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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Equation to calculate homogeneity of environment

It's been so long since I've done anything more complicated than balancing my checkbook that, not only do I not know how to do this, I don't even know what to search for to see if it's already been answered. I have a linear environment with evenly…
zenzic
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Analysis Open and Closed Sets

I have the following question and i'm not sure how to go about proving whether sets are open closed, or both. Which of the following sets are open and which are closed? $1)\ [1,2] \cup [3,4] \ in \ ( \mathbb{R},|\cdot|)$ $2)[1.2] \ in…
Mathsstudent147
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Glue Together smooth functions

Let's say that $f(x)$ is a $C^{1}$ function defined on a closed interval $I\subset \mathbb{R^{+}}$ and $g(x)\equiv c$ ($c$=constant) on an open interval $J\subset \mathbb{R^{+}}$ where $\overline{J}∩I\neq \emptyset$. Is there a way to "glue"…
Fraleigh
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How to extend the property of non-negative functions, to $f \in L^1$?

Countable additivity. Suppose that $\{g_n\}$ is a sequence of measurable functions on X such that $$\int_X \Big( \sum_{i=1}^{\infty} |g_n| \Big) d\mu < \infty .$$ Then $$\int \Big (\sum_{n=1}^\infty g_n \Big) d\mu = \sum_{n=1}^\infty \int g_n…
dxdydz
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Evaluate $ \int_0^\frac{\pi}{2}\frac{\cos\theta}{\cos\theta+\sin\theta}\,d\theta. $

Evaluate $$ \int_0^\frac{\pi}{2}\frac{\cos\theta}{\cos\theta+\sin\theta}\,d\theta.\qquad\text{(1)} $$ By letting $t=\tan\theta$, $(1)$ equals $$ \int_0^\infty\frac{1}{(1+t)(1+t^2)} \,dt,$$ and then? By letting $t=\tan\frac{\theta}{2}$, (1)…
Knt
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Evaluate $ \lim_{n\to+\infty}\sum_{k=1}^{n}\frac{k}{n^2+k}\ln(1+\frac{k}{n})$.

Evaluate $$ \lim_{n\to+\infty}\sum_{k=1}^{n}\frac{k}{n^2+k}\ln(1+\frac{k}{n}). $$ I think it may be associated with the definition of Riemann integration. Since we know that…
Knt
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Prove that $ f(x,\,y)>0\quad\text{ for all $(x,\,y)\in\mathbb R^2$}$.

Suppose $f(x,\,y)\in C^1(\mathbb R^2)$. If $\displaystyle\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}$ and $f(x,\,0)>0$ for all $x\in\mathbb R$. Prove that $$ f(x,\,y)>0\quad\text{ for all $(x,\,y)\in\mathbb R^2$}. $$ I find that…
Knt
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Simpler way to check whether or not a sequence is uniformly convergent

Let $f_n(x)=(1+x^n)^\frac{1}{n}$ on $[0, \infty)$. I want to check if this is uniformly convergent. It's pointwise limit is $$f(x)=\begin{cases} x \text{ if } |x|\geq1\\ 1 \text{ if } |x|<1\\ \end{cases}$$ My first attempt was to use the Weierstrass…
emka
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Prove that $f(x)$ is uniformly continuous on $I$ if and only if the image of each Cauchy sequence under $f$ is also a Cauchy sequence.

Suppose $f(x)$ defines on the bounded interval $I$. Prove that $f(x)$ is uniformly continuous on $I$ if and only if the image of each Cauchy sequence under $f$ is also a Cauchy sequence. The "only if" part is easy. For the "if" part I can prove…
Knt
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Evaluate $\lim_{t\to0+}\frac{\iint_\Omega P(x,\,y,\,z)\,dydz+Q(x,\,y,\,z)\,dzdx+R(x,\,y,\,z)\,dxdy}{t^4}$.

Suppose $P(x,\,y,\,z)=Q(x,\,y,\,z)=R(x,\,y,\,z)=f((x^2+y^2)z)$ and $f$ has continuous derivative. Evaluate $$ \lim_{t\to0+}\frac{\iint_\Omega P(x,\,y,\,z)\,dydz+Q(x,\,y,\,z)\,dzdx+R(x,\,y,\,z)\,dxdy}{t^4}, $$ where $\Omega$ is the outside…
Knt
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Correctness and help with Union and intersection proof of Open Sets

I need to prove the following: Let $A$ and $B$ be subsets of a metric space $(X, d)$ show that $A^o \cup B^o \subset (A \cup B)^o$. $(A \cap B)^o = A^o \cap B^o$ Here is my attempt: For 1) Let $x\in (A^o \cup B^o)$. Then by definition of open set…
CodeKingPlusPlus
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Showing a sequence of functions converges uniformly on any bounded interval

Question: Let $\{f_n\}$ be a sequence of continuous functions on $\mathbb{R}$. Let $f_n \to f$ uniformly on $\mathbb{R}$. Let $g_n(x):=f_n(x+\frac{1}{n})$ for $n=1,2,3,....$ Then $g_n \to f$ uniformly on any bounded interval $[a,b]$. I think…
emka
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Any power of a uniform convergence function

I have problem showing this: Suppose $ f: X \to X $ is a continuous surjective function and there exists a sequence in $ \mathbb{N} $ such as $ \lbrace n_{k} \rbrace_{k \geq 1} $ such that $ n_{k} \to \infty$ and $ f^{n_{k}} \to I_{X} $ uniformly .…
Reza Yaghmaeian
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Generalization of commutativity for unions of families of sets (Halmos)

So, I'm unraveling the Halmos's book "Naive Set Theory", just for fun. But I stumbled upon the following problem: "Let $\{I_j\}$, $j \in J$ be a family. Write $K = \displaystyle\bigcup_{j \in J} I_j$ and let $\{A_k\}$, $k \in K$, be a family. It's…
Fractal Admirer
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Easy way of proving continuity of a convex function

Is there an easy and fast way to prove that a convex function $f : I \to \mathbb{R}, \, I \subset \mathbb{R}$ an interval, is continuous? All the proofs that I know are very long and technical. Thanks in advance for any answer!
astrobarrel
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