Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

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).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Evalute $\int_0^1 \sqrt{1-x^2} \cos (tx) \, dx$

I can't solve this problem. Can anyone help me? Let $t \in \mathbb{R}$. Evaluate $\int_0^1 \sqrt{1-x^2} \cos (tx) \, dx.$ Thank you very much!
tunguyen
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If limit of $f'(t)$ approaches $0$, then limit of $f(t)/t$ approaches $0$.

Let $f: (0, \infty) \longrightarrow \mathbb{R}$ be differentiable. Show that if $\lim_{t \to \infty}f'(t) = 0$, then $\lim_{t \to \infty}\frac{f(t)}{t} = 0$. I don't really even know where to start. If we take $\epsilon > 0$, there is $M \in…
Sprinkle
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"Conservative Fields" in Inifinite Dimesions

Let $X$ be a infinite dimensional Banach space and $X^\star$ its dual. Let $f:X\rightarrow X^\star$ be a continuous function. What is a necessary and sufficient condition to find $F:X\rightarrow\mathbb{R}$ such that $$F'(x)=f(x)$$ where $F'$ is the…
Tomás
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Why the double integral does not exist?

Consider $([-1,1]\times[-1,1],\mathcal{B}[-1,1]\times\mathcal{B}[-1,1],dx\,dy)$. Let $f(x,y)=\dfrac{xy}{(x^2+y^2)^2}$. Show the double integral does not exist. My…
Sam
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Proving limits using $\epsilon-\delta$ definition

Would this be the proper way to prove a limit of a function using the $\epsilon-\delta$ definition? Use the definition of limit to show that $\lim_\limits{x\to2}x^2+4x=12$ Let $\epsilon>0$, then there exists $\delta>0$, such that $x\in…
Michelle Drolet
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Please Explain Baby Rudin Theorem 1.20 (b) By Using Statements on Baby Rudin

I have a question in Baby Rudin Theorem 1.20 (b). I have checked other Q and A's of this theorem in mathstackexchange (and I can understand this theorem). But, those answers did not explain statements in Baby Rudin (such as Baby Rudin Theorem 1.20…
user472102
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Space-Filling Curve

How can I prove that there is no closed smooth space filling curve from [0,1] to 2-dimensional sphere. I can construct a space filling curve to a cube in R^3, but why I can not expand it to sphere?
Ekber
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If $f$ is continuous and satisfies $\lim_{x \to \infty} f(x + y) - f(x) = 0$ for all $y \in [0, 1]$, then $f$ is uniformly continuous.

If $f: [0, \infty[ \to \mathbb{R}^p$ is continuous and satisfies $\lim_{x \to \infty} f(x + y) - f(x) = 0$ for all $y \in [0, 1]$, then $f$ is uniformly continuous. Attempt: suppose $f$ is not uniformly continuous. Then $\exists \epsilon' > 0,…
user388557
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How to prove that gcd of two numbers is one.

For $l \in \mathbb{N}$ I'm to prove that the greatest common divisor, $\gcd(8l^2+20l+13,4l+2) = 1$. I've tried induction, but I couldn't pull off, now I'm at a loss of how to even begin to solve this. PS: It's also possible that it's not possible…
Mathaniel
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How to show this function is injective?

Define $h: \mathbb Z \to [0,1]$ by $h(z)= z \pi - [z\pi]$ where $[.]$ denotes the floor function. I want to show this function is injective, I have tried both the approaches using $h(z_1)= h(z_2)$ and try to get $z_1= z_2$ and the another one…
User
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Is this functional series differentiable?

Is this functional series differentiable on (-1,1)? If it is could you give the full proof? $$\sum_{n=1}^{\infty}x^{n!}$$
J.John
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Integral of an analytic function

Is the integral of a real analytic function analytic? If it is so, how do you go about the proof? In particular, what guarantees that the Taylor series expansion of this integral converges to itself?
user233467
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2 answers

Uniformly continuity theorem proof.

Let $f$ be continuous on [a,b]. Then $f$ is uniformly continuous on [a,b] and there exists $\delta >0$ such that $|f(s)-f(t)|<\epsilon$ if $|s-t|<\delta$. Let P={$x_0,x_1,...,x_n$} is a partition of [a,b] with $x_i-x_{i-1}<\delta$ for all i. If…
niagara
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Proving convergence

Let f be a function defined at x. Suppose that every sequence $p_1, p_2, p_3 \dots$ in the domain of f converging to $x$ has the property that $f(p_1), f(p_2),f(p_3), \dots$ converges to $f(x)$. Prove that $f$ is continuous at $x$. So I could…
RockChalk
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Why is $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ rational?

For this question, Show the following irrational-looking expressions are actually rational numbers. (a) $\sqrt{4+2\sqrt{3}}-\sqrt{3}$, and (b) ... I solved it as follows: $$\begin{align} x &= \sqrt{4+2\sqrt{3}}-\sqrt{3},\\ x+\sqrt{3} &=…
eca2ed291a2f572f66f4a5fcf57511
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