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
votes
2answers
177 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 ...
2
votes
1answer
64 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 ...
0
votes
1answer
38 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 ...
0
votes
2answers
81 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 ≤ ...
0
votes
2answers
39 views

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

I'm having trouble integrating $B_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}\sin^{3}t \,\sin(nt)\,dt$.
0
votes
1answer
32 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 ...
1
vote
0answers
36 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} ...
0
votes
2answers
50 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
votes
1answer
72 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
votes
1answer
42 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 ...
0
votes
1answer
63 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 $f$ ...
3
votes
0answers
122 views

Is there a simple and fast way of computing the residue at an essential singularity?

Is there a simple and fast way of computing the residue at an essential singularity ? I mean if we have a pole of order $n$ at $c$ we can use the formula : $$\mathrm{Res}(f,c) = \frac{1}{(n-1)!} ...
2
votes
1answer
2k views

How to show $\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$

How to show $$\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$$ I begin with $$\sin(x+iy) = \frac{e^{x+iy}-e^{-x-iy}}{2i} = \frac{e^xe^{iy}-e^{-x}e^{-iy}}{2i}$$ $$ = ...
0
votes
1answer
59 views

Plot of a domain in the complex plane

I am trying to plot the following domain in the complex plane: $\lbrace x\in\mathbb{C}|\: |x^{2}-1|<r\rbrace$ for some $r>1$. I know that in general to take a square root of a complex number ...
2
votes
2answers
208 views

Real Analysis - Uniform Convergence Problem

So I screwed up a problem on my exam. I know that now. But pure mathematics is as difficult and terrifying as it is rewarding for me, and I can't let this go! If someone could tell me if the following ...
1
vote
1answer
49 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
-1
votes
2answers
511 views

Proving uniform continuity of absolute value

Prove that the function $f(x) = |x-a| - |x-b|$ is uniformly continuous on $\mathbb{R}$.
0
votes
3answers
59 views

Show a function is not continuous

let $g(x) = x - \lfloor{x}\rfloor$ and I want to show that the function is not continuous. I want to use this definition im pretty sure: "For every open set U in $R$, $f^{-1}$ U is open" But I am ...
0
votes
1answer
63 views

Complex number, strangely written

Find all the complex solutions of the equation: $$\frac{z^3}{i} = 1$$ I mean is this the same thing as $$z^3 = i$$? Because I don't understand why my teacher would put it like that on a test. At ...
2
votes
2answers
132 views

Dual Mapping Preserves Linear Independence if and only if Original Mapping is Surjective

Here is my question: Let $V$ and $W$ be finite-dimensional vectors spaces over a field $F$ and $f:V \rightarrow W$ a linear map. Show that $f$ is surjective if and only if the image under ...
0
votes
0answers
42 views

Computing $\int_\gamma \overline{f(z)}\ dz$

Background: My question concerns a calculation involving the integral $\int_\gamma \overline{f(z)}\ dz$. Consider that if we write $f = g + ih$ with real-valued functions $g$ and $h$, and similarly ...
1
vote
1answer
33 views

Does $\int_a^b \overline{f(z)}\ dz = \int_a^b u(t)\ dt - i \int_a^b v(t)\ dt$?

Hypothesis: Let $[a,b] \subseteq \mathbb{R}$ and $f = u + iv$ with domain $[a,b]$. Question: Do we have that $$\int_a^b \overline{f(z)}\ dz = \int_a^b u(t)\ dt + i \int_a^b -v(t)\ dt = \int_a^b ...
1
vote
3answers
40 views

How to prove $ z^n - z^n_0 = (z-z_0) \sum_0^{n-1} z^kz_0^{n-1-k} $ [duplicate]

I want to prove that with $z_0$ a root of $1+z^n$, I have $$ z^n - z^n_0 = (z-z_0)\sum_0^{n-1} z^kz_0^{n-1-k}$$
2
votes
1answer
153 views

Equicontinuous family of sequence of functions

We are given a sequence of real valued functions $\{g_n\}$ that are defined and continuous on the unit sphere $S$ and differentiable inside it (except at the boundary of the sphere $S$ Also, it is ...
0
votes
1answer
43 views

Maximize arccos-function

I need to find a maximum of the function $$y=\arccos\left(\frac{29+12x\sin(22)+6x\cos(22)+x^2} {\sqrt{x^2+6x\cos(22)-20x\sin(22)+109}\sqrt{x^2+6x\cos(22)-4x\sin(22)+13)}} \right) $$ between x=0 and ...
0
votes
3answers
34 views

How to find the $n$ zeros of $\displaystyle1+z^n$?

How to find the $n$ zeros of $1+z^n$?
1
vote
0answers
24 views

Using the chain rule in $\mathbb{R}^n$

Suppose that the function $\psi:\mathbb{R}^2 \to \mathbb{R}$ is continuously differentiable. Define the function $g:\mathbb{R}^2 \to \mathbb{R}$ by $g(s,t) = \psi(s^2t,s)$ for $(s,t) \in ...
0
votes
1answer
26 views

Is $\sum_0^\infty (-1)^k z^{k-1}$ equal to $\sum_{-1}^\infty (-1)^{k+1} z^{k}$

Is $$\sum_0^\infty (-1)^k z^{k-1}$$ equal to $$\sum_{-1}^\infty (-1)^{k+1} z^{k}$$ i.e am I allowed to reindex the beginning of series ?
1
vote
0answers
34 views

Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
3
votes
1answer
31 views

bessel function maximizer

I try to find global maximum for $ \frac{J_2(x)}{x^2} $ I suspect it happens at x=0 ( plotting the graph) where the value of the function is $ \frac{1}{8} $ I know local maximizers are at zeros of ...
0
votes
2answers
27 views

How to compute $f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$

How to compute this serie : $$f(z) = \sum_0^{\infty} (1+2i+(2+i)(-1)^k)^{-k}z^k$$ The serie is convergent if $|z| < \sqrt{2} $ I can find that $$f(z) = \sum_0^{\infty} 3^{-2k}(1+i)^{-2k}z^{2k} + ...
2
votes
1answer
145 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
0
votes
2answers
40 views

Finding numbers $a$ and $b$ for a complex number

Problem. Given a complex number $$z=2-2i$$ Find numbers $a$ and $b$ such that $$a+ib = \frac{1}{z}$$ I tried multiplying both sides by $z$ and got $$(a+ib)(2-2i)$$ $$= 2a-2ai+2bi-2bi^2$$ ...
1
vote
0answers
52 views

Curve with second derivative identically zero

I just solved the following exercise: Let $I = (a,b)$ be an open interval, $a,b$ possibly equal to $\pm \infty$ and $\alpha: I \to \mathbb R^3$ a smooth parametrized curve that has the property that ...
3
votes
1answer
207 views

Some questions about proof of Theorem 2.43 in Baby Rudin

I will include the proof here and highlight the parts that are giving me trouble. Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof ...
1
vote
0answers
59 views

Why is it an interior minimum

I just solved the following exercise: (Here $I=(a,b)$ is an open interval, possibly $a,b=\pm \infty$ and $\alpha : I \to \mathbb R^3$ is smooth.) Let $\alpha$ be a parametrized curve that does not ...
2
votes
1answer
84 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
5
votes
5answers
225 views

infinitely descending natural numbers

Show that there is no infinitely descending sequence of natural numbers. I was thinking that there exists no infinite descending chain on the natural numbers, since every chain of natural numbers has ...
3
votes
1answer
145 views

Density of a set of functions in Schwartz space

I have a difficulty doing the following problem: Let $S(\mathbb{R}^n)$ be the Schwartz space. I need to determine whether the following set of functions $A$: $$A= \{f\in S(\mathbb{R}^n): ...
5
votes
2answers
336 views

The phrases “has … in ” vs. “contains … of” in Baby Rudin

Consider the following two statements. (Assume $E \subseteq K$.) $E$ has a limit point in $K$. vs. $E$ contains a limit point of $K$. What do they each mean and how are they different?
2
votes
0answers
70 views

how to prove this curious identity with the Chebyshev polinomials

we defined the Tm like this (where Tm are the Chebyshev polinomials) Then I showed this: And now I have no idea how to proove this: I also have to make the remark that I also proved that the ...
1
vote
3answers
101 views

Enumeration of rational numbers

If $\Bbb Q=\{q_n:n\in \Bbb N\}$ be an enumeration of $\Bbb Q$, is it true that $|q_n|<1/n$ for infinitely many $n$? I just come up with this question, it seemed simple but I can't solve it. Is ...
0
votes
1answer
29 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
50 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
0
votes
1answer
52 views

How to compute this integral : $\oint \bar{z}^n dz$

How to compute this integral : $$\oint_{|z|=a} \; \bar{z}\;^n dz$$ I choose $z = ae^{i \theta}$, and so $\bar{z}\;^n = a^n e^{-i\theta}$ And $$\oint_{|z|=a} \; \bar{z}\;^n dz = ...
1
vote
2answers
339 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
1
vote
0answers
31 views

Integraion of the function $1/r$ over a sphere in $\mathbb{R}^3$

Assume in $\mathbb{R}^3$ there is a sphere $S=S(A,R)$ centered at a point $A$ with radius $R$, and $|A|=a$, where $|A|$ is the Euclid norm of $A$. Now let a $X$ be a uniform distributed random point ...
1
vote
1answer
49 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
1
vote
0answers
148 views

Predictor-Corrector for Adams-Moulton

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation ...
5
votes
1answer
83 views

integration in five dimensions space

I am doing this problem: Consider the differential form $$a=p_1 \, dq_1+p_2 \, dq_2-(p_1^2+p_2^2+q_1^2+q_2^2) \, dt\text{ in }\mathbb R^5=(p_1,p_2,q_1,q_2,t).$$ (a) Compute the differential $da$ and ...