. Now it seems to me that no matter ...
3
votes
1answer
77 views

Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$.

I want to Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$. Where $A$ is a matrix. I have a proof that involves Jordan Blocks. The proof is long and involved but it not ...
2
votes
2answers
69 views

Proving that an operator $K$ is bounded and $||K|| = \max_{0\leq x\leq 1}\bigg\{\int_0^1|k(x,y)|dy\bigg\}$

Define $K:C([0,1])\rightarrow C([0,1])$ by $$Kf(x) = \int_0^1 k(x,y)f(y)dy,$$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. Prove that $K$ is bounded and $$||K|| = \max_{0\leq ...
2
votes
1answer
75 views

Prove that $\frac{n^2+(-1)^nn+2}{7n^2+3}$ converges to $\frac{1}{7}$

I want to show that $\frac{n^2+(-1)^nn+2}{7n^2+3}$ converges to $\frac{1}{7}$ using the definition of convergence. Skratch work: I need ...
3
votes
1answer
232 views

The 2nd total derivative (Hessian) of a composite function -Version 1

Let $f\in C^2(\mathbb R^n,\mathbb R)$ and $Df:A\subset\mathbb R^n\to L(\mathbb R^n,\mathbb R)$ so that $Df_x:\mathbb R^n\to\mathbb R$ is $f$'s total derivative at $x\in\mathbb R^n$. ...
5
votes
1answer
105 views

Continuity of Green's function

Suppose $\Omega \subset \mathbb C$ is a region (open and connected set) and let $$g(z,z_0)=G(z,z_0)-\log|z-z_0| $$ be its Green's function with pole at $z_0 \in \Omega$. Here $G(z,z_0)$ is the ...
2
votes
0answers
73 views

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$? In this paper on quantum computing (last paragraph of page 25), Dorit ...
0
votes
2answers
3k views

Prove that inf(A+B) = infA + infB

A+B={a+b} I proved that the set A+B is bounded below. Now I'm stuck on how to prove that inf(A+B) = infA + infB
2
votes
1answer
151 views

Filling in a small detail in Evans' PDE (chap 6 - second-order elliptic equations)

I'm reading Evans' PDE book. There's a tiny detail in one of his proofs that I'm not understanding. The proof in question is Theorem 1, chapter 6.3.1, on interior regularity of second-order elliptic ...
1
vote
1answer
55 views

Borel sets on the plane

Let's say we have two sigma algebras $D_1$ and $D_2$ both of which contain open intervals. We know that the Borel sigma algebra $B(R)\subset D_{1}\cap D_{2}$. I'm having difficulty proving that ...
0
votes
1answer
44 views

Showing that a set $D$ is closed and open

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) \mid |t-t_0|\leq T, |u-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
1
vote
1answer
58 views

Showing that the sequence converges knowing that three other sequences converge

I have a question in Analysis. Knowing that $x_{2n}$, $x_{2n-1}$, $x_{3n}$ converge, how can I show that $x_{n}$ converges?
2
votes
1answer
394 views

Properties of $||f||_{\infty}$ - the infinity norm

Prove that $||f||_{\infty}$ is the smallest of all numbers of the form $\sup\{|g(x)|: x\in X\}$, where $f=g$ ($\mu$ almost everywhere). In addition, if $f$ is a continuous function on the measure ...
3
votes
0answers
73 views

For the sequence $u_n$, $u_n \to +\infty \iff \frac{1}{u_n} \to 0$

Let $u=(u_n)_{n \in \mathbb{N}}$ be a sequence such that $u_n \neq 0$, $u_n \to +\infty$, for $ n \to +\infty$. Proof that $u_n \to + \infty , ( n \to +\infty) \iff \left(( \exists n_0 , ...
2
votes
1answer
136 views

Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$

Let $1<p_0<\infty$. Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$, but $f_k$ does not converge in $L^{p_0}$. ...
2
votes
3answers
134 views

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$.

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$. Also suppose there exist $g\in L^{p_1}$ and $h\in ...
0
votes
1answer
109 views

Derive asymptotic behavior of inverse of the normal cdf with respect to 2^n

I have a normal distribution $\mu = 0$ and $\sigma = 0.58n$ where $n > 0 $ and I am trying to derive the asymptotic behavior of the following equation: ...
2
votes
1answer
59 views

Help with understanding a proof on Ordinary Differential Equations

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) | |t-t_0|\leq T, |u(t)-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
3
votes
2answers
88 views

French metro metric: difficulty to prove that $d(x, y) = 0\iff x = y$.

I think that it is related to the special definition of the metric in my book: $$d(x, y) = \begin{cases}||x - y||,\mbox{ if }\exists \alpha\in\mathbb{R}: \alpha x + (1-\alpha) y = 0;\\ ||x|| + ||y||, ...
0
votes
2answers
97 views

Different ways of defining Absolute Value

Calculus I presents this definition of absolute value: $$f(x)=y=|x|=\left\{\begin{array}{}\;\;\;x&\text{ if}\,\,\,x\geq 0\\-x&\text{ if}\,\,\,x<0\end{array}\right.$$ But you can also ...
-2
votes
2answers
36 views

Proving -(a+b) = (-a) + (-b) using A1-A4 only.

A1: + is commutative. A2: + is associative. A3: 0 is an additive identity. A4: -a is an additive inverse of a. Thanks.
0
votes
3answers
62 views

Calculate the limit:

I need to calculate: $$\lim_{x\rightarrow \frac{\pi }{4}}\frac{\sin2x-\cos^{2}2x-1}{\cos^{2}2x+2\cos^{2}x-1}$$ I replaced $2\cos^{2}x-1=\cos2x$ and $\cos^{2}2x=1-\sin^{2}2x$, so this limit equals ...
0
votes
1answer
249 views

Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed. To prove that $S+T$ is closed I have to show that for any limit point $x$ ...
2
votes
2answers
234 views

Accumulation points of the set $S=\{(\frac {1} {n}, \frac {1} {m}) \space m, n \in \mathbb N\}$

The exercise is to find the accumulation points of the set $S=\{(\frac {1} {n}, \frac {1} {m}) \space m, n \in \mathbb N\}$ I'm trying to prove that if $A$={accumulation points of the set $S$}, then ...
2
votes
2answers
61 views

Let $f : X \to Y$ be a function and $E \subseteq X$ and $F \subseteq X$. Show that in general

Let $f:X\to Y$ be a function and $E\subseteq X$ and $F\subseteq X$. Show that in general $f(E − F)\nsubseteq f(E) − f(F)$. I have no idea about how to prove this; and could anyone please explain ...
4
votes
2answers
105 views

Clarification on a proof involving cluster point

Definition of cluster point- Let $A \subseteq \mathbb{R}$. A point $c\in\mathbb{R}$ is a cluster point of $A$ if for evert $\delta>0$ there exists at least one point $x\in A$, $x\neq c$ such that ...