Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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7 views

Computing ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$ for $U(z)= \int_\gamma (z - a)^n\ dz$

Goal: Let $$ U(z)= \int_\gamma (z - a)^n\ dz $$ I'm trying to compute ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$. Attempt: I know that $(z-a)^n$ is the derivative of ...
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1answer
15 views

How to know that $(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$

How to know that $$(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$$ with $z_0$ a root of $z^4+1$. I can check that it is true, but is there a way to tell, by seeing the LHS expression, ...
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2answers
23 views

Infinite series of a function involving an enumeration of rationals on $0,1$

Let $\{ q_n : n \in \mathbb{N} \}$ be an enumeration of the rational numbers in $(0,1)$ and define $f_n(x) = \begin{cases} 0 \qquad \text{if} \; x \in (0, q_n) \\ 2^{-n} \quad \: \text{if} \; x \in ...
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1answer
20 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
2
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0answers
13 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 ...
2
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2answers
28 views

Derivative of a Matrix to a Power

Fix a positive interger $k$ and let $F: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ be the map on $n \times n$ matrices defined by $F(A)= A^k$. Show that $F$ is differentiable at ...
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1answer
24 views

some problem about countable I think? [on hold]

I cannot solve this: Let $\left \{ G_{\alpha } \right \}_{\alpha \in A}$ be a collection of open sets in $\mathbb{R}$ such that $\bigcup _{\alpha \in A}G_{\alpha }=\mathbb{R}$. Show that there is a ...
2
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3answers
24 views

Some problem about Cauchy sequence.

I cannot solve this: Let X be a complete metric space with a metric d. (a) Suppose that the sequence $x_{n}$ in X satisfies $\sum_{n=0}^{\infty}d(x_{n},x_{n+1})<\infty$ Show that $x_{n}$ ...
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1answer
17 views

What are the values of the parameters that make the function differentiable at zero?

I think I might have found a way to solve this problem but I'm not sure if this is correct, if someone could tell me if this is the correct approach or not that would be nice. If it's not the correct ...
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0answers
10 views

about lp estimate of schwarz function

A homework question that I really couldnt find how to start. Prove that for any f in schwarz class $ \lVert f \rVert_{q} \leq C_{p,q} \lVert \nabla f \rVert_{2}^{a} \lVert f \rVert_{2}^{1-a} $ $ ...
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1answer
37 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
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1answer
17 views

Is the inverse of a symmetric positive semidefinite matrix also a symmetric positive semidefinite matrix?

If we let $$S_{++}^n(\mathbb{R})$$ denote the set of all square symmetric positive definite matrix over the real numbers, then is it true if $A\in S_{++}(\mathbb{R}) \implies A^{-1} \in ...
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2answers
21 views

The Lebesgue Integral and the Dirichlet function

I am considering the function $$f(x)=\begin{cases} 1 &\text{if } x\in [0,1]-Q \\{}\\ 0 &\text{if } x\in [0,1] \cap Q\end{cases}$$ I am trying to evaluate this using the Lebesgue integral. ...
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0answers
16 views

integration in five dimensions space part three

I am following this: integration in five dimensions space part two Maybe I need to simplify my question: Find the integration of $\int_{\partial S}-p_1dq_1\wedge dp_2\wedge dq_2$, where $S$ is the ...
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1answer
3 views

Question about outer regularity and inf

Let $\mu$ be a measure. Suppose for every $\varepsilon > 0$, there exists an open set $U \supset E$ such that $\mu(U) < \mu(E) + \varepsilon$. Then must $\mu(E) = \inf\{\mu(U): U \supset E, U ...
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1answer
19 views

Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
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1answer
31 views

Advanced Calc 1 Help [on hold]

Suppose $u,v,u',v'$ are continuous on $R$ and $uv'-u'v$ is never zero. Show that between any two consecutive zeros of $u$ there is a zero of $v$, and similarly for $v$.
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0answers
7 views

Jordan decomposition of sum of two measures

Let $\mu$ and $\nu$ be finite signed measures. Then by the Jordan Decomposition Theorem, we can write $\mu = \mu^{+} - \mu^{-}$ and $\nu = \nu^{+} - \nu^{-}$ where $\mu^{\pm}, \nu^{\pm}$ are unsigned ...
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1answer
27 views

A supremum/Inf property?

Can something justify this equality to me? $$\sup_y \{ \langle y,x\rangle + \inf_z\{f(z) - \langle y,z\rangle \} \} = \sup_y \{ \inf_z \{ f(z) - \langle y,x - z\rangle\} \}$$ I don't understand ...
3
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1answer
42 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
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38 views

Find a map $T$ such that $T (D^∗ ) = D$.

$$D = \left\{(x, y, z)| (7x − 3y − z)^2 + (−3x + 7y − z)^2 + (−x − y + 3z)^2 ≤ 100\right\}$$ $$D^∗ = \left\{(u, v, w)| u^2 + v^2 + w^2 ≤ 1\right\}$$ I'm pretty stuck. My 1st thought is that $u=7x − 3y ...
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0answers
10 views

scale transformation is invariant for H_1

Consider the subspace $H_1$ of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to ...
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0answers
6 views

Integration of characteristic function with varying boundaries

I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals? $$ A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx $$ and $$ ...
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23 views

What is meant by a “general line integral of form $\int_\gamma p\ dx + q\ dy$”?

In his text on complex analysis, Ahlfors speaks of "general line integrals of form $\int_\gamma p\ dx + q\ dy$". I'm curious exactly what is meant by this. I take it that $p$ and $q$ are not ...
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0answers
17 views

An easy (I guess) exercise about limit inferior

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: lim inf$_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq$ lim inf$_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
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1answer
23 views

How to prove that $\pi \frac{e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}}}{1 - e^{2\pi it}} = \frac{\pi}{2\cos\left(\frac{\pi t}{2}\right)}$

How to prove that : $$\pi \frac{e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}}}{1 - e^{2\pi it}} = \frac{\pi}{2\cos\left(\frac{\pi t}{2}\right)}$$ I start with $$e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}} = ...
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0answers
16 views

Computing a complex line integral $dz$ in terms of line integrals $dx$ and $dy$

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ...
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1answer
55 views

question about compactness proof in complete metric space!

I cannot solve this: Let $X$ be a complete metric space. Suppose that for any $r> 0$ there are finite points $x_1, x_2, \dots ,x_n$ such that $N_r(x_1),\dots, N_r(x_n)$ cover $X$. Show ...
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3answers
642 views

How do I solve this definite integral?

$$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$ I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows: $\sin^{4}x + \cos^{4}x = (\sin^{2}x + ...
2
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1answer
21 views

Is the dual of a complete topological vector space always complete?

Let $X$ be a complete topological vector space (over $\mathbb{C}$ say), and $X'$ its dual with the weak*-topology. Then is $X'$ always complete? You may assume $X$ is locally convex if you like.
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1answer
21 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
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0answers
9 views

Difference of two finite Radon measures

Is the difference of two finite unsigned Radon measures still Radon?
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0answers
17 views

any theory about determinant as a function of the elements of a matrix

Let $X=(x_{ij})_{1\le i,j\le n}$ be a n by n matrix where $n\ge3$. Consider the function $f:(R^n)^n\to R$ given by the formula $f(X):=\det(X)$. (a) Assume that $rank(X)=n-1$. Is it true that X is a ...
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0answers
8 views

Finite difference scheme and its stability

The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if ...
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0answers
10 views

How to prove that $ T_t=\exp(tA)$ is a semigroup?

How to prove that $ T_t=\exp(tA)$ ,$t\ge 0$ is a semigroup, where $-A$ is a non-negative operator on a Banach space.
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0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
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1answer
47 views

Absolute convergence when all the rotated series converge

The question here might be standard in some textbook. Let $a_n, n\ge1$ be a series of real numbers. It is evident that if $\displaystyle \sum_{n\ge 1} |a_n|<+\infty$, then $\displaystyle ...
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1answer
46 views

integration in five dimensions space part two

I am following the discussion here: integration in five dimensions space I am doing this problem: Consider the differential form $$a=p_1 \, dq_1+p_2 \, dq_2-(p_1^2+p_2^2+q_1^2+q_2^2) \, dt\text{ in ...
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1answer
30 views

Simple question about a complex valued function

This is taken from an exam. One and only one of the answers is true. Let $f:\mathbb R\longrightarrow\mathbb C$ such that $\lim_{x\rightarrow0}|f(x)|=+\infty$. Hence: a)There exists ...
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1answer
34 views

General conceptual confusion relating to vacuous proofs and quantifier help

I need to prove the statement: Let $x \in \mathbb{R}$. Prove that $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. So I start with the forward implication: If $1 ≤ ...
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2answers
31 views

Find the Fourier series of $\sin^3(x)$ on $[-\pi,\pi]$

I'm having trouble integrating $B_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}\sin^{3}t \,\sin(nt)\,dt$.
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1answer
22 views

Analysis, showing that something is equal 0

Sorry, for the title, but I'm not native English speaker, and actually, I have no idea how to name it in my own language. So, I have $\left[-x^{4}e^{\frac{-x^2}{2}}\right]_{-\infty}^{\infty}$ And I ...
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1answer
30 views

Solution to a stochastic differential equation

I could really do with some help on this question, have no idea where to start. Any advice would be much appreciated, thank u in advance. I am given $$\begin{align}dx(t)&=(1+x(t))dt + x(t) ...
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0answers
35 views

Does $\int_a^b f(z)\ \overline{dz} = \int_a^b f(z)\ dz$? [duplicate]

Question: Attempted Answer: Yes, for if $f = u + iv$ where $u$ and $v$ are real-valued functions, then we have that $$ \int_a^b f(z)\ \overline{dz} = \overline{\int_a^b \overline{f(z)}\ dz} ...
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2answers
39 views

Analysis: Prove the converse

It can be shown that if $\lim_{n\to\infty} a_n = L$, then $\lim_{n\to\infty} |a_n| = |L|$. Is the converse of this result true?
2
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1answer
31 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
0
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0answers
17 views

When does $\int_\gamma f(z) \,dz = \int_\gamma f(z)\, \overline{dz}$?

Suppose $f:[a,b]\rightarrow \mathbb{C}$ satisfies $f = u + iv = u$ (i.e., $v = 0$). Then is it correct to assert that $$ \int_\gamma f(z)\ dz = \int_a^b f(\gamma(t)) \gamma'(t)\ dt = \int_a^b ...
15
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5answers
1k views

Limit is found using polar coordinates but it is not supposed to exist.

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of ...
0
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0answers
23 views

Visual representation of the complex line integral

Wikipedia has a visual depiction of a line integral of a scalar field: http://en.wikipedia.org/wiki/Line_integral#Vector_calculus I'm curious if this graphical representation could be used to ...
0
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1answer
57 views

why do we take this partition?

I am looking at the following exercise: Let $f:[0,1] \to \mathbb{R}$ with: $f(x)=\left\{\begin{matrix} 1 &,x=0 \\ 0 &,0<x \leq 1 \end{matrix}\right.$ Show with the definition that $f$ ...