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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11
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6answers
4k views

Companions to Rudin?

I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I ...
5
votes
1answer
36 views

Prob. 8, Sec. 3.5 in Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications

Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications Prob. 8, Sec. 3.5 $\DeclareMathOperator{\span}{span}$Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$, and let $M = ...
1
vote
1answer
17 views

homeomorphism inbetween the $\mathbb{R}^n$ and an open unit cube

I want to find a homeomorphism that maps the open cube $W = (-1,1)^n\subseteq \mathbb{R}^n$ to the $\mathbb{R}^n$. I know that these two are homeomorphic, but I don't know where to start when it ...
1
vote
2answers
28 views

Closed subset of $\mathbb{R}^2$ induced by the graph of a function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. I want to show that the set $$A := \{(x, y) \in \mathbb{R}^2|y ≤ f(x)\}$$ induced by $f$ is a closed subset of $\mathbb{R}^2$. Now ...
2
votes
2answers
176 views

Measure on Baire space

Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$? Some ...
0
votes
1answer
17 views

incomplete vector space of continously differentiable functions

Consider the vector space $C^1[a, b] := \{f: [a, b] \to \mathbb{C} \space |\space f$ continuously differentiable$\}$. I now want to show that ($C^1[-1, 1]$, $||.||_\infty)$ is not complete (using ...
1
vote
2answers
406 views

Tricks to solve inequalities

I am wondering if there are some tricks to solve inequalities which are not manageable analytically. For example consider the inequality (say we restrict on positive $x$): $\displaystyle \frac {\text ...
2
votes
0answers
12 views

How do you nondimensionalize/rescale the following equation? [on hold]

$$N1' = a(1- N1/k)N1 - bN1N2 $$ $$N2' = -cN2 + dN1N2$$ To be the dimensionless form (Where $t$ is $\tau$ here) $$dx/dt = (1-x)x - β1xy $$ $$dy/dt = -αy + β2xy $$ where $β1 = b/a$, $β2 = dk/a$ ...
0
votes
0answers
27 views

Function not equal a.e. to continous function on real line and on circle

I am looking for a proof of the following fact: Suppose that $H: \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with period $1$. Suppose further that there is no continuous function ...
0
votes
1answer
38 views

When Cauchy-Riemann equations hold, WHERE does it tell you $f$ is analytic?

So I've been asked 'Is $f$ analytic anywhere? Everywhere? Justify your answer.' My function is $f(z)=z^2-iz+iz^2$ which I've expressed as $f=u+iv$ so I can use Cauchy-Riemann equations to test to ...
0
votes
0answers
25 views

A special argument to derive the derivative of the determinant

In this short note, in the section "a better result" the author says: [...] if $\Phi(t)$ is the identity [...] then $$ \frac{d}{d t} \operatorname{det} \Phi(t) = \operatorname{tr} \dot{\Phi}(t) ...
0
votes
0answers
25 views

The Analysis of Linear Partial Differential Operators I Prerequisites

I am a graduate level student in Mathematics and I would like to study the books titled "the analysis of linear partial differential operators I-IV" by Hörmander. As I have been away from mathematics ...
2
votes
0answers
24 views

Proof on Riemann's Theorem that any conditionally convergent series can be rearranged to yield a series which converges to any given sum.

I am looking at the proof of the following theorem from Apostol's Mathematical Analysis. I am having trouble showing the last part that the author left to the reader. I'm trying to show that $y$ is ...
1
vote
2answers
241 views

Express complex Bessel function in terms of functions taking real arguements

I want to use the Bessel function in C++. Since this one is not implemented there for complex arguments, I am looking for a way to express the bessel function(first and second kind) as: ...
84
votes
19answers
19k views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin x}{x} \ dx = \frac{\pi}{2}$$ Well, can anyone ...
1
vote
0answers
35 views

$f(t,x)$ defined and continuous implies $dx/dt=f$ has solution

Let $f(t,x)$ be defined and continuous for $a\leq t\leq b$ and $x \in \mathbb{R}^n$. Show that the problem \begin{equation} \begin{cases} \frac{dx}{dt} &=f(t,x) ,\\ x(a) & = x_0 ...
0
votes
2answers
17 views

Proving a norm on the space of differentiable functions

I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that $$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$ defines a norm on ...
2
votes
1answer
23 views

Prove or disprove: $p(x)$ diverges to infinity for $a_{n}>0$

Prove or disprove that for any $n$ degree polynomial, $p(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{1}x+a_{0}$, if $a_{n}>0$, then $p(x)$ diverges to infinity as x tends to infinity. This is not homework.
-1
votes
4answers
57 views

One-one and continuous $\implies$ strictly monotonic [on hold]

If $f$ is one-one and continuous on $[a,b],$ then prove that $f$ must be strictly monotonic. Hints will be appreciated
1
vote
2answers
41 views

Integral over a sequence of sets whose measures $\to 0.$

If $ f \in L_p$ with $1 \leq p \leq \infty $ and ${A_n}$ is a sequence of measurable sets such that $ \mu (An) \rightarrow 0,$ then $ \int_{A_n} f \rightarrow 0$. Can someone give me a hint?
2
votes
3answers
57 views

Show complex solutions exist

Let A be a complex number and B a real number. Show that the equation $\,\lvert z^2\rvert+ \mathrm{Re}\, (Az) + B = 0\,$ has a solution iff $\,\lvert A^2\rvert \geq 4B$. If this is so, show that the ...
-1
votes
0answers
23 views

Weierstrass approximation.

Show that the algebra generated by the pair of functions $\{l,2\}$ is dense in the set of all even functions that are continuous on $[—1, 1]$ I only know the above algebra dense in continuous function ...
1
vote
2answers
31 views

Proving the convergence of the improper integral $\int_0^1 \operatorname{ln}(\operatorname{sin}x)dx$

I'm trying to prove that \begin{equation*} \int_0^1 \operatorname{ln}(\operatorname{sin}x)dx \end{equation*} converges. I tried to show this by decomposing \begin{equation*} ...
1
vote
1answer
30 views

Fundamental solution and Green's function

I am currently dealing with Poisson's equation $- \Delta u= f $ on some open domain $U$ and $u =g$ on the boundary $\partial U.$ Now a fundamental solution is a solution to $- \Delta u(x) = ...
0
votes
0answers
7 views

Union and intersections of $L_p$ spaces and proper subsets.

Let $X= [0,1), S= \mathcal{B}_{[0,1)}, \lambda = $Lebesgue measure in $\mathcal{B}_{[0,1)} $ Prove (a) $L_p(\lambda) \subsetneq \bigcap_{0<r<p} L_r(\lambda) $ for every fixed p. (b) ...
2
votes
1answer
83 views

How do I solve this integral ? $\lfloor \rfloor$

I haven't come across this yet, but the question is to solve this definite integral: $$\int_{e}^{4+e} (3x- \lfloor 3x \rfloor)dx$$ What is obviously causing problems is the whole part of the number ...
0
votes
0answers
43 views

How to show the following inequality with f(x) = 1/(1+x^2)?

How to show the following inequality : Let, $f(x) = \frac{1}{1+x^2}$ . Then show that $$\int_{\mathbb{R}\setminus (-1,1)} \left( \sqrt{f(x+y)}-\sqrt{f(x)}\ \right)^2 \ \frac{dy}{y^2} \leq C f(x) ...
-1
votes
1answer
29 views

Prove whether the following subset is open, closed, or neither

We've sketched the subset, and we now need to prove whether it's closed, open or neither. We have no idea where to start so any help is appreciated, thanks. $S\subset \mathbb C$
1
vote
1answer
54 views

Is it or not a continuous embedding?

Please I have this two spaces $$C_{\theta}=\{u\in C(\overline{\Omega}), \sup (|x|^{\theta} |u(x)|)<\infty\}$$ with the norm $\displaystyle||u||_{\theta}=\sup_{\Omega}(|x|^{\theta} |u(x)|)$ and ...
3
votes
3answers
109 views

Is there an algebraic solution to $e^{-x/a}+e^{-x/b}=1$ ($a\neq b$, $a,b$ constants)?

Is there an algebraic solution for the to find the intersection of the following two functions for values of $x\geq 0$: $$f_1(x)=1-2e^{-x/a}=f_2(x)=-1+2e^{-x/b}$$ $a$ and $b$ are positive constants. ...
1
vote
2answers
187 views

Uniform convergence for $x\arctan(nx)$

I am to check the uniform convergence of this sequence of functions : $f_{n}(x) = x\arctan(nx)$ where $x \in \mathbb{R} $. I came to a conclusion that $f_{n}(x) \rightarrow \frac{\left|x\right|\pi}{2} ...
3
votes
3answers
60 views

What the terms “basis functions” and “orthogonal” denote in the case of signals?

I am a beginner. I have read that any given signal whether it is a simple or complex one, can be represented as summation of orthogonal basis functions. Also, there are many ways through which basis ...
0
votes
0answers
13 views

Differntiating matrix functions $f : \mathbb R^{n\times m} \to \mathbb R^{p\times q}$

How would you differentiate matrix functions $f : \mathbb R^{n\times m} \to \mathbb R^{p\times q}$ like for example $f(X) = X^T \cdot X$? There are no directional derivatives in the usual sense, and ...
1
vote
0answers
16 views

help with an asymptotic for a certain product

I'm having difficulty finding an asymptotic formula for the following product: $$ k^{\alpha}\prod_{1 \leq i \leq N \atop i \neq k} (i^{\alpha}-k^{\alpha})$$ where $N$ is a parameter tending to ...
1
vote
1answer
8 views

forms of the Romberg Method equation

My teacher wrote the this equation for the Romberg method $ I_{j,k}=\frac{4^j I_{j-1/k+1}-I_{j-1/k}}{4^j-1} $ Is this the right equation? Most the equations I looked at online for the Romberg ...
0
votes
1answer
39 views

Zeroes of a complex function

Let $\mathcal{O} = \{z: f$ has zero of order $\infty$$\}$. We claim that $\mathcal{O}$ is open. To prove this we note that, if $f$ has a zero of infinite order at $z_{0}$, then all the Taylor ...
1
vote
1answer
28 views

Closed point problems in $l$-spaces, namely $ l^1$ and $l^\inf$

a) Give an example of a closed convex set $C$ and point $x$ in $l^\infty$ such that the closed point in $C$ to $x$ is not unique. Solution: So I was thinking that a closed convex set would be the ...
2
votes
1answer
28 views

High Dimensional Rotation Matrices As Product of In-Plane Rotations

Lately I've been thinking a lot about how to find high-dimensional rotation matrices. In particular, can any rotation in $n$-dimensional space be represented as the product of $2$D plane rotations? ...
2
votes
1answer
32 views

Reduce PDE to ODE

Maybe you don't want to check all the details, but could look at a few equations here. Would you mind leaving a comment that you at least some part looks okay?- This way, I know that at least somebody ...
3
votes
0answers
12 views

Are not all neighborhoods of $0$ in a locally convex space absorbent?

A locally convex space (LCS) can be defined as a topological vector space (i.e. scalar product and sum are continuous) whose topology is generated by translation of a family of balanced and absorbent ...
0
votes
1answer
21 views

Determining Bounded Variation

I want to check if $f(x) = \cos(\frac{x}{2})$, $x \in [0,2\pi]$, is of bounded variation. I am following the definition here: http://www.math.ubc.ca/~feldman/m321/variation.pdf However, i'm not sure ...
1
vote
2answers
839 views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in ...
1
vote
1answer
88 views

Rudin's Proof on Riesz Representation Theorem

The proof is from Rudin's Real and Complex Analysis. I am having a hard time understanding part of the Riesz Representation Theorem The Theorem states: Every open set $E$ satisfies ...
3
votes
2answers
52 views

Example 5, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: Is this map always continuous?

Let $(X, \Vert \cdot \Vert)$ be a given normed space that has elements other than the zero vector $\theta_X$. And let $T \colon X-\{\theta_X \} \to X$ be defined by $$T(x) \colon= \frac{1}{\Vert x ...
2
votes
1answer
67 views

Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=1+|a_{1}|+\cdots ...
0
votes
0answers
28 views

Rudin's Riesz Representation Theorem

The proof is from Rudin's Real and Complex Analysis. The Theorem states: Every open set $E$ satisfies $$\mu(E)=\sup\,\{\mu(K):K\subset E,\,K\,\,compact\}\,\,\,\,\,\,(3)$$ (Note: Here $\mu$ is a ...
3
votes
2answers
14 views

Total Variation of Constant Function

I want to prove that the total variation of, $f:[a,b] \to \mathbb{R}$, is $0$ iff $f$ is a constant function, but i'm not entirely sure how. I can intuitively see why that it would be zero since the ...
2
votes
1answer
26 views

Reference for the theory of analytic functions

Question: Are there any good references for a theory on analytic functions? Lagrange attempted to develop analysis from this vantage point. Are there any texts that take a similar approach but, ...
-1
votes
0answers
18 views

Increase in sales losses relative to price

If a price increase on a product from $7.25 to $9 was projected to have a sales loss of 200,000 units; and a price increase from $7.25 to $10.10 was projected to have a sales loss of 1,000,000 units; ...
1
vote
2answers
20 views

If $f \in L_4([0,1])$ then $f \in L_2([0,1])$ and $||f||_2 \leq ||f||_4$

If $f \in L_4([0,1])$ then $f \in L_2([0,1])$ and $||f||_2 \leq ||f||_4$ I am not sure how to prove the first statement, we say that $f \in L_P$ if $\int |f|^p < \infty$. Then if $f \in ...