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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0
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1answer
161 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 ...
2
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0answers
48 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
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4answers
288 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
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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
542 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
75 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 ...
7
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2answers
1k 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 ...
2
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1answer
3k 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
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2answers
47 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
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1answer
174 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
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1answer
57 views

Finding extremal of function $J(x,y,y')=\int\left[y'(x)\right]^{2} + 12x\,y\left(x\right)\,{\rm d}x$

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} + ...
2
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2answers
63 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 ...
12
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0answers
492 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
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1answer
41 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
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1answer
88 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
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2answers
73 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
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1answer
79 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
294 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
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1answer
135 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
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0answers
75 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
4k 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
172 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
65 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
54 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
59 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
498 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
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0answers
74 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
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1answer
146 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
158 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
121 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
61 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
103 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
108 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
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2answers
67 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
322 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
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2answers
269 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
62 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
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2answers
134 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 ...
2
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0answers
94 views

Line Segments for a Triangle

What are the requirements for three line segments to be able to form a triangle? What would be the proof of that? Thanks!
0
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0answers
118 views

Q: Constructing Matrix for a Tight Frame

Consider a frame $\Phi = \{\varphi_1, ..., \varphi_m\}$ for $\Bbb C ^n$ consisting of $m \geq n$ vectors. Denote the frame bounds by $A \leq B$, the Bessel map and its associated $m \times n$ matrix ...
0
votes
1answer
166 views

Which is the correct definition of stationary point for real-valued functions in Euclidean space?

Given a multivariable real-valued function $f$ whose first partials all exist (but which aren't all continuous) at $p$, it is possible that $f$ is not (totally) differentiable at $p$. But since the ...
1
vote
1answer
51 views

Show why given set is not a frame

I am rather new to this material and an explanation of what is happening would be greatly appreciated. At first glance, it seems like the sum of squares is bounded at both ends but I guess I'm ...
3
votes
2answers
118 views

How does one make real functions a differentiable field?

If you want to apply the results of differential field theory to actual $\Bbb R\to\Bbb R$ functions, then first of all you have to find operations that make these functions a field. The trouble is ...
1
vote
2answers
110 views

Why does determining the nature of local extrema for $\mathbb R \to\mathbb R$ functions require twice continuous-differentiability?

In the text Elementary Classical Analysis, why does Marsden specify the condition "twice continuously differentiable" here? Isn't mere twice-differentiability sufficient for the purpose indicated? ...
2
votes
1answer
64 views

Choosing the right regression

I'm trying to analyze my sleep using regression analysis. Each night is rated (dependent variable). I'm trying to explain this rating with, for example, my sleep duration and each night's bed time's ...
2
votes
1answer
24 views

Is $f(n) = 2^{\frac{1}{2}(n^2-n)} / n!$ polynomially bounded?

The numerator counts the number of different adjacency matrices. I think Sterlings approximation helps to anwser my question but I fail to derive the answer. So, is there a polynomial function $g(x)$ ...
0
votes
1answer
36 views

All Partial sums of two given sequences are bounded by a positive constant

Let $\theta \in \mathbb R$ be a non-integer multiple of $2\pi$. Prove that the sequences $(\sin(n\theta))_{n \in \mathbb N}$ and $(\cos(n\theta))_{n \in \mathbb N}$ verify $|S_N|\leq K$ where $K>0$ ...
0
votes
1answer
50 views

Define $f$ on $\Bbb R$ by $f(x)=x^3$ for $x\ge0$ and $f(x)=0$ for $x\lt 0$. Find all $n\in\mathbb N$ such that $f^{(n)}$ exists on all of $\Bbb R$.

Define $f$ on $\Bbb R$ by $f(x)=x^3$ for $x\ge0$ and $f(x)=0$ for $x\lt 0$. Find all $n\in\mathbb N$ such that $f^{(n)}$ exists on all of $\Bbb R$. I am studying for Analysis midterm. I saw this ...