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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By applying the second version of the Fundamental Theorem of Calculus find the integral:

The second version of the Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ then $\int_{a}^{b} f(x) dx = F(b)-F(a)$. I need to use this to find a) $\int_{-2}^{-1} \frac{1}{x^3} dx $ and b) ...
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2answers
19 views

Show that $(g\circ f)':[a,b] \to R$ is differentiable and find the derivative. (Chain Rule Proof)

I know I probably shouldn't ask two questions in a short amount of time but this is a rather simple one. Question: Suppose $f : [a, b] \to [c, d]$ and $g : [c, d] \to \mathbb{R}$ are differentiable ...
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1answer
9 views

Association of a vector space to metric, normed and inner product spaces

There is a nice visual representation of mathematical spaces from this post: I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but ...
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5answers
128 views

Prove that g is differentiable

Question: Suppose f, g, and h are defined on (a,b) and $a < x_0 < b$. Assume f and h are differentiable at $x_0$, $f(x_0) = h(x_0)$, and $f(x) \le g(x) \le h(x)$ for all x in (a,b). Prove that g ...
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0answers
27 views

Proof of taking derivatives of both equal sides

I am curious about the proof of the following or whether the statement is true in general Assume that I have the following property: $f(x,y)=g(x,y,z)$ Can I assert that $D_xf=D_xg$ at any point ...
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1answer
33 views

Uniform convergence

Determine whether or not the given se ries of functions converges uniformly on the indicated interval (set) $\sum_{n=1}^\infty\frac{(1)}{(nx)^2}$ where x $\in (0,1]$ I don't know if we can apply can ...
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1answer
10 views

invertibility, derivative, and difference quotient

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 $a$ and that $f^{-1}$ is continuous at $a$. ...
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0answers
19 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
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0answers
16 views

A basic ergodic question

I know that irrational number can be approximated by p/q and error less than 1/q^2. But I still cannot give a rigorous proof to this problem. And how to show that the difference between the left ...
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3answers
24 views

how to find the smallest s to make f continuous at (0,0)

$$ f(x,y)=\left\{ \begin{array}{lll} \frac{|x|^s|y|^{2s}}{x^2+y^2} & \text{if}& (x,y) \neq (0,0)\\ 0 & \text{otherwise} \end{array} \right. $$ what is the smallest s to make f(x,y) ...
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0answers
17 views

Unbounded Function

I am trying to find vales of $a >0$ s.t the function is unbounded on $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
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1answer
21 views

Find $f$ such that the contraction $\phi$ has a fixed-point $\rho= \sqrt{2}$

I use the Newton method and the Banach fixed-point theorem and have: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous and $f: I \rightarrow \mathbb{R}$ a ...
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2answers
14 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
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1answer
44 views

How to prove this sequence is convergent?

Suppose that the series $\sum_{k=1}^\infty{a_k}$ converges. Prove that $$\lim_{n→\infty}\frac{1}{n}\sum_{k=1}^{n}ka_k=0$$ I tried to use the definition of convergence of $\sum a_k$ ...
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0answers
16 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
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0answers
24 views

how can I give an elementary proof of Maximum Modulus Theorem for polynomials?

how can I give an elementary proof of Maximum Modulus Theorem for polynomials? I got proof, but not elementary. This question in this book Conway.
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0answers
9 views

composition of Riemann integrable functions.

I have two functions: f:[a,b]->R and g:[c,d]->R where a My question is if it follows that g o f (the composite function of f,g) is Riemann integrable as well?
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1answer
60 views

Find this maximum of this $\frac{\int_{0}^{\pi}f(x) \, dx}{\int_{0}^{\pi} f(x)\sin x\,dx}$

Question: Assmue that $\int_0^\pi f(x)\,dx$ and $\int_0^\pi f(x)\sin x\,dx$ is convergence,and $f(x)>0,\forall x\in(0,\pi)$ Find this maximum as possible for all function $f$ ...
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0answers
17 views

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$?

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$ ?. I know that zero is an essential singularity, and so $f(D(0,\delta)-\{ 0 \})$ is dense in $\mathbb{C}$. This question ...
2
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0answers
45 views

What is known about$\sum\limits_{p\text{ prime}} \frac{1}{p^2-1}$?

Are there some known results for $\sum\limits_{p\text{ prime}} \dfrac{1}{p^2-1}$? I wasn't able to find anything about this sum in the internet or in my books!
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0answers
30 views

A complicated question

I have the following operator $A: H^1_{0,p}\longrightarrow H^1_{0,p}$ be defined by \begin{equation} Au(t)=\int_0^{+\infty} G(t,s)q(s)f(s,u(s))\,ds-\sum_{k=0}^{+\infty}G(t,t_k)h(t_k)I(u(t_k)), ...
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0answers
21 views

Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
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1answer
18 views

$\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$

I can't compute this $\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$. I have separate it into 2 integrals but i can't continue.
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1answer
21 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
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1answer
32 views

How find this sum $\sum_{n=1}^{\infty}\frac{n}{[(2n)!]^2}$

Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{n}{[(2n)!]^2}$$ I think we can note $$\dfrac{n}{((2n)!)^2}=\dfrac{1}{2}\dfrac{2n}{((2n)!)^2)}=\dfrac{1}{2}\cdot\dfrac{1}{(2n)!\cdot(2n-1)!}$$
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0answers
23 views

Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
3
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1answer
22 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
2
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1answer
32 views

Weighted sum of cosines

Consider $$f(x) = \sum_{k=1}^\infty \cos(kx) k^\alpha.$$ The first question is: does this have a name (Mathematica gives it as a sum of polylogs of complex arguments, but this seems unnatural). Also, ...
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2answers
28 views

Calculate an integral that has a sum within.

Im trying to calculate this integral: $\displaystyle \int_{0}^{\pi} \sum_{n=1}^{\infty} \frac{n \sin(nx)}{2^n}$ The only thing I have been able to do is switch the integral and the sum, and in the ...
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2answers
24 views

Example of $x$ being adherent point but not accumulation point?

So I was just reading Apostol and I see that if $x$ is an accumulation point of set $S$, it has to be an adherent point as well. I guess it's possible for $x$ to be an adherent point only, not an ...
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0answers
19 views

What can the Weierstrass therem say about an arbitrary continuous function being analytic?

Please describe the conditions and why they do not match up, as clearly not every continuous function is analytic. Thank you!
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2answers
49 views

The product of uniformly continuous functions is not necessarily uniformly continuous

I was asked to show that given two functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ which are both uniformly continuous, to show that the product ...
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2answers
30 views

Theorem 3.29 in Baby Rudin

Theorem 3.29 in Walter Rudin's Principles of Mathematical Analysis, 3rd ed., states that If $p>1$, then the series $$\sum_{n=2}^\infty \frac{1}{n (\log n)^p} $$ converges; if $p \leq 1$, the ...
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1answer
26 views

Fejer's theorem with Riemann integrable function

If $f$ is integrable and $f(x+), f(x-)$ exists for some $x$, then $$ \lim_{N \rightarrow \infty} {\frac{1}{{2\pi }}\int_{ - \pi }^\pi {f\left( {x - t} \right){K_N}\left( t \right)dt} } = ...
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2answers
68 views

What branch of analysis deals most with sequences and series?

I'm really interested in sequences and series (mainly series). What kind of math branch should I look more into? I understand that sequences and series mostly point toward analysis, but what ...
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0answers
25 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
3
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2answers
49 views

Tedious undefined limit without L'Hospital $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$

When I try to calculate this limit: $$\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$$ I find this: $$\begin{array}{l} L = \mathop {\lim }\limits_{x \to ...
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0answers
16 views

Showing a fact about $\sigma$-algebras and Borel sets

Let $(\Omega,\mathcal{A})$ be a measurable space, $(A_n)_{n\in\mathbb{N}}\subset\mathcal{A}$ and $f_n:\Omega\to [-\infty,\infty]$ be a $\mathcal{A}-\overline{\mathcal{B}}$ measurable function, where ...
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2answers
48 views

$\int_1^e \! \frac{\mathrm{d}x}{x\,\sqrt{-\ln \ln x}}$

$$\int\limits_1^e \! \frac{\mathrm{d}x}{x\,\sqrt{-\ln \ln x}}$$ I can't find any antiderivative, is it possible to calculate the definite integral?
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0answers
39 views

MERGE(L1,L2) two sorted list [on hold]

Merge(L1, L2), below, is a function which merges two sorted linked lists L1 and L2, and outputs a single sorted linked list. By "sorted", we mean increasing order. ...
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0answers
21 views

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k ...
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0answers
26 views

Uniform convergence on subintervals

(a) Fix a positive integer $M$ and let $\{f_{n} : [0, M]\rightarrow \mathbb{R}\}$ be a sequence of functions. Suppose that $f_{n}\rightarrow f$ pointwise on $[0, M]$ and that $f_{n}\rightarrow f$ ...
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2answers
45 views

How to find the limit without L'Hospital rule

Find the limit $$\lim_{x\to\infty} x\left[\left(1+\frac{1}{2x}+\frac{1}{4x^2}+\frac{1}{8x^3}\right)^{1/3}-1\right].$$ I assume that L'Hospital rule here is useless and something else must be done.
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Analysis, about Riemann-intrability [on hold]

Consider the following function $$ f(x) = \begin{cases} x & \text{if } x \in \mathbb{Q} \cap [0, 1] \\ x^3 & \text{if } x \in \mathbb{Q}^c \cap [0, 1] \end{cases} $$ Is the ...
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1answer
21 views

Proof of a limit formula

If $h(x) = f(x)/g(x)$ $lim(x->b) f(x) = L$ $lim(x->b) g(x) = M$ Prove that $lim(x->b) h(x) = L/M$ Sorry for the terrible latex. ONLY FORMAL PROOFS! For every $\epsilon > 0$ Since ...
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0answers
16 views

Riemann Integrable on Circle implies $L^2$ Convergence of linear interpolation.

I was reading the proof of well-known theorem. If $f$ is Riemann integrable on $[0,2\pi]$, then its Fourier series converges to $f$ in $L^2$ norm. The proof was depending on the $L^2$convergence ...
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1answer
26 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
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2answers
33 views

How prove $0\le \lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{i+\frac{1}{i}}-\ln{\frac{n}{\sqrt{2}}}$

let $$x_{n}=\dfrac{1}{1+1}+\dfrac{1}{2+\frac{1}{2}}+\cdots+\dfrac{1}{n+\frac{1}{n}} -\ln{\dfrac{n}{\sqrt{2}}},n=1,2,\cdots$$ show that $a=\displaystyle\lim_{n\to\infty}x_{n}$ is exsit,and $0\le ...
1
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1answer
29 views

Fundamental solution of heat equation on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold of $m$ dimensional. Then there exists a sequence $(\phi_i, \lambda_i)_{i\in\mathbb{N}}\subset C^\infty(M)\times\mathbb{R}_{\geq0}$ such that ...
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0answers
18 views

If $\limsup_{t\to \infty} \int_{0}^{t}Tr(A(s))ds = \infty$ then $\limsup_{t\to \infty} |x(t)|=\infty$

For a homogeneous linear system of differential equations: $x'=Ax$ : Suppose that $\limsup_{t\to \infty} \int_{0}^{t}tr(A(s))ds = \infty$ ($tr(A):=$ trace of the matrix A). Then there exists solution ...